Tutorial: Minimum Time Problems
This tutorial shows how to solve time-optimal problems where trajectory duration is minimized alongside other objectives.
Problem Description
Find the fastest trajectory from start to goal with bounded controls.
Dynamics:
\[\dot{x} = (G_0 + u_1 G_1) x\]
Objective: Minimize total time + control effort
Constraints: |u| ≤ 1
using DirectTrajOpt
using NamedTrajectories
using LinearAlgebra
using Statistics
using PrintfFixed Time vs Free Time
println("="^50)
println("MINIMUM TIME TRAJECTORY OPTIMIZATION")
println("="^50)
println("""
Two approaches:
1. **Fixed time**: All Δt equal and constant
2. **Free time**: Each Δt is a variable (what we'll use)
Free time allows the optimizer to adjust trajectory duration.
""")==================================================
MINIMUM TIME TRAJECTORY OPTIMIZATION
==================================================
Two approaches:
1. **Fixed time**: All Δt equal and constant
2. **Free time**: Each Δt is a variable (what we'll use)
Free time allows the optimizer to adjust trajectory duration.Step 1: System Definition
G_drift = [
-0.1 1.0;
-1.0 -0.1
]
G_drives = [[
0.0 1.0;
1.0 0.0
]]
G = u -> G_drift + sum(u .* G_drives)
println("System: 2D damped oscillator with symmetric control coupling")System: 2D damped oscillator with symmetric control couplingStep 2: Trajectory Setup
N = 40# Number of time steps (fewer than before)
Δt_init = 0.15 # Initial guess for time step
x_init = [0.0, 0.0]
x_goal = [1.0, 0.0]2-element Vector{Float64}:
1.0
0.0Initial guess
x_guess = hcat([x_init + (x_goal - x_init) * (t/(N-1)) for t = 0:(N-1)]...)
u_guess = 0.1 * randn(1, N)
Δt_guess = fill(Δt_init, N)
println("\nProblem setup:")
println(" Time steps: $N")
println(" Initial guess for Δt: $Δt_init")
println(" Initial total time: ", sum(Δt_guess))
Problem setup:
Time steps: 40
Initial guess for Δt: 0.15
Initial total time: 6.000000000000002Step 3: Create Free-Time Trajectory
Key: timestep=:Δt makes time steps decision variables
traj_mintime = NamedTrajectory(
(x = x_guess, u = u_guess, Δt = Δt_guess);
timestep = :Δt, # Time is a variable!
controls = :u,
initial = (x = x_init,),
final = (x = x_goal,),
bounds = (
u = 1.0, # -1 ≤ u ≤ 1
Δt = (0.01, 0.5), # 0.01 ≤ Δt ≤ 0.5
),
)
println("\nTrajectory bounds:")
println(" Control: ", traj_mintime.bounds.u)
println(" Time step: ", traj_mintime.bounds.Δt)
Trajectory bounds:
Control: ([-1.0], [1.0])
Time step: ([0.01], [0.5])Step 4: Define Objectives
Time Minimization Weight
The key parameter: balance speed vs control effort
w_time = 1.0 # Weight on total time
w_control = 1e-2 # Weight on control effort
obj_mintime = (
w_control * QuadraticRegularizer(:u, traj_mintime, 1.0) +
w_time * MinimumTimeObjective(traj_mintime, 1.0)
)
println("\nObjective weights:")
println(" Control effort: $w_control")
println(" Time: $w_time")
println(" → Emphasizes minimizing time")
Objective weights:
Control effort: 0.01
Time: 1.0
→ Emphasizes minimizing timeStep 5: Solve Minimum Time Problem
integrator_mintime = BilinearIntegrator(G, :x, :u, traj_mintime)
prob_mintime = DirectTrajOptProblem(traj_mintime, obj_mintime, integrator_mintime)
prob_mintimeDirectTrajOptProblem
Trajectory
Timesteps: 40
Duration: 5.85
Knot dim: 4
Variables: x (2), u (1), Δt (1)
Controls: u, Δt
Objective (2 terms)
0.01 * QuadraticRegularizer on :u (R = [1.0], all)
1.0 * MinimumTimeObjective (D = 1.0)
Dynamics (1 integrators)
BilinearIntegrator: :x = exp(Δt G(:u)) :x (dim = 2)
Constraints (4 total: 2 equality, 2 bounds)
EqualityConstraint: "initial value of x"
EqualityConstraint: "final value of x"
BoundsConstraint: "bounds on u"
BoundsConstraint: "bounds on Δt"println("Solving minimum time problem...")
println("="^50)
solve!(prob_mintime; max_iter = 200, verbose = false)
println("="^50)
println("Minimum time solution found!")
println("="^50)Solving minimum time problem...
==================================================
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 616
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 994
Total number of variables............................: 156
variables with only lower bounds: 0
variables with lower and upper bounds: 80
variables with only upper bounds: 0
Total number of equality constraints.................: 78
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 5.8500289e+00 1.39e-01 1.00e+00 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.4351193e+00 6.91e-02 1.08e+00 -0.8 5.08e-01 - 9.95e-01 8.66e-01h 1
2 2.1847072e+00 4.97e-02 2.58e+00 -4.0 2.07e+00 - 3.09e-01 2.77e-01f 1
3 3.1636077e+00 7.11e-02 5.27e+00 -0.4 2.09e+00 0.0 5.06e-01 2.30e-01f 1
4 3.3567640e+00 6.63e-02 6.33e+00 -0.7 1.76e+00 0.4 1.68e-01 6.86e-02h 1
5 3.3337604e+00 6.75e-02 6.17e+00 -0.3 4.11e+01 - 1.09e-02 7.54e-03f 2
6 3.4582139e+00 8.26e-02 6.01e+00 -4.0 1.44e+01 - 4.12e-02 5.20e-02h 1
7 3.5289906e+00 8.92e-02 1.61e+01 -0.8 2.29e+00 0.9 7.22e-01 1.85e-01h 1
8 3.5699447e+00 8.07e-02 1.41e+01 -4.0 4.85e+00 - 1.55e-01 9.50e-02h 1
9 3.7561027e+00 7.32e-02 1.64e+02 0.1 1.90e+00 1.3 6.65e-01 5.71e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 4.3439319e+00 6.16e-02 1.35e+02 -0.2 2.02e+00 - 3.39e-01 1.54e-01h 1
11 4.4724613e+00 5.64e-02 1.20e+02 -4.0 1.61e+00 - 1.18e-01 8.40e-02h 1
12 4.7405862e+00 5.25e-02 1.14e+02 -4.0 3.49e+00 - 5.27e-02 6.93e-02h 1
13 4.8933240e+00 5.11e-02 8.08e+02 -0.5 6.42e+00 - 9.97e-01 2.67e-02h 1
14 4.9563712e+00 5.06e-02 3.62e+03 0.7 5.93e+00 - 1.00e+00 1.02e-02h 1
15 4.5141657e+00 4.79e-02 2.80e+04 1.1 6.58e+00 - 1.00e+00 1.15e-01h 1
16 4.5020070e+00 5.93e-02 2.28e+04 -4.0 1.81e+01 - 8.52e-03 1.23e-02h 1
17 4.5714059e+00 5.90e-02 2.31e+04 2.0 1.27e+01 3.5 2.59e-02 6.12e-03h 1
18 4.5968031e+00 5.89e-02 3.02e+04 1.1 5.37e+00 3.9 1.35e-02 2.48e-03h 1
19 4.7191449e+00 5.81e-02 2.28e+04 1.1 4.44e+00 4.4 6.08e-03 1.31e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 5.1839619e+00 5.82e-02 3.89e+04 2.8 6.06e+00 3.9 5.02e-02 2.53e-02f 1
21 5.4865378e+00 5.65e-02 1.28e+05 2.8 4.09e+00 4.3 8.12e-02 2.21e-02f 1
22 5.7501943e+00 5.53e-02 3.42e+05 2.8 6.37e+00 3.8 1.16e-01 1.82e-02h 1
23 6.2851221e+00 5.36e-02 3.34e+05 2.8 1.20e+01 - 2.18e-02 3.06e-02f 1
24 6.6468843e+00 5.26e-02 3.21e+05 2.8 1.26e+01 - 2.49e-02 1.90e-02h 1
25 6.7515791e+00 5.24e-02 3.22e+05 -3.8 3.74e+01 - 1.79e-03 3.24e-03h 1
26 7.0577179e+00 5.18e-02 1.51e+06 2.8 2.50e+01 - 3.66e-01 1.19e-02h 1
27 7.3549435e+00 5.17e-02 1.55e+06 2.8 1.29e+02 - 2.48e-04 2.39e-03h 1
28 7.4971036e+00 5.14e-02 3.37e+06 2.8 3.38e+01 - 1.43e-01 4.51e-03h 1
29 7.2696859e+00 5.06e-02 1.17e+07 2.8 1.79e+01 - 8.01e-02 1.63e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 7.3279086e+00 5.02e-02 3.64e+06 -3.1 2.28e+01 - 1.52e-03 5.61e-03h 1
31 7.3709360e+00 5.01e-02 3.77e+06 -3.3 3.65e+01 5.2 1.07e-03 8.42e-04h 1
32 7.3940698e+00 5.01e-02 3.85e+06 2.8 1.00e+02 - 7.27e-04 4.90e-04h 1
33 7.5343229e+00 5.00e-02 4.40e+06 2.8 4.12e+01 - 4.05e-03 2.61e-03h 1
34 7.9234960e+00 4.96e-02 1.41e+07 2.8 3.88e+01 - 2.69e-02 6.70e-03h 1
35 8.1936612e+00 4.95e-02 1.34e+07 2.8 7.70e+01 - 2.88e-03 3.53e-03h 1
36 8.2013728e+00 4.95e-02 1.84e+07 -3.2 2.72e+01 - 8.52e-04 9.99e-05h 1
37 8.2476881e+00 4.94e-02 2.08e+07 2.8 2.39e+02 - 1.02e-04 3.29e-04h 1
38 8.3448854e+00 4.94e-02 1.69e+08 2.8 1.04e+02 - 1.09e-01 9.54e-04h 1
39 8.3808989e+00 4.94e-02 1.74e+08 2.8 1.12e+02 - 1.84e-03 3.24e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 8.6270759e+00 4.93e-02 2.93e+08 2.8 1.13e+02 - 2.37e-02 2.21e-03h 1
41 8.7790268e+00 4.92e-02 2.82e+08 2.8 2.69e+02 - 2.89e-05 5.80e-04h 1
42 8.8962487e+00 4.92e-02 2.69e+08 -2.7 1.36e+02 5.6 6.12e-05 5.42e-04h 1
43 8.9354921e+00 4.92e-02 2.69e+08 2.8 2.74e+02 5.1 3.13e-04 2.41e-04h 1
44 9.0668080e+00 4.92e-02 8.74e+08 2.8 1.78e+02 - 5.18e-02 6.83e-04h 1
45 9.2545985e+00 4.91e-02 8.77e+08 2.8 1.91e+02 - 8.88e-04 7.36e-04h 2
46 9.4478353e+00 4.91e-02 9.31e+08 2.8 2.23e+02 - 2.05e-03 6.39e-04h 1
47 9.4712771e+00 4.91e-02 1.21e+09 2.8 2.13e+02 - 1.86e-03 7.13e-05h 1
48 9.4796385e+00 4.91e-02 2.43e+10 2.8 1.93e+02 - 4.52e-02 5.22e-05h 1
49 9.4463563e+00 4.90e-02 1.30e+11 2.8 8.95e+01 - 7.75e-03 1.24e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 9.4582242e+00 4.90e-02 5.25e+10 -0.9 4.39e+01 - 4.90e-05 1.27e-04h 1
51 9.4728972e+00 4.90e-02 5.65e+10 2.8 9.56e+02 - 4.90e-06 1.95e-05h 1
52 9.4985227e+00 4.90e-02 5.59e+10 2.8 6.96e+02 - 3.32e-05 3.71e-05h 1
53 9.5402065e+00 4.90e-02 5.85e+10 2.8 7.22e+02 - 6.90e-05 5.78e-05h 1
54 9.6117997e+00 4.90e-02 1.76e+11 2.8 7.87e+02 - 4.96e-04 9.10e-05h 1
55 9.6494139e+00 4.90e-02 1.59e+11 -1.2 9.43e+02 - 3.13e-06 3.99e-05h 4
56 9.6461820e+00 4.90e-02 1.61e+11 -1.2 3.47e+03 - 4.68e-09 6.19e-06f 1
57 9.6576403e+00 4.90e-02 1.61e+11 2.8 8.60e+02 - 2.14e-05 2.02e-05h 1
58 9.6802093e+00 4.90e-02 5.90e+11 2.8 8.71e+02 - 1.13e-03 3.78e-05h 1
59 9.7159347e+00 4.90e-02 5.67e+11 2.8 9.48e+02 - 9.11e-06 5.50e-05f 4
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 9.7783589e+00 4.90e-02 5.09e+11 2.8 1.07e+03 - 6.49e-08 8.55e-05f 3
61 9.8140725e+00 4.90e-02 4.57e+11 2.8 1.42e+03 - 5.54e-07 4.86e-05f 3
62 9.8583358e+00 4.90e-02 4.57e+11 2.1 1.67e+03 - 5.09e-05 5.09e-05s 9
63 9.9222230e+00 4.90e-02 4.56e+11 1.4 2.14e+03 - 5.72e-05 5.72e-05s 8
64r 9.9222230e+00 4.90e-02 9.97e+02 0.7 0.00e+00 - 0.00e+00 1.82e-07R 2
65r 1.0179592e+01 2.24e-02 2.55e+02 1.4 9.37e-01 - 8.70e-01 3.87e-01f 1
66 1.0094637e+01 2.23e-02 8.67e+01 -0.0 2.82e+00 - 6.24e-01 2.61e-02f 1
67 8.0841272e+00 3.00e-02 9.00e+01 -0.0 3.39e+00 - 1.45e-01 1.33e-01f 1
68 8.1094033e+00 2.92e-02 1.01e+03 -0.0 2.16e-01 4.6 2.05e-01 2.51e-02h 1
69 8.1488110e+00 2.89e-02 3.95e+03 -0.0 5.10e-01 4.2 7.11e-02 1.04e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 8.1794139e+00 2.88e-02 8.06e+03 -0.0 1.33e+00 3.7 1.08e-01 4.57e-03h 1
71 8.4741930e+00 2.79e-02 7.25e+03 -0.0 1.52e+00 3.2 2.22e-02 3.29e-02h 1
72 8.5241241e+00 2.78e-02 6.48e+04 -0.0 1.64e+00 3.6 2.88e-01 5.65e-03h 1
73 8.8168562e+00 2.76e-02 6.07e+04 -0.0 3.68e+00 3.2 8.98e-03 2.04e-02h 1
74 8.9307438e+00 2.74e-02 6.27e+04 -0.0 1.21e+01 2.7 1.88e-02 8.70e-03h 1
75 9.0150845e+00 2.74e-02 7.94e+04 -0.0 1.08e+01 3.1 2.44e-02 4.16e-03h 1
76 9.2601916e+00 2.76e-02 9.00e+04 -0.0 1.91e+01 2.6 2.31e-02 8.90e-03h 1
77 9.2660232e+00 2.76e-02 9.94e+04 -0.0 1.92e+01 2.1 1.03e-02 3.72e-03h 1
78 9.4132549e+00 2.75e-02 1.36e+05 -0.0 2.68e+01 2.6 8.42e-03 2.33e-03h 1
79 9.4418614e+00 2.75e-02 1.34e+05 -0.0 2.04e+02 - 4.96e-04 8.01e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 9.5490076e+00 2.75e-02 1.41e+05 -0.0 7.68e+01 2.1 1.96e-03 1.44e-03h 1
81 9.6282211e+00 2.75e-02 1.84e+05 -0.0 1.33e+02 1.6 2.81e-03 1.06e-03h 1
82 9.7235673e+00 2.74e-02 4.77e+05 -0.0 1.91e+02 2.0 6.62e-03 1.21e-03h 1
83 9.7362068e+00 2.74e-02 5.31e+05 -0.0 1.04e+03 1.6 6.33e-04 1.84e-04h 1
84 9.8185033e+00 2.74e-02 2.67e+06 -0.0 3.82e+02 2.0 1.27e-02 1.10e-03h 1
85 9.8189819e+00 2.74e-02 1.26e+07 -0.0 3.19e+02 - 4.74e-03 3.90e-05h 1
86 9.8860352e+00 2.74e-02 7.97e+07 -0.0 1.10e+02 - 6.78e-03 8.54e-04h 1
87 9.8589566e+00 2.74e-02 8.21e+07 -0.0 1.45e+03 - 3.04e-05 1.90e-05h 5
88 9.8180145e+00 2.74e-02 8.68e+07 -0.0 1.51e+03 - 4.74e-05 2.98e-05h 3
89 9.7535169e+00 2.74e-02 1.13e+08 -0.0 7.27e+02 - 1.67e-04 1.03e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90 9.7975889e+00 2.74e-02 2.35e+10 -0.0 5.48e+02 - 2.01e-02 9.01e-05h 1
91r 9.7975889e+00 2.74e-02 1.00e+03 -0.0 0.00e+00 - 0.00e+00 3.84e-07R 7
92r 9.4593430e+00 2.26e-02 9.66e+02 1.2 3.31e+01 - 7.22e-01 1.04e-02f 1
93r 9.4734180e+00 2.16e-02 1.51e+03 0.6 1.61e+00 - 2.08e-01 3.51e-02f 1
94 9.4734353e+00 2.15e-02 3.09e+02 -0.5 1.09e+00 - 9.86e-01 6.62e-03f 1
95 8.3981406e+00 2.12e-02 8.28e+02 -0.2 4.22e+01 - 4.77e-02 1.14e-02f 1
96 8.4875182e+00 2.12e-02 1.19e+03 -4.0 4.21e+01 - 6.75e-03 1.18e-03h 1
97 8.9783073e+00 2.12e-02 1.04e+05 1.3 2.74e+02 - 1.00e+00 1.74e-03f 1
98 9.5488134e+00 2.11e-02 1.16e+05 2.8 2.18e+02 - 1.78e-02 2.18e-03f 1
99 9.5423319e+00 2.10e-02 1.49e+05 1.4 3.23e+00 - 2.60e-02 4.46e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
100 9.4233857e+00 2.23e-02 9.63e+04 1.4 9.14e+00 - 1.80e-03 1.03e-02h 1
101 9.6504403e+00 2.22e-02 3.28e+05 2.8 1.68e+01 - 1.14e-02 4.04e-03h 1
102 9.8095335e+00 2.22e-02 1.79e+06 1.6 5.13e+01 - 1.14e-02 1.81e-03h 1
103 9.8544517e+00 2.22e-02 1.24e+07 2.8 2.33e+02 - 7.92e-03 8.73e-04h 1
104 9.8639167e+00 2.22e-02 2.97e+08 2.8 4.29e+02 - 1.38e-02 5.77e-04h 1
105 9.7899506e+00 2.21e-02 5.66e+08 -3.6 1.48e+03 - 7.83e-05 3.92e-05f 1
106 9.8678472e+00 2.21e-02 5.65e+08 2.6 2.18e+03 - 3.06e-05 3.06e-05s 7
107r 9.8678472e+00 2.21e-02 9.95e+02 1.9 0.00e+00 - 0.00e+00 3.58e-07R 7
108r 9.8618359e+00 1.54e-02 9.07e+00 0.6 1.16e-01 - 9.92e-01 9.11e-01f 1
109 9.8368452e+00 1.49e-02 6.13e+01 -0.6 8.11e-01 - 6.40e-01 3.02e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
110 8.4915145e+00 1.44e-02 1.77e+02 -0.5 1.19e+01 - 1.42e-01 3.97e-02f 1
111 9.1078127e+00 1.40e-02 3.79e+02 -4.0 3.89e+00 - 7.53e-02 2.41e-02h 1
112 9.2575654e+00 1.40e-02 1.71e+03 0.3 8.26e+01 - 7.51e-02 1.59e-03h 1
113 9.6896750e+00 1.39e-02 1.69e+03 -0.6 8.01e+01 - 1.04e-02 1.00e-02h 1
114 9.8459169e+00 1.39e-02 1.84e+03 1.3 3.79e+02 - 3.48e-03 6.98e-04f 1
115 9.8473115e+00 1.39e-02 2.05e+03 -0.4 2.54e+02 - 2.06e-03 7.34e-05h 6
116 9.8528227e+00 1.38e-02 3.62e+03 -0.4 1.36e+02 - 8.85e-03 2.52e-04h 4
117 9.8646631e+00 1.38e-02 5.01e+03 -0.4 6.76e+01 - 4.73e-03 4.65e-04h 5
118 9.8788163e+00 1.38e-02 8.88e+03 -0.4 6.71e+01 - 9.50e-03 5.25e-04h 5
119 9.8930589e+00 1.38e-02 1.30e+04 -0.4 6.16e+01 - 6.28e-03 5.25e-04h 5
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
120 9.9079072e+00 1.38e-02 2.29e+04 -0.4 6.33e+01 - 9.91e-03 5.67e-04h 5
121 9.9429184e+00 1.38e-02 2.38e+04 -0.4 5.95e+01 - 1.25e-03 7.37e-04h 4
122 9.9662303e+00 1.38e-02 4.12e+04 -0.4 6.91e+01 - 9.68e-03 4.62e-04h 4
123 9.9891010e+00 1.38e-02 5.13e+04 -0.4 5.79e+01 - 3.54e-03 5.69e-04h 5
124 1.0009960e+01 1.38e-02 4.31e+06 0.2 6.03e+01 - 1.00e+00 5.46e-04h 5
125 1.0085605e+01 1.38e-02 5.50e+06 2.8 5.55e+01 - 4.14e-03 8.89e-04h 4
126 1.0087111e+01 1.38e-02 1.64e+07 2.8 4.89e+01 - 2.34e-02 5.27e-04h 5
127 1.0092475e+01 1.38e-02 4.59e+07 2.8 4.54e+01 - 2.11e-02 6.56e-04h 5
128 1.0116435e+01 1.38e-02 6.92e+07 2.8 4.50e+01 - 6.45e-03 6.20e-04h 5
129 1.0133782e+01 1.37e-02 1.70e+09 2.8 4.58e+01 - 2.76e-01 5.72e-04h 5
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
130 1.0152081e+01 1.37e-02 3.77e+09 2.8 3.65e+01 - 1.41e-02 6.46e-04h 5
131 1.0173080e+01 1.37e-02 8.44e+09 2.8 3.84e+01 - 1.47e-02 6.02e-04h 5
132 1.0188182e+01 1.37e-02 1.95e+11 2.8 3.66e+01 - 2.56e-01 5.92e-04h 5
133 1.0214163e+01 1.37e-02 3.21e+11 2.8 3.16e+01 - 7.76e-03 6.43e-04h 5
134 1.0221807e+01 1.37e-02 1.55e+12 2.8 3.21e+01 - 4.57e-02 5.94e-04h 5
135 1.0244608e+01 1.37e-02 2.94e+12 2.8 2.83e+01 - 1.08e-02 6.31e-04h 5
136 1.0255007e+01 1.37e-02 4.96e+13 2.8 2.75e+01 - 1.92e-01 6.11e-04h 5
137 1.0280419e+01 1.37e-02 8.31e+13 2.8 2.37e+01 - 8.34e-03 6.25e-04h 5
138 1.0286201e+01 1.37e-02 2.34e+14 2.8 2.32e+01 - 2.38e-02 1.28e-03h 4
139 1.0310797e+01 1.36e-02 4.18e+14 2.8 2.26e+01 - 1.21e-02 6.27e-04h 5
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
140 1.0340035e+01 1.36e-02 2.98e+16 2.8 1.99e+01 - 1.00e+00 1.36e-03h 4
141 1.0343091e+01 1.36e-02 3.09e+16 2.8 3.02e+01 - 7.38e-03 4.95e-04h 5
142 1.0355198e+01 1.42e-02 3.22e+16 2.8 2.34e+01 - 2.33e-02 1.24e-03h 4
143 1.0365902e+01 1.45e-02 5.95e+16 2.8 2.48e+01 - 3.66e-01 1.10e-03h 4
144 1.0384243e+01 1.55e-02 5.95e+16 2.8 2.36e+01 - 1.48e-02 2.03e-03h 3
145 1.0411850e+01 1.72e-02 5.87e+16 2.8 2.18e+01 - 3.31e-02 3.31e-03h 2
146 1.0441610e+01 1.93e-02 5.83e+16 2.8 1.15e+01 - 9.05e-03 6.33e-03h 1
147 1.0443916e+01 1.92e-02 5.83e+16 2.8 4.06e+00 - 8.94e-03 4.11e-03h 1
148 1.0447399e+01 1.91e-02 5.97e+16 2.8 3.99e+00 - 1.32e-02 5.11e-03h 1
149 1.0452779e+01 1.90e-02 7.92e+16 2.8 3.70e+00 - 1.53e-02 2.74e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
150 1.0470367e+01 1.90e-02 9.05e+16 2.8 1.08e+01 - 2.14e-03 1.47e-03h 1
151 1.0497049e+01 1.90e-02 1.37e+17 2.8 2.56e+01 - 1.55e-03 8.85e-04h 1
152 1.0498861e+01 1.90e-02 3.29e+17 2.8 1.18e+01 15.8 1.34e-03 6.27e-04h 1
153 1.0494484e+01 1.90e-02 3.35e+17 2.8 2.55e+01 - 3.65e-04 4.09e-04f 1
154 1.0505972e+01 1.90e-02 2.84e+17 2.8 3.13e+02 15.4 4.30e-05 7.72e-05h 1
155 1.0501537e+01 1.90e-02 2.65e+17 2.8 1.89e+02 - 2.44e-04 1.78e-04h 1
156 1.0496220e+01 1.90e-02 2.66e+17 2.8 1.54e+03 - 5.14e-05 5.36e-05h 1
157 1.0497426e+01 1.91e-02 3.09e+17 2.8 5.05e+02 - 3.62e-04 7.98e-05f 3
158r 1.0497426e+01 1.91e-02 9.71e+02 2.8 0.00e+00 14.9 0.00e+00 3.82e-07R 3
159r 1.0493432e+01 1.63e-02 9.23e+00 0.6 6.35e-01 - 9.91e-01 9.82e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
160 1.0469426e+01 1.53e-02 7.34e+00 -2.1 4.90e-01 - 4.75e-01 6.43e-02h 1
161 9.4808605e+00 1.51e-02 5.60e+02 -0.9 8.29e+00 - 8.72e-01 1.60e-02f 1
162 9.9072568e+00 1.51e-02 4.76e+03 1.5 9.68e+02 - 5.15e-02 2.03e-03f 1
163 9.7277414e+00 1.50e-02 5.03e+03 -4.0 1.03e+01 - 2.27e-02 1.00e-02f 1
164 1.0340337e+01 1.49e-02 5.06e+03 1.1 1.10e+02 - 1.07e-01 4.38e-03f 1
165 1.0917084e+01 1.47e-02 7.29e+03 1.4 3.04e+01 - 3.49e-02 1.47e-02h 1
166 1.0947427e+01 1.46e-02 1.47e+05 1.4 1.90e+01 - 1.00e+00 5.47e-03h 1
167 1.1075494e+01 1.44e-02 1.39e+06 2.8 3.26e+01 - 2.39e-01 1.56e-02h 1
168 1.1445499e+01 1.42e-02 1.04e+06 -3.7 1.51e+01 - 4.59e-03 1.14e-02h 1
169 1.1572376e+01 1.41e-02 1.40e+06 2.8 2.40e+01 - 8.55e-02 5.51e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
170 1.1707425e+01 1.41e-02 2.07e+07 2.8 2.05e+01 - 4.58e-01 5.60e-03h 1
171 1.1892931e+01 1.39e-02 1.52e+07 2.8 5.93e+01 - 7.53e-03 9.99e-03h 1
172 1.1927791e+01 1.39e-02 1.72e+07 2.8 3.03e+01 - 5.72e-04 1.25e-03f 1
173 1.1986612e+01 1.39e-02 2.82e+07 2.8 2.66e+01 - 1.20e-01 2.19e-03h 1
174 1.1986670e+01 1.39e-02 2.58e+09 2.8 1.89e-01 14.4 1.80e-02 6.90e-05h 1
175 1.1986786e+01 1.39e-02 7.74e+11 2.8 2.48e-01 13.9 2.41e-02 9.46e-05h 1
176 1.1986900e+01 1.39e-02 8.33e+11 2.8 5.44e-01 13.5 8.29e-04 3.45e-05h 1
177 1.1987095e+01 1.39e-02 1.37e+12 2.8 7.66e-01 13.0 1.15e-02 4.29e-05h 1
178r 1.1987095e+01 1.39e-02 9.36e+02 2.8 0.00e+00 12.5 0.00e+00 3.09e-07R 6
179r 1.1987146e+01 1.08e-02 1.08e+01 -3.4 6.35e-01 - 9.88e-01 9.81e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
180 1.1995126e+01 1.08e-02 7.98e+00 -4.0 5.06e+00 - 1.39e-02 1.56e-03h 1
181 1.2430075e+01 9.89e-03 6.28e+00 -1.2 5.73e+00 - 5.58e-01 8.47e-02f 1
182 1.2303409e+01 9.42e-03 7.80e+00 -2.2 1.97e+00 - 5.74e-02 4.80e-02h 1
183 1.2559731e+01 9.05e-03 1.47e+01 -2.4 6.31e+00 - 3.59e-02 4.00e-02h 1
184 1.2852584e+01 8.76e-03 1.48e+01 -1.5 7.87e+00 - 2.10e-02 3.16e-02f 1
185 1.3082369e+01 8.59e-03 1.90e+01 -2.6 6.41e+00 - 2.75e-02 1.97e-02h 1
186 1.3083859e+01 8.52e-03 2.05e+02 0.2 4.68e+01 - 9.22e-02 5.27e-03f 1
187 1.3607270e+01 8.48e-03 9.35e+03 1.0 1.00e+02 - 9.29e-01 4.45e-03f 1
188 1.4070755e+01 8.34e-03 3.95e+05 2.2 1.60e+01 - 9.88e-01 1.67e-02h 1
189 1.4391139e+01 8.11e-03 6.47e+05 2.8 1.88e+01 - 5.65e-02 2.55e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
190 1.4823545e+01 8.03e-03 7.26e+05 2.8 2.41e+01 - 1.20e-02 9.28e-03h 1
191 1.4972496e+01 7.99e-03 1.77e+06 -3.7 4.69e+01 - 1.73e-02 5.09e-03h 1
192 1.4986885e+01 7.99e-03 1.78e+06 2.8 3.25e+02 - 5.21e-04 3.31e-04f 3
193 1.5007602e+01 7.99e-03 1.54e+06 2.2 3.57e+02 - 5.63e-03 4.53e-04f 2
194 1.5190191e+01 7.95e-03 1.24e+06 1.5 2.06e+01 - 2.54e-03 4.52e-03h 2
195 1.5245114e+01 7.94e-03 3.86e+05 1.1 7.43e+01 - 5.19e-03 1.36e-03f 1
196 1.5541191e+01 7.87e-03 3.19e+06 2.0 2.71e+01 - 1.49e-01 9.00e-03h 1
197 1.5690437e+01 7.84e-03 8.15e+07 2.8 1.76e+01 - 1.39e-01 3.60e-03h 1
198 1.5688104e+01 7.82e-03 1.69e+08 2.8 6.14e+01 - 6.16e-03 1.89e-03h 1
199 1.5860011e+01 7.80e-03 2.46e+07 2.8 3.62e+01 - 1.89e-04 2.10e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
200 1.5849339e+01 7.80e-03 7.22e+08 -3.5 5.25e+02 - 4.25e-04 2.72e-05f 1
Number of Iterations....: 200
(scaled) (unscaled)
Objective...............: 1.5849339467302581e+01 1.5849339467302581e+01
Dual infeasibility......: 7.2171132001666582e+08 7.2171132001666582e+08
Constraint violation....: 7.8037068786840358e-03 7.8037068786840358e-03
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.6670615089360967e+03 2.6670615089360967e+03
Overall NLP error.......: 1.7450866344943236e+02 7.2171132001666582e+08
Number of objective function evaluations = 431
Number of objective gradient evaluations = 200
Number of equality constraint evaluations = 431
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 206
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 200
Total seconds in IPOPT = 7.250
EXIT: Maximum Number of Iterations Exceeded.
==================================================
Minimum time solution found!
==================================================Step 6: Analyze Time-Optimal Solution
x_sol_mintime = prob_mintime.trajectory.x
u_sol_mintime = prob_mintime.trajectory.u
Δt_sol_mintime = prob_mintime.trajectory.Δt
total_time_mintime = sum(Δt_sol_mintime)
println("\nMinimum time solution:")
println(" Total time: $total_time_mintime seconds")
println(" Average Δt: ", mean(Δt_sol_mintime))
println(" Min Δt: ", minimum(Δt_sol_mintime))
println(" Max Δt: ", maximum(Δt_sol_mintime))
println("\nControl statistics:")
println(" Max |u|: ", maximum(abs.(u_sol_mintime)))
println(" Mean |u|: ", mean(abs.(u_sol_mintime)))
println(" ||u||: ", norm(u_sol_mintime))
Minimum time solution:
Total time: 16.065664500738613 seconds
Average Δt: 0.4016416125184653
Min Δt: 0.010144558843697379
Max Δt: 0.4999993756816071
Control statistics:
Max |u|: 0.9999998381866488
Mean |u|: 0.9491249342178758
||u||: 6.158665675311596Check if controls saturate
u_saturated = sum(abs.(u_sol_mintime) .> 0.99)
println(" Time steps with |u| > 0.99: $u_saturated / $N") Time steps with |u| > 0.99: 37 / 40Step 7: Comparison with Fixed-Time Solution
println("\n" * "="^50)
println("COMPARISON: MINIMUM TIME vs FIXED TIME")
println("="^50)
==================================================
COMPARISON: MINIMUM TIME vs FIXED TIME
==================================================Solve fixed-time problem with same total time
Δt_fixed = total_time_mintime / N
traj_fixed = NamedTrajectory(
(x = x_guess, u = u_guess, Δt = fill(Δt_fixed, N));
timestep = :Δt,
controls = :u,
initial = (x = x_init,),
final = (x = x_goal,),
bounds = (u = 1.0,),
)
obj_fixed = QuadraticRegularizer(:u, traj_fixed, 1.0)
integrator_fixed = BilinearIntegrator(G, :x, :u, traj_fixed)
prob_fixed = DirectTrajOptProblem(traj_fixed, obj_fixed, integrator_fixed)
println("\nSolving fixed-time problem with T = $total_time_mintime seconds...")
solve!(prob_fixed; max_iter = 150, verbose = false)
u_sol_fixed = prob_fixed.trajectory.u
println("\nComparison:")
println(" Minimum time:")
println(" Total time: $total_time_mintime s")
println(" ||u||: ", norm(u_sol_mintime))
println(" Max |u|: ", maximum(abs.(u_sol_mintime)))
println(" Fixed time:")
println(" Total time: ", sum(prob_fixed.trajectory.Δt), " s")
println(" ||u||: ", norm(u_sol_fixed))
println(" Max |u|: ", maximum(abs.(u_sol_fixed)))┌ Warning: Trajectory has timestep variable :Δt but no bounds on it.
│ Adding default lower bound of 0 to prevent negative timesteps.
│
│ Recommended: Add explicit bounds when creating the trajectory:
│ NamedTrajectory(...; Δt_bounds=(min, max))
│ Example:
│ NamedTrajectory(qtraj, N; Δt_bounds=(1e-3, 0.5))
│
│ Or use timesteps_all_equal=true in problem options to fix timesteps.
└ @ DirectTrajOpt.Problems ~/work/DirectTrajOpt.jl/DirectTrajOpt.jl/src/problems.jl:66
Solving fixed-time problem with T = 16.065664500738613 seconds...
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 616
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 994
Total number of variables............................: 156
variables with only lower bounds: 40
variables with lower and upper bounds: 40
variables with only upper bounds: 0
Total number of equality constraints.................: 78
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 2.0717831e-02 3.54e-01 1.02e+00 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 2.0505183e-02 9.40e-02 4.55e-01 -4.0 4.60e-01 0.0 8.11e-01 1.00e+00h 1
2 2.3646015e-02 6.64e-02 8.91e-01 -1.3 5.03e-01 -0.5 7.08e-01 4.68e-01h 1
3 1.0193629e-01 9.47e-02 3.07e+00 -1.4 6.19e+00 -0.1 5.64e-02 1.03e-01h 1
Number of Iterations....: 4
Number of objective function evaluations = 5
Number of objective gradient evaluations = 5
Number of equality constraint evaluations = 5
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 4
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 4
Total seconds in IPOPT = 0.017
EXIT: Invalid number in NLP function or derivative detected.
Comparison:
Minimum time:
Total time: 16.065664500738613 s
||u||: 6.158665675311596
Max |u|: 0.9999998381866488
Fixed time:
Total time: 12.780483511623359 s
||u||: 1.5235157385546705
Max |u|: 0.9999358770050446Step 8: Effect of Time Weight
println("\n" * "="^50)
println("EXPLORING TIME WEIGHT EFFECTS")
println("="^50)
==================================================
EXPLORING TIME WEIGHT EFFECTS
==================================================Try different time weights
time_weights = [0.1, 1.0, 10.0]
results = []
for w_t in time_weights
traj_test = NamedTrajectory(
(x = x_guess, u = u_guess, Δt = Δt_guess);
timestep = :Δt,
controls = :u,
initial = (x = x_init,),
final = (x = x_goal,),
bounds = (u = 1.0, Δt = (0.01, 0.5)),
)
obj_test = (
1e-2 * QuadraticRegularizer(:u, traj_test, 1.0) +
w_t * MinimumTimeObjective(traj_test, 1.0)
)
integrator_test = BilinearIntegrator(G, :x, :u, traj_test)
prob_test = DirectTrajOptProblem(traj_test, obj_test, integrator_test)
solve!(prob_test; max_iter = 200, verbose = false)
push!(
results,
(
weight = w_t,
time = sum(prob_test.trajectory.Δt),
control_norm = norm(prob_test.trajectory.u),
max_control = maximum(abs.(prob_test.trajectory.u)),
),
)
end
println("\nTime weight effects:")
println("Weight | Total Time | ||u|| | Max |u|")
println("-"^45)
for r in results
println(
@sprintf(
"%.1f | %.4f s | %.4f | %.4f",
r.weight,
r.time,
r.control_norm,
r.max_control
)
)
end
println("\nObservations:")
println(" - Lower weight → slower trajectory, gentler controls")
println(" - Higher weight → faster trajectory, more aggressive controls")This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 616
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 994
Total number of variables............................: 156
variables with only lower bounds: 0
variables with lower and upper bounds: 80
variables with only upper bounds: 0
Total number of equality constraints.................: 78
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 5.8502890e-01 1.39e-01 1.00e-01 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 5.8276207e-01 9.61e-02 2.29e+00 -0.8 7.86e-01 - 9.95e-01 9.64e-01h 1
2 6.6674230e-01 5.43e-02 1.82e+00 -0.9 2.09e+00 - 4.57e-01 4.75e-01f 1
3 7.1914146e-01 4.34e-02 1.42e+00 -0.7 1.53e+00 - 6.29e-01 2.11e-01h 1
4 7.7628022e-01 2.44e-02 1.45e+00 -1.7 1.02e+00 - 4.61e-01 4.88e-01h 1
5 7.8199098e-01 2.37e-02 1.43e+00 -4.0 1.60e+01 - 2.96e-02 2.66e-02h 1
6 8.1487037e-01 1.94e-02 3.01e+00 -1.1 1.28e+00 0.0 9.21e-01 1.53e-01h 1
7 8.6192880e-01 1.56e-02 2.17e+00 -3.1 2.21e+00 - 2.88e-01 3.09e-01h 1
8 9.0971355e-01 1.26e-02 3.34e+00 -1.0 1.16e+00 - 4.31e-01 1.45e-01h 1
9 1.0369456e+00 1.32e-02 8.71e+00 -1.4 3.48e-01 0.4 9.90e-01 5.86e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 1.0458508e+00 1.27e-02 8.48e+00 -4.0 7.88e+00 -0.1 2.18e-02 2.71e-02h 1
11 1.0693129e+00 1.25e-02 7.92e+00 -2.8 7.46e-01 0.4 3.55e-01 9.45e-02h 1
12 1.1299711e+00 1.66e-02 2.73e+01 -1.7 6.91e-01 0.8 9.98e-01 2.17e-01h 1
13 1.1305763e+00 1.66e-02 2.73e+01 -4.0 6.38e+01 0.3 4.89e-03 3.19e-04h 5
14 1.1484051e+00 1.63e-02 2.52e+02 -0.5 4.33e+00 0.7 1.00e+00 2.71e-02h 1
15 1.1630991e+00 1.63e-02 2.29e+02 1.4 4.13e+02 - 3.90e-03 7.84e-04f 1
16 1.1775898e+00 1.58e-02 2.20e+02 -0.5 7.20e+00 0.3 8.75e-02 3.16e-02h 1
17 1.1934751e+00 1.53e-02 2.07e+02 -1.5 1.07e+01 -0.2 1.06e-01 3.07e-02h 1
18 1.2469310e+00 1.44e-02 2.37e+02 -0.1 5.89e+00 - 3.08e-01 5.81e-02f 1
19 1.2618382e+00 1.42e-02 4.67e+02 0.4 1.72e+01 - 1.00e+00 1.14e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 1.2705186e+00 1.39e-02 5.97e+02 -4.0 4.47e+00 - 1.70e-01 2.58e-02h 1
21 1.2742077e+00 1.37e-02 6.46e+02 -4.0 3.47e+01 - 3.86e-02 1.68e-02h 1
22 1.2772761e+00 1.35e-02 9.68e+02 -4.0 6.09e+00 - 7.48e-02 1.10e-02h 1
23 1.3286240e+00 2.31e-02 2.08e+03 0.7 8.72e+00 - 1.00e+00 5.14e-02f 1
24 1.3456607e+00 2.15e-02 3.93e+03 0.6 3.45e+00 - 1.32e-01 8.34e-02h 1
25 1.3577577e+00 2.13e-02 3.91e+03 -0.1 2.05e+01 - 1.63e-02 8.63e-03h 2
26 1.3615999e+00 2.12e-02 3.90e+03 -0.1 3.16e+01 - 7.06e-03 2.82e-03f 2
27 1.3883123e+00 2.08e-02 4.08e+03 -0.1 7.98e+00 - 6.39e-02 2.04e-02h 1
28 1.4211547e+00 2.02e-02 5.33e+03 -0.1 8.80e+00 - 3.99e-02 2.94e-02h 1
29 1.4228977e+00 2.02e-02 6.28e+03 1.2 2.76e+02 - 3.44e-03 8.76e-04f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 1.4543184e+00 1.96e-02 6.64e+03 0.4 1.52e+01 - 7.10e-02 2.78e-02h 1
31 1.4589066e+00 1.95e-02 6.86e+03 1.5 6.72e+01 - 2.48e-02 1.25e-03h 1
32 1.4748528e+00 1.94e-02 7.21e+04 2.0 3.15e+01 - 1.00e+00 4.04e-03f 1
33 1.4817377e+00 1.93e-02 1.21e+05 1.8 3.76e+00 - 6.25e-02 8.55e-03h 1
34 1.4972266e+00 1.90e-02 1.58e+05 -4.0 7.24e+00 - 2.41e-02 1.52e-02h 2
35 1.5170860e+00 1.88e-02 5.33e+05 2.8 3.89e+01 - 4.39e-02 8.72e-03h 2
36 1.5362610e+00 1.87e-02 7.65e+05 -3.9 6.09e+01 - 7.40e-03 5.63e-03h 1
37 1.5625798e+00 1.86e-02 9.22e+05 2.8 2.44e+01 - 1.21e-01 7.35e-03h 1
38 1.5786821e+00 1.85e-02 2.05e+06 2.8 2.02e+01 - 4.35e-02 4.41e-03h 3
39 1.5879372e+00 1.84e-02 3.06e+06 2.8 1.78e+02 - 6.64e-03 2.37e-03h 2
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 1.5864895e+00 1.84e-02 3.21e+06 2.8 7.16e+02 2.9 6.87e-04 5.93e-04h 1
41 1.5860247e+00 1.84e-02 3.63e+06 2.8 3.99e+01 - 1.06e-02 2.25e-03h 1
42 1.5936271e+00 1.83e-02 4.07e+06 2.8 2.76e+01 - 3.27e-02 3.28e-03h 1
43 1.6073858e+00 1.82e-02 1.38e+07 2.8 1.73e+01 - 1.48e-01 5.43e-03h 2
44 1.6146789e+00 1.82e-02 2.16e+08 2.8 2.74e+01 - 3.75e-01 2.79e-03h 4
45 1.6213651e+00 1.81e-02 6.57e+08 2.8 2.68e+01 - 5.09e-02 2.64e-03h 4
46 1.6332305e+00 1.80e-02 8.15e+09 2.8 2.33e+01 - 2.39e-01 4.32e-03h 3
47 1.6518662e+00 1.79e-02 4.87e+10 2.8 2.17e+01 - 5.98e-02 6.76e-03h 2
48 1.6730153e+00 1.78e-02 5.29e+11 0.2 1.90e+01 - 8.24e-04 6.67e-03h 1
49 1.6773851e+00 1.78e-02 5.29e+11 2.8 1.05e+02 8.8 5.66e-05 3.83e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 1.6788162e+00 1.78e-02 5.29e+11 2.8 1.32e+02 - 3.49e-04 1.10e-04h 1
51 1.6847092e+00 1.78e-02 5.28e+11 2.8 1.26e+02 - 2.18e-03 4.62e-04h 1
52 1.7210786e+00 1.77e-02 5.28e+11 2.8 1.29e+02 - 5.67e-02 2.74e-03h 1
53 1.7303774e+00 1.77e-02 5.28e+11 2.8 6.40e+01 9.2 9.42e-05 1.13e-03h 1
54 1.7325892e+00 1.77e-02 5.27e+11 2.8 7.03e+01 - 2.08e-04 6.12e-04h 1
55 1.7381850e+00 1.77e-02 5.27e+11 2.8 5.70e+01 - 2.10e-03 9.12e-04h 1
56 1.7561853e+00 1.76e-02 5.25e+11 2.8 5.72e+01 - 3.13e-03 2.81e-03h 1
57 1.7571982e+00 1.76e-02 5.25e+11 2.8 5.90e+01 - 6.59e-02 1.46e-04h 1
58 1.7704457e+00 1.76e-02 5.24e+11 2.8 6.31e+01 - 5.95e-03 2.03e-03h 1
59 1.7724480e+00 1.76e-02 8.51e+11 2.2 6.56e+01 - 1.00e+00 2.92e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 1.7732953e+00 1.76e-02 3.77e+12 1.2 4.02e+01 - 1.04e-02 1.77e-04h 1
61 1.8209782e+00 1.76e-02 5.13e+13 2.8 2.97e+02 - 8.51e-05 1.55e-03h 1
62 1.8214568e+00 1.76e-02 5.13e+13 2.8 2.63e+02 - 1.82e-03 1.75e-05h 1
63r 1.8214568e+00 1.76e-02 9.54e+02 2.8 0.00e+00 - 0.00e+00 8.20e-08R 2
64r 1.8060502e+00 1.55e-02 8.94e+00 0.7 6.35e-01 - 9.91e-01 9.84e-01f 1
65 1.8086247e+00 1.44e-02 2.40e+00 -4.0 2.64e-01 - 2.39e-01 7.07e-02h 1
66 1.8219702e+00 1.38e-02 9.66e+01 -0.3 1.84e+01 - 1.16e-01 2.56e-02f 1
67 1.8292750e+00 1.31e-02 3.55e+02 -0.4 7.01e+00 - 8.95e-01 4.87e-02f 1
68 1.8288781e+00 1.26e-02 1.34e+03 0.4 6.65e+00 - 1.00e+00 4.12e-02h 1
69 1.8952384e+00 1.03e-02 1.19e+04 0.1 3.43e-01 - 1.00e+00 1.60e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 1.9150252e+00 1.00e-02 1.74e+05 1.0 3.01e+00 - 1.00e+00 2.53e-02h 1
71 1.9186871e+00 9.91e-03 7.90e+06 1.8 1.26e+01 - 1.00e+00 8.65e-03h 2
72 1.9218119e+00 9.84e-03 5.44e+08 2.7 1.50e+01 - 1.00e+00 7.15e-03h 2
73 1.9248940e+00 9.77e-03 8.33e+09 2.8 1.63e+01 - 1.15e-01 7.15e-03h 1
74 1.9250173e+00 9.76e-03 9.40e+12 2.8 5.85e+01 - 4.16e-01 3.71e-04h 1
75 1.9250275e+00 3.44e-02 8.89e+12 2.8 4.69e-01 9.6 5.49e-02 5.49e-02s 18
76r 1.9250275e+00 3.44e-02 9.16e+02 2.8 0.00e+00 - 0.00e+00 2.69e-07R 5
77r 1.9250415e+00 3.19e-02 1.34e+01 -3.4 6.58e-01 - 9.85e-01 9.63e-01f 1
78r 1.8959659e+00 6.30e-04 1.79e-01 -1.0 3.82e-02 - 9.88e-01 9.90e-01f 1
79 1.8929347e+00 5.59e-04 4.94e+00 -1.9 2.61e-01 - 5.77e-01 1.14e-01H 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 1.8138750e+00 6.66e-04 3.09e+03 -0.3 4.71e+01 - 1.00e+00 4.19e-02f 1
81 1.8244873e+00 6.50e-04 3.56e+03 -0.0 3.64e+00 - 5.65e-01 2.90e-02h 2
82 1.8458166e+00 6.44e-04 4.24e+03 -0.5 1.64e-01 - 1.00e+00 5.22e-02h 3
83 1.8840887e+00 6.75e-04 1.68e+04 -0.6 2.86e-01 - 1.00e+00 9.23e-02h 2
84 1.9237094e+00 7.05e-04 1.65e+05 0.3 5.42e-01 - 1.00e+00 9.63e-02h 1
85 1.9242287e+00 7.04e-04 3.37e+06 0.4 1.67e+01 - 1.19e-01 1.04e-03h 3
86 1.9244296e+00 7.04e-04 1.03e+09 0.4 2.33e+01 - 1.00e+00 4.10e-04h 4
87 1.9247820e+00 7.03e-04 2.98e+10 0.4 2.33e+01 - 8.25e-02 7.22e-04h 3
88 1.9250480e+00 7.03e-04 1.38e+13 0.4 2.33e+01 - 1.00e+00 5.41e-04h 3
89 1.9258487e+00 7.03e-04 6.96e+14 0.4 2.34e+01 - 8.95e-02 1.75e-03H 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90r 1.9258487e+00 7.03e-04 1.00e+03 0.4 0.00e+00 - 0.00e+00 2.65e-07R 7
91r 1.9250711e+00 5.70e-04 9.90e+00 -0.6 3.24e-03 - 9.90e-01 9.30e-01f 1
92 1.9242594e+00 5.70e-04 1.98e+01 -0.3 1.49e+00 - 4.82e-03 4.84e-04f 1
93 1.8943778e+00 4.86e-04 2.34e+04 -0.3 4.99e+01 - 2.25e-02 3.97e-02f 1
94 1.8921646e+00 4.81e-04 2.32e+04 -0.8 7.36e-01 - 4.48e-01 9.89e-03h 1
95 1.8935625e+00 4.79e-04 2.31e+04 0.5 5.02e+00 - 1.00e+00 3.10e-03f 1
96 1.9215864e+00 7.46e-04 4.29e+04 0.7 2.30e+00 - 1.00e+00 7.54e-02h 1
97 1.9237312e+00 7.73e-04 4.82e+06 1.7 6.82e+00 - 1.00e+00 4.91e-03h 2
98 1.9237982e+00 7.73e-04 8.32e+08 2.1 3.00e+01 - 1.00e+00 1.80e-04h 6
99 1.9238633e+00 7.74e-04 9.12e+09 1.4 3.07e+01 - 5.70e-02 1.76e-04h 6
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
100 1.9239266e+00 7.75e-04 7.80e+11 1.4 3.05e+01 - 4.66e-01 1.70e-04h 6
101 1.9239880e+00 7.75e-04 1.08e+13 1.4 3.06e+01 - 6.87e-02 1.65e-04h 6
102 1.9240475e+00 7.76e-04 2.09e+15 1.4 3.05e+01 - 1.00e+00 1.60e-04h 6
103 1.9241053e+00 7.76e-04 5.83e+15 1.4 3.05e+01 - 6.01e-02 1.55e-04h 6
104 1.9245342e+00 7.79e-04 4.19e+14 0.7 1.28e+01 - 6.75e-01 9.54e-04h 3
105 1.9248813e+00 7.83e-04 9.89e+14 -0.0 1.54e+01 - 1.12e-01 8.11e-04h 3
106 1.9251604e+00 7.85e-04 1.32e+15 -0.7 1.59e+01 - 7.75e-01 6.71e-04h 3
107 1.9256006e+00 7.87e-04 1.29e+15 -0.7 1.12e+01 - 1.00e+00 1.05e-03h 2
108 1.9256578e+00 7.88e-04 1.29e+15 -0.7 3.39e+01 - 5.59e-02 1.62e-04h 4
109 1.9256817e+00 7.88e-04 1.29e+15 -0.7 4.13e+01 - 4.83e-01 7.17e-05h 5
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
110 1.9257034e+00 7.88e-04 1.29e+15 -0.7 4.52e+01 - 4.46e-02 6.75e-05h 5
111 1.9257233e+00 7.89e-04 1.29e+15 -0.7 4.93e+01 - 1.00e+00 6.35e-05h 5
112 1.9257413e+00 7.89e-04 1.29e+15 -0.7 5.36e+01 - 3.55e-02 5.98e-05h 5
113 1.9257494e+00 7.89e-04 1.29e+15 -0.7 5.81e+01 - 3.44e-01 2.81e-05h 6
114 1.9257572e+00 7.89e-04 1.29e+15 -0.7 6.02e+01 - 3.47e-02 2.73e-05h 6
115 1.9257646e+00 7.89e-04 1.29e+15 -0.7 6.25e+01 - 1.00e+00 2.65e-05h 6
116 1.9257716e+00 7.89e-04 1.29e+15 -0.7 6.47e+01 - 2.95e-02 2.57e-05h 6
117 1.9257783e+00 7.89e-04 1.29e+15 -0.7 6.69e+01 - 2.97e-01 2.50e-05h 6
118 1.9257846e+00 7.89e-04 1.29e+15 -0.7 6.91e+01 - 3.08e-02 2.42e-05h 6
119 1.9257906e+00 7.89e-04 1.29e+15 -0.7 7.14e+01 - 1.00e+00 2.35e-05h 6
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
120 1.9257964e+00 7.89e-04 1.29e+15 -0.7 7.36e+01 - 2.60e-02 2.28e-05h 6
121 1.9258018e+00 7.89e-04 1.29e+15 -0.7 7.58e+01 - 2.62e-01 2.21e-05h 6
122r 1.9258018e+00 7.89e-04 1.00e+03 -0.7 0.00e+00 - 0.00e+00 3.36e-07R 12
123r 1.9238624e+00 8.88e-04 1.13e+01 -0.1 1.63e-02 - 9.90e-01 1.00e+00f 1
124r 1.8097962e+00 2.76e-03 2.34e+02 -0.3 1.80e-01 - 9.11e-01 6.16e-01f 1
125r 1.7525775e+00 2.75e-02 1.03e+03 -0.3 1.05e+01 - 6.69e-03 6.61e-02f 1
126r 1.4339621e+00 1.17e-02 7.98e+01 -0.3 1.36e-01 2.0 1.00e+00 1.00e+00f 1
127r 1.3654400e+00 4.91e-03 6.27e+01 -0.3 1.11e+00 - 5.56e-01 6.17e-01h 1
128r 1.3316131e+00 3.19e-03 1.84e+02 -0.3 1.28e+00 - 6.15e-01 4.67e-01f 1
129r 1.3325760e+00 4.32e-03 4.43e+01 -0.3 1.66e-01 2.4 1.00e+00 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
130r 1.2879617e+00 8.81e-03 3.62e+01 -0.3 1.07e+00 - 1.00e+00 8.39e-01f 1
131r 1.2627708e+00 1.90e-02 2.03e+01 -0.3 9.58e-01 - 5.57e-01 8.38e-01F 1
132r 1.3087157e+00 1.47e-02 1.42e+02 -1.0 3.79e+00 - 1.28e-01 2.35e-01f 1
133r 1.3155555e+00 4.62e-03 3.26e+00 -1.0 3.82e-02 1.9 1.00e+00 1.00e+00f 1
134r 1.3696726e+00 2.87e-03 1.03e+00 -2.0 3.69e-02 1.5 9.83e-01 1.00e+00f 1
135r 1.4440394e+00 1.43e-03 2.07e+00 -3.1 6.01e-02 1.0 1.00e+00 1.00e+00f 1
136r 1.5193229e+00 2.27e-03 6.05e+00 -4.0 9.11e-02 0.5 1.00e+00 8.00e-01f 1
137r 1.5613820e+00 2.05e-03 2.73e+01 -4.0 1.35e-01 0.0 1.00e+00 3.46e-01f 1
138r 1.6011223e+00 1.33e-03 6.69e+00 -4.0 5.87e-02 0.5 1.00e+00 5.51e-01h 1
139r 1.6293231e+00 1.28e-03 2.27e+01 -4.0 1.18e-01 -0.0 1.00e+00 2.72e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
140r 1.6368146e+00 1.27e-03 1.75e+01 -4.0 2.42e+01 -0.5 1.74e-02 7.21e-03f 1
141r 1.6464779e+00 1.18e-03 2.93e+01 -4.0 1.20e-01 -0.1 1.00e+00 1.02e-01h 1
142r 1.6630519e+00 1.09e-03 7.23e+01 -3.8 7.05e-01 -0.5 1.00e+00 9.73e-02h 1
143r 1.7065207e+00 1.24e-03 1.76e+01 -4.0 1.19e-01 -0.1 1.00e+00 5.20e-01h 1
144r 1.7292647e+00 1.51e-03 9.74e+01 -3.9 8.90e-01 -0.6 1.00e+00 1.53e-01h 1
145r 1.7578974e+00 2.01e-03 1.14e+01 -4.0 1.23e-01 -0.2 1.00e+00 5.72e-01h 1
146r 1.7665166e+00 4.06e-03 3.89e+01 -3.7 4.39e-01 -0.6 1.00e+00 1.18e-01h 1
147r 1.7702528e+00 3.76e-03 3.28e+01 -3.9 5.89e+00 -1.1 1.75e-01 7.37e-02h 1
148r 1.7789480e+00 2.83e-03 2.50e+01 -4.0 2.53e-01 -0.7 1.00e+00 2.48e-01h 1
149r 1.8042206e+00 5.15e-03 5.03e+01 -4.0 1.00e+00 -1.2 8.22e-01 5.50e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
150r 1.8046803e+00 4.28e-03 2.48e+01 -3.5 2.41e-01 -0.7 1.00e+00 1.61e-01h 1
151r 1.8175080e+00 1.81e-03 1.10e+01 -3.8 2.52e-01 -1.2 7.56e-01 3.67e-01h 1
152r 1.8348319e+00 6.95e-04 1.21e+00 -4.0 1.04e-01 -0.8 1.00e+00 8.40e-01h 1
153 1.8364876e+00 6.80e-04 1.41e+01 -4.0 2.55e+00 - 3.33e-01 2.20e-02h 1
154 1.8398249e+00 1.61e-03 9.47e+01 -1.3 6.67e+00 - 4.97e-01 9.62e-02f 1
155 1.8399393e+00 1.61e-03 1.06e+02 -0.7 2.68e+01 - 4.95e-02 2.25e-04h 7
156 1.8401072e+00 1.61e-03 3.74e+02 -0.7 2.07e+01 - 1.41e-01 2.37e-04h 8
157 1.8402065e+00 1.61e-03 5.07e+02 -0.7 1.68e+01 - 2.13e-02 9.80e-05h 10
158 1.8403323e+00 1.61e-03 1.26e+03 -0.7 1.65e+01 - 9.27e-02 1.06e-04h 10
159 1.8404145e+00 1.61e-03 1.50e+03 -0.7 2.03e+01 - 1.32e-02 4.90e-05h 11
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
160 1.8404947e+00 1.61e-03 2.89e+03 -0.7 2.03e+01 - 6.46e-02 4.57e-05h 11
161 1.8405693e+00 1.61e-03 3.43e+03 -0.7 2.02e+01 - 1.36e-02 3.61e-05h 11
162 1.8406461e+00 1.61e-03 5.54e+03 -0.7 2.05e+01 - 4.41e-02 3.80e-05h 11
163 1.8407204e+00 1.61e-03 6.78e+03 -0.7 2.07e+01 - 1.60e-02 3.68e-05h 11
164 1.8407980e+00 1.61e-03 1.04e+04 -0.7 2.10e+01 - 3.77e-02 4.08e-05h 11
165 1.8408776e+00 1.61e-03 1.33e+04 -0.7 2.13e+01 - 1.93e-02 4.45e-05h 11
166 1.8409612e+00 1.61e-03 2.29e+04 -0.7 2.16e+01 - 4.90e-02 5.00e-05h 11
167 1.8410470e+00 1.61e-03 2.95e+04 -0.7 2.17e+01 - 1.95e-02 5.40e-05h 11
168 1.8410918e+00 1.61e-03 8.28e+04 -0.7 2.19e+01 - 1.20e-01 3.00e-05h 12
169 1.8411354e+00 1.61e-03 1.03e+05 -0.7 2.19e+01 - 1.62e-02 2.82e-05h 12
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
170 1.8411802e+00 1.61e-03 1.65e+06 -0.7 2.20e+01 - 1.00e+00 3.01e-05h 12
171 1.8412241e+00 1.61e-03 2.01e+06 -0.7 2.20e+01 - 1.46e-02 2.86e-05h 12
172 1.8412679e+00 1.61e-03 6.94e+06 -0.7 2.21e+01 - 1.63e-01 2.99e-05h 12
173 1.8413116e+00 1.61e-03 8.92e+06 -0.7 2.21e+01 - 1.89e-02 2.99e-05h 12
174 1.8413533e+00 1.61e-03 1.45e+08 -0.7 2.22e+01 - 1.00e+00 2.96e-05h 12
175 1.8413981e+00 1.61e-03 1.77e+08 -0.7 2.21e+01 - 1.49e-02 2.98e-05h 12
176 1.8414403e+00 1.61e-03 2.82e+08 -0.7 2.22e+01 - 3.90e-02 2.96e-05h 12
177 1.8414816e+00 1.61e-03 4.67e+08 -0.7 2.23e+01 - 4.27e-02 2.94e-05h 12
178 1.8415227e+00 1.61e-03 1.03e+09 -0.7 2.23e+01 - 7.91e-02 2.94e-05h 12
179 1.8415647e+00 1.61e-03 1.33e+09 -0.7 2.23e+01 - 1.90e-02 2.94e-05h 12
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
180 1.8416050e+00 1.61e-03 2.18e+10 -0.7 2.24e+01 - 1.00e+00 2.92e-05h 12
181 1.8416492e+00 1.61e-03 2.68e+10 -0.7 2.20e+01 - 1.51e-02 2.92e-05h 12
182 1.8416911e+00 1.61e-03 4.03e+10 -0.7 2.24e+01 - 3.28e-02 2.93e-05h 12
183 1.8417315e+00 1.61e-03 7.37e+10 -0.7 2.24e+01 - 5.36e-02 2.92e-05h 12
184 1.8417721e+00 1.61e-03 1.40e+11 -0.7 2.25e+01 - 5.82e-02 2.91e-05h 12
185 1.8418127e+00 1.61e-03 1.86e+11 -0.7 2.25e+01 - 2.12e-02 2.91e-05h 12
186 1.8418785e+00 1.61e-03 1.14e+12 -0.7 2.16e+01 - 3.45e-01 5.10e-05h 11
187 1.8419343e+00 1.61e-03 1.33e+12 -0.7 4.60e+01 - 3.56e-02 9.07e-05h 9
188 1.8419892e+00 1.61e-03 1.56e+12 -0.7 4.71e+01 - 4.18e-02 8.83e-05h 9
189 1.8420426e+00 1.61e-03 2.16e+12 -0.7 4.80e+01 - 1.15e-01 8.63e-05h 9
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
190 1.8420906e+00 1.61e-03 2.27e+12 -0.7 4.97e+01 - 2.18e-02 8.30e-05h 9
191 1.8421395e+00 1.61e-03 5.67e+12 -0.7 4.98e+01 - 1.00e+00 8.26e-05h 9
192 1.8421868e+00 1.61e-03 5.67e+12 -0.7 5.21e+01 - 1.79e-02 7.86e-05h 9
193 1.8422354e+00 1.61e-03 5.67e+12 -0.7 5.19e+01 - 3.59e-02 7.86e-05h 9
194 1.8422853e+00 1.61e-03 5.67e+12 -0.7 5.17e+01 - 5.61e-02 7.87e-05h 9
195 1.8423363e+00 1.61e-03 5.67e+12 -0.7 5.14e+01 - 4.94e-02 7.87e-05h 9
196 1.8423882e+00 1.61e-03 5.67e+12 -0.7 5.12e+01 - 2.90e-02 7.88e-05h 9
197 1.8424409e+00 1.61e-03 5.67e+12 -0.7 5.10e+01 - 2.96e-01 7.88e-05h 9
198 1.8424943e+00 1.61e-03 5.67e+12 -0.7 5.07e+01 - 1.98e-02 7.89e-05h 9
199 1.8425483e+00 1.61e-03 5.67e+12 -0.7 5.05e+01 - 1.00e+00 7.89e-05h 9
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
200 1.8426030e+00 1.61e-03 5.67e+12 -0.7 5.03e+01 - 1.92e-02 7.90e-05h 9
Number of Iterations....: 200
(scaled) (unscaled)
Objective...............: 1.8426029533491142e+00 1.8426029533491142e+00
Dual infeasibility......: 5.6745228616168818e+12 5.6745228616168818e+12
Constraint violation....: 1.6071542169465713e-03 1.6071542169465713e-03
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 1.8450883631699016e+09 1.8450883631699016e+09
Overall NLP error.......: 3.7766778807665526e+02 5.6745228616168818e+12
Number of objective function evaluations = 968
Number of objective gradient evaluations = 171
Number of equality constraint evaluations = 968
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 205
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 200
Total seconds in IPOPT = 1.231
EXIT: Maximum Number of Iterations Exceeded.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 616
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 994
Total number of variables............................: 156
variables with only lower bounds: 0
variables with lower and upper bounds: 80
variables with only upper bounds: 0
Total number of equality constraints.................: 78
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 5.8500289e+00 1.39e-01 1.00e+00 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.4351193e+00 6.91e-02 1.08e+00 -0.8 5.08e-01 - 9.95e-01 8.66e-01h 1
2 2.1847072e+00 4.97e-02 2.58e+00 -4.0 2.07e+00 - 3.09e-01 2.77e-01f 1
3 3.1636077e+00 7.11e-02 5.27e+00 -0.4 2.09e+00 0.0 5.06e-01 2.30e-01f 1
4 3.3567640e+00 6.63e-02 6.33e+00 -0.7 1.76e+00 0.4 1.68e-01 6.86e-02h 1
5 3.3337604e+00 6.75e-02 6.17e+00 -0.3 4.11e+01 - 1.09e-02 7.54e-03f 2
6 3.4582139e+00 8.26e-02 6.01e+00 -4.0 1.44e+01 - 4.12e-02 5.20e-02h 1
7 3.5289906e+00 8.92e-02 1.61e+01 -0.8 2.29e+00 0.9 7.22e-01 1.85e-01h 1
8 3.5699447e+00 8.07e-02 1.41e+01 -4.0 4.85e+00 - 1.55e-01 9.50e-02h 1
9 3.7561027e+00 7.32e-02 1.64e+02 0.1 1.90e+00 1.3 6.65e-01 5.71e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 4.3439319e+00 6.16e-02 1.35e+02 -0.2 2.02e+00 - 3.39e-01 1.54e-01h 1
11 4.4724613e+00 5.64e-02 1.20e+02 -4.0 1.61e+00 - 1.18e-01 8.40e-02h 1
12 4.7405862e+00 5.25e-02 1.14e+02 -4.0 3.49e+00 - 5.27e-02 6.93e-02h 1
13 4.8933240e+00 5.11e-02 8.08e+02 -0.5 6.42e+00 - 9.97e-01 2.67e-02h 1
14 4.9563712e+00 5.06e-02 3.62e+03 0.7 5.93e+00 - 1.00e+00 1.02e-02h 1
15 4.5141657e+00 4.79e-02 2.80e+04 1.1 6.58e+00 - 1.00e+00 1.15e-01h 1
16 4.5020070e+00 5.93e-02 2.28e+04 -4.0 1.81e+01 - 8.52e-03 1.23e-02h 1
17 4.5714059e+00 5.90e-02 2.31e+04 2.0 1.27e+01 3.5 2.59e-02 6.12e-03h 1
18 4.5968031e+00 5.89e-02 3.02e+04 1.1 5.37e+00 3.9 1.35e-02 2.48e-03h 1
19 4.7191449e+00 5.81e-02 2.28e+04 1.1 4.44e+00 4.4 6.08e-03 1.31e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 5.1839619e+00 5.82e-02 3.89e+04 2.8 6.06e+00 3.9 5.02e-02 2.53e-02f 1
21 5.4865378e+00 5.65e-02 1.28e+05 2.8 4.09e+00 4.3 8.12e-02 2.21e-02f 1
22 5.7501943e+00 5.53e-02 3.42e+05 2.8 6.37e+00 3.8 1.16e-01 1.82e-02h 1
23 6.2851221e+00 5.36e-02 3.34e+05 2.8 1.20e+01 - 2.18e-02 3.06e-02f 1
24 6.6468843e+00 5.26e-02 3.21e+05 2.8 1.26e+01 - 2.49e-02 1.90e-02h 1
25 6.7515791e+00 5.24e-02 3.22e+05 -3.8 3.74e+01 - 1.79e-03 3.24e-03h 1
26 7.0577179e+00 5.18e-02 1.51e+06 2.8 2.50e+01 - 3.66e-01 1.19e-02h 1
27 7.3549435e+00 5.17e-02 1.55e+06 2.8 1.29e+02 - 2.48e-04 2.39e-03h 1
28 7.4971036e+00 5.14e-02 3.37e+06 2.8 3.38e+01 - 1.43e-01 4.51e-03h 1
29 7.2696859e+00 5.06e-02 1.17e+07 2.8 1.79e+01 - 8.01e-02 1.63e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 7.3279086e+00 5.02e-02 3.64e+06 -3.1 2.28e+01 - 1.52e-03 5.61e-03h 1
31 7.3709360e+00 5.01e-02 3.77e+06 -3.3 3.65e+01 5.2 1.07e-03 8.42e-04h 1
32 7.3940698e+00 5.01e-02 3.85e+06 2.8 1.00e+02 - 7.27e-04 4.90e-04h 1
33 7.5343229e+00 5.00e-02 4.40e+06 2.8 4.12e+01 - 4.05e-03 2.61e-03h 1
34 7.9234960e+00 4.96e-02 1.41e+07 2.8 3.88e+01 - 2.69e-02 6.70e-03h 1
35 8.1936612e+00 4.95e-02 1.34e+07 2.8 7.70e+01 - 2.88e-03 3.53e-03h 1
36 8.2013728e+00 4.95e-02 1.84e+07 -3.2 2.72e+01 - 8.52e-04 9.99e-05h 1
37 8.2476881e+00 4.94e-02 2.08e+07 2.8 2.39e+02 - 1.02e-04 3.29e-04h 1
38 8.3448854e+00 4.94e-02 1.69e+08 2.8 1.04e+02 - 1.09e-01 9.54e-04h 1
39 8.3808989e+00 4.94e-02 1.74e+08 2.8 1.12e+02 - 1.84e-03 3.24e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 8.6270759e+00 4.93e-02 2.93e+08 2.8 1.13e+02 - 2.37e-02 2.21e-03h 1
41 8.7790268e+00 4.92e-02 2.82e+08 2.8 2.69e+02 - 2.89e-05 5.80e-04h 1
42 8.8962487e+00 4.92e-02 2.69e+08 -2.7 1.36e+02 5.6 6.12e-05 5.42e-04h 1
43 8.9354921e+00 4.92e-02 2.69e+08 2.8 2.74e+02 5.1 3.13e-04 2.41e-04h 1
44 9.0668080e+00 4.92e-02 8.74e+08 2.8 1.78e+02 - 5.18e-02 6.83e-04h 1
45 9.2545985e+00 4.91e-02 8.77e+08 2.8 1.91e+02 - 8.88e-04 7.36e-04h 2
46 9.4478353e+00 4.91e-02 9.31e+08 2.8 2.23e+02 - 2.05e-03 6.39e-04h 1
47 9.4712771e+00 4.91e-02 1.21e+09 2.8 2.13e+02 - 1.86e-03 7.13e-05h 1
48 9.4796385e+00 4.91e-02 2.43e+10 2.8 1.93e+02 - 4.52e-02 5.22e-05h 1
49 9.4463563e+00 4.90e-02 1.30e+11 2.8 8.95e+01 - 7.75e-03 1.24e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 9.4582242e+00 4.90e-02 5.25e+10 -0.9 4.39e+01 - 4.90e-05 1.27e-04h 1
51 9.4728972e+00 4.90e-02 5.65e+10 2.8 9.56e+02 - 4.90e-06 1.95e-05h 1
52 9.4985227e+00 4.90e-02 5.59e+10 2.8 6.96e+02 - 3.32e-05 3.71e-05h 1
53 9.5402065e+00 4.90e-02 5.85e+10 2.8 7.22e+02 - 6.90e-05 5.78e-05h 1
54 9.6117997e+00 4.90e-02 1.76e+11 2.8 7.87e+02 - 4.96e-04 9.10e-05h 1
55 9.6494139e+00 4.90e-02 1.59e+11 -1.2 9.43e+02 - 3.13e-06 3.99e-05h 4
56 9.6461820e+00 4.90e-02 1.61e+11 -1.2 3.47e+03 - 4.68e-09 6.19e-06f 1
57 9.6576403e+00 4.90e-02 1.61e+11 2.8 8.60e+02 - 2.14e-05 2.02e-05h 1
58 9.6802093e+00 4.90e-02 5.90e+11 2.8 8.71e+02 - 1.13e-03 3.78e-05h 1
59 9.7159347e+00 4.90e-02 5.67e+11 2.8 9.48e+02 - 9.11e-06 5.50e-05f 4
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 9.7783589e+00 4.90e-02 5.09e+11 2.8 1.07e+03 - 6.49e-08 8.55e-05f 3
61 9.8140725e+00 4.90e-02 4.57e+11 2.8 1.42e+03 - 5.54e-07 4.86e-05f 3
62 9.8583358e+00 4.90e-02 4.57e+11 2.1 1.67e+03 - 5.09e-05 5.09e-05s 9
63 9.9222230e+00 4.90e-02 4.56e+11 1.4 2.14e+03 - 5.72e-05 5.72e-05s 8
64r 9.9222230e+00 4.90e-02 9.97e+02 0.7 0.00e+00 - 0.00e+00 1.82e-07R 2
65r 1.0179592e+01 2.24e-02 2.55e+02 1.4 9.37e-01 - 8.70e-01 3.87e-01f 1
66 1.0094637e+01 2.23e-02 8.67e+01 -0.0 2.82e+00 - 6.24e-01 2.61e-02f 1
67 8.0841272e+00 3.00e-02 9.00e+01 -0.0 3.39e+00 - 1.45e-01 1.33e-01f 1
68 8.1094033e+00 2.92e-02 1.01e+03 -0.0 2.16e-01 4.6 2.05e-01 2.51e-02h 1
69 8.1488110e+00 2.89e-02 3.95e+03 -0.0 5.10e-01 4.2 7.11e-02 1.04e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 8.1794139e+00 2.88e-02 8.06e+03 -0.0 1.33e+00 3.7 1.08e-01 4.57e-03h 1
71 8.4741930e+00 2.79e-02 7.25e+03 -0.0 1.52e+00 3.2 2.22e-02 3.29e-02h 1
72 8.5241241e+00 2.78e-02 6.48e+04 -0.0 1.64e+00 3.6 2.88e-01 5.65e-03h 1
73 8.8168562e+00 2.76e-02 6.07e+04 -0.0 3.68e+00 3.2 8.98e-03 2.04e-02h 1
74 8.9307438e+00 2.74e-02 6.27e+04 -0.0 1.21e+01 2.7 1.88e-02 8.70e-03h 1
75 9.0150845e+00 2.74e-02 7.94e+04 -0.0 1.08e+01 3.1 2.44e-02 4.16e-03h 1
76 9.2601916e+00 2.76e-02 9.00e+04 -0.0 1.91e+01 2.6 2.31e-02 8.90e-03h 1
77 9.2660232e+00 2.76e-02 9.94e+04 -0.0 1.92e+01 2.1 1.03e-02 3.72e-03h 1
78 9.4132549e+00 2.75e-02 1.36e+05 -0.0 2.68e+01 2.6 8.42e-03 2.33e-03h 1
79 9.4418614e+00 2.75e-02 1.34e+05 -0.0 2.04e+02 - 4.96e-04 8.01e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 9.5490076e+00 2.75e-02 1.41e+05 -0.0 7.68e+01 2.1 1.96e-03 1.44e-03h 1
81 9.6282211e+00 2.75e-02 1.84e+05 -0.0 1.33e+02 1.6 2.81e-03 1.06e-03h 1
82 9.7235673e+00 2.74e-02 4.77e+05 -0.0 1.91e+02 2.0 6.62e-03 1.21e-03h 1
83 9.7362068e+00 2.74e-02 5.31e+05 -0.0 1.04e+03 1.6 6.33e-04 1.84e-04h 1
84 9.8185033e+00 2.74e-02 2.67e+06 -0.0 3.82e+02 2.0 1.27e-02 1.10e-03h 1
85 9.8189819e+00 2.74e-02 1.26e+07 -0.0 3.19e+02 - 4.74e-03 3.90e-05h 1
86 9.8860352e+00 2.74e-02 7.97e+07 -0.0 1.10e+02 - 6.78e-03 8.54e-04h 1
87 9.8589566e+00 2.74e-02 8.21e+07 -0.0 1.45e+03 - 3.04e-05 1.90e-05h 5
88 9.8180145e+00 2.74e-02 8.68e+07 -0.0 1.51e+03 - 4.74e-05 2.98e-05h 3
89 9.7535169e+00 2.74e-02 1.13e+08 -0.0 7.27e+02 - 1.67e-04 1.03e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90 9.7975889e+00 2.74e-02 2.35e+10 -0.0 5.48e+02 - 2.01e-02 9.01e-05h 1
91r 9.7975889e+00 2.74e-02 1.00e+03 -0.0 0.00e+00 - 0.00e+00 3.84e-07R 7
92r 9.4593430e+00 2.26e-02 9.66e+02 1.2 3.31e+01 - 7.22e-01 1.04e-02f 1
93r 9.4734180e+00 2.16e-02 1.51e+03 0.6 1.61e+00 - 2.08e-01 3.51e-02f 1
94 9.4734353e+00 2.15e-02 3.09e+02 -0.5 1.09e+00 - 9.86e-01 6.62e-03f 1
95 8.3981406e+00 2.12e-02 8.28e+02 -0.2 4.22e+01 - 4.77e-02 1.14e-02f 1
96 8.4875182e+00 2.12e-02 1.19e+03 -4.0 4.21e+01 - 6.75e-03 1.18e-03h 1
97 8.9783073e+00 2.12e-02 1.04e+05 1.3 2.74e+02 - 1.00e+00 1.74e-03f 1
98 9.5488134e+00 2.11e-02 1.16e+05 2.8 2.18e+02 - 1.78e-02 2.18e-03f 1
99 9.5423319e+00 2.10e-02 1.49e+05 1.4 3.23e+00 - 2.60e-02 4.46e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
100 9.4233857e+00 2.23e-02 9.63e+04 1.4 9.14e+00 - 1.80e-03 1.03e-02h 1
101 9.6504403e+00 2.22e-02 3.28e+05 2.8 1.68e+01 - 1.14e-02 4.04e-03h 1
102 9.8095335e+00 2.22e-02 1.79e+06 1.6 5.13e+01 - 1.14e-02 1.81e-03h 1
103 9.8544517e+00 2.22e-02 1.24e+07 2.8 2.33e+02 - 7.92e-03 8.73e-04h 1
104 9.8639167e+00 2.22e-02 2.97e+08 2.8 4.29e+02 - 1.38e-02 5.77e-04h 1
105 9.7899506e+00 2.21e-02 5.66e+08 -3.6 1.48e+03 - 7.83e-05 3.92e-05f 1
106 9.8678472e+00 2.21e-02 5.65e+08 2.6 2.18e+03 - 3.06e-05 3.06e-05s 7
107r 9.8678472e+00 2.21e-02 9.95e+02 1.9 0.00e+00 - 0.00e+00 3.58e-07R 7
108r 9.8618359e+00 1.54e-02 9.07e+00 0.6 1.16e-01 - 9.92e-01 9.11e-01f 1
109 9.8368452e+00 1.49e-02 6.13e+01 -0.6 8.11e-01 - 6.40e-01 3.02e-02f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
110 8.4915145e+00 1.44e-02 1.77e+02 -0.5 1.19e+01 - 1.42e-01 3.97e-02f 1
111 9.1078127e+00 1.40e-02 3.79e+02 -4.0 3.89e+00 - 7.53e-02 2.41e-02h 1
112 9.2575654e+00 1.40e-02 1.71e+03 0.3 8.26e+01 - 7.51e-02 1.59e-03h 1
113 9.6896750e+00 1.39e-02 1.69e+03 -0.6 8.01e+01 - 1.04e-02 1.00e-02h 1
114 9.8459169e+00 1.39e-02 1.84e+03 1.3 3.79e+02 - 3.48e-03 6.98e-04f 1
115 9.8473115e+00 1.39e-02 2.05e+03 -0.4 2.54e+02 - 2.06e-03 7.34e-05h 6
116 9.8528227e+00 1.38e-02 3.62e+03 -0.4 1.36e+02 - 8.85e-03 2.52e-04h 4
117 9.8646631e+00 1.38e-02 5.01e+03 -0.4 6.76e+01 - 4.73e-03 4.65e-04h 5
118 9.8788163e+00 1.38e-02 8.88e+03 -0.4 6.71e+01 - 9.50e-03 5.25e-04h 5
119 9.8930589e+00 1.38e-02 1.30e+04 -0.4 6.16e+01 - 6.28e-03 5.25e-04h 5
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
120 9.9079072e+00 1.38e-02 2.29e+04 -0.4 6.33e+01 - 9.91e-03 5.67e-04h 5
121 9.9429184e+00 1.38e-02 2.38e+04 -0.4 5.95e+01 - 1.25e-03 7.37e-04h 4
122 9.9662303e+00 1.38e-02 4.12e+04 -0.4 6.91e+01 - 9.68e-03 4.62e-04h 4
123 9.9891010e+00 1.38e-02 5.13e+04 -0.4 5.79e+01 - 3.54e-03 5.69e-04h 5
124 1.0009960e+01 1.38e-02 4.31e+06 0.2 6.03e+01 - 1.00e+00 5.46e-04h 5
125 1.0085605e+01 1.38e-02 5.50e+06 2.8 5.55e+01 - 4.14e-03 8.89e-04h 4
126 1.0087111e+01 1.38e-02 1.64e+07 2.8 4.89e+01 - 2.34e-02 5.27e-04h 5
127 1.0092475e+01 1.38e-02 4.59e+07 2.8 4.54e+01 - 2.11e-02 6.56e-04h 5
128 1.0116435e+01 1.38e-02 6.92e+07 2.8 4.50e+01 - 6.45e-03 6.20e-04h 5
129 1.0133782e+01 1.37e-02 1.70e+09 2.8 4.58e+01 - 2.76e-01 5.72e-04h 5
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
130 1.0152081e+01 1.37e-02 3.77e+09 2.8 3.65e+01 - 1.41e-02 6.46e-04h 5
131 1.0173080e+01 1.37e-02 8.44e+09 2.8 3.84e+01 - 1.47e-02 6.02e-04h 5
132 1.0188182e+01 1.37e-02 1.95e+11 2.8 3.66e+01 - 2.56e-01 5.92e-04h 5
133 1.0214163e+01 1.37e-02 3.21e+11 2.8 3.16e+01 - 7.76e-03 6.43e-04h 5
134 1.0221807e+01 1.37e-02 1.55e+12 2.8 3.21e+01 - 4.57e-02 5.94e-04h 5
135 1.0244608e+01 1.37e-02 2.94e+12 2.8 2.83e+01 - 1.08e-02 6.31e-04h 5
136 1.0255007e+01 1.37e-02 4.96e+13 2.8 2.75e+01 - 1.92e-01 6.11e-04h 5
137 1.0280419e+01 1.37e-02 8.31e+13 2.8 2.37e+01 - 8.34e-03 6.25e-04h 5
138 1.0286201e+01 1.37e-02 2.34e+14 2.8 2.32e+01 - 2.38e-02 1.28e-03h 4
139 1.0310797e+01 1.36e-02 4.18e+14 2.8 2.26e+01 - 1.21e-02 6.27e-04h 5
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
140 1.0340035e+01 1.36e-02 2.98e+16 2.8 1.99e+01 - 1.00e+00 1.36e-03h 4
141 1.0343091e+01 1.36e-02 3.09e+16 2.8 3.02e+01 - 7.38e-03 4.95e-04h 5
142 1.0355198e+01 1.42e-02 3.22e+16 2.8 2.34e+01 - 2.33e-02 1.24e-03h 4
143 1.0365902e+01 1.45e-02 5.95e+16 2.8 2.48e+01 - 3.66e-01 1.10e-03h 4
144 1.0384243e+01 1.55e-02 5.95e+16 2.8 2.36e+01 - 1.48e-02 2.03e-03h 3
145 1.0411850e+01 1.72e-02 5.87e+16 2.8 2.18e+01 - 3.31e-02 3.31e-03h 2
146 1.0441610e+01 1.93e-02 5.83e+16 2.8 1.15e+01 - 9.05e-03 6.33e-03h 1
147 1.0443916e+01 1.92e-02 5.83e+16 2.8 4.06e+00 - 8.94e-03 4.11e-03h 1
148 1.0447399e+01 1.91e-02 5.97e+16 2.8 3.99e+00 - 1.32e-02 5.11e-03h 1
149 1.0452779e+01 1.90e-02 7.92e+16 2.8 3.70e+00 - 1.53e-02 2.74e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
150 1.0470367e+01 1.90e-02 9.05e+16 2.8 1.08e+01 - 2.14e-03 1.47e-03h 1
151 1.0497049e+01 1.90e-02 1.37e+17 2.8 2.56e+01 - 1.55e-03 8.85e-04h 1
152 1.0498861e+01 1.90e-02 3.29e+17 2.8 1.18e+01 15.8 1.34e-03 6.27e-04h 1
153 1.0494484e+01 1.90e-02 3.35e+17 2.8 2.55e+01 - 3.65e-04 4.09e-04f 1
154 1.0505972e+01 1.90e-02 2.84e+17 2.8 3.13e+02 15.4 4.30e-05 7.72e-05h 1
155 1.0501537e+01 1.90e-02 2.65e+17 2.8 1.89e+02 - 2.44e-04 1.78e-04h 1
156 1.0496220e+01 1.90e-02 2.66e+17 2.8 1.54e+03 - 5.14e-05 5.36e-05h 1
157 1.0497426e+01 1.91e-02 3.09e+17 2.8 5.05e+02 - 3.62e-04 7.98e-05f 3
158r 1.0497426e+01 1.91e-02 9.71e+02 2.8 0.00e+00 14.9 0.00e+00 3.82e-07R 3
159r 1.0493432e+01 1.63e-02 9.23e+00 0.6 6.35e-01 - 9.91e-01 9.82e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
160 1.0469426e+01 1.53e-02 7.34e+00 -2.1 4.90e-01 - 4.75e-01 6.43e-02h 1
161 9.4808605e+00 1.51e-02 5.60e+02 -0.9 8.29e+00 - 8.72e-01 1.60e-02f 1
162 9.9072568e+00 1.51e-02 4.76e+03 1.5 9.68e+02 - 5.15e-02 2.03e-03f 1
163 9.7277414e+00 1.50e-02 5.03e+03 -4.0 1.03e+01 - 2.27e-02 1.00e-02f 1
164 1.0340337e+01 1.49e-02 5.06e+03 1.1 1.10e+02 - 1.07e-01 4.38e-03f 1
165 1.0917084e+01 1.47e-02 7.29e+03 1.4 3.04e+01 - 3.49e-02 1.47e-02h 1
166 1.0947427e+01 1.46e-02 1.47e+05 1.4 1.90e+01 - 1.00e+00 5.47e-03h 1
167 1.1075494e+01 1.44e-02 1.39e+06 2.8 3.26e+01 - 2.39e-01 1.56e-02h 1
168 1.1445499e+01 1.42e-02 1.04e+06 -3.7 1.51e+01 - 4.59e-03 1.14e-02h 1
169 1.1572376e+01 1.41e-02 1.40e+06 2.8 2.40e+01 - 8.55e-02 5.51e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
170 1.1707425e+01 1.41e-02 2.07e+07 2.8 2.05e+01 - 4.58e-01 5.60e-03h 1
171 1.1892931e+01 1.39e-02 1.52e+07 2.8 5.93e+01 - 7.53e-03 9.99e-03h 1
172 1.1927791e+01 1.39e-02 1.72e+07 2.8 3.03e+01 - 5.72e-04 1.25e-03f 1
173 1.1986612e+01 1.39e-02 2.82e+07 2.8 2.66e+01 - 1.20e-01 2.19e-03h 1
174 1.1986670e+01 1.39e-02 2.58e+09 2.8 1.89e-01 14.4 1.80e-02 6.90e-05h 1
175 1.1986786e+01 1.39e-02 7.74e+11 2.8 2.48e-01 13.9 2.41e-02 9.46e-05h 1
176 1.1986900e+01 1.39e-02 8.33e+11 2.8 5.44e-01 13.5 8.29e-04 3.45e-05h 1
177 1.1987095e+01 1.39e-02 1.37e+12 2.8 7.66e-01 13.0 1.15e-02 4.29e-05h 1
178r 1.1987095e+01 1.39e-02 9.36e+02 2.8 0.00e+00 12.5 0.00e+00 3.09e-07R 6
179r 1.1987146e+01 1.08e-02 1.08e+01 -3.4 6.35e-01 - 9.88e-01 9.81e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
180 1.1995126e+01 1.08e-02 7.98e+00 -4.0 5.06e+00 - 1.39e-02 1.56e-03h 1
181 1.2430075e+01 9.89e-03 6.28e+00 -1.2 5.73e+00 - 5.58e-01 8.47e-02f 1
182 1.2303409e+01 9.42e-03 7.80e+00 -2.2 1.97e+00 - 5.74e-02 4.80e-02h 1
183 1.2559731e+01 9.05e-03 1.47e+01 -2.4 6.31e+00 - 3.59e-02 4.00e-02h 1
184 1.2852584e+01 8.76e-03 1.48e+01 -1.5 7.87e+00 - 2.10e-02 3.16e-02f 1
185 1.3082369e+01 8.59e-03 1.90e+01 -2.6 6.41e+00 - 2.75e-02 1.97e-02h 1
186 1.3083859e+01 8.52e-03 2.05e+02 0.2 4.68e+01 - 9.22e-02 5.27e-03f 1
187 1.3607270e+01 8.48e-03 9.35e+03 1.0 1.00e+02 - 9.29e-01 4.45e-03f 1
188 1.4070755e+01 8.34e-03 3.95e+05 2.2 1.60e+01 - 9.88e-01 1.67e-02h 1
189 1.4391139e+01 8.11e-03 6.47e+05 2.8 1.88e+01 - 5.65e-02 2.55e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
190 1.4823545e+01 8.03e-03 7.26e+05 2.8 2.41e+01 - 1.20e-02 9.28e-03h 1
191 1.4972496e+01 7.99e-03 1.77e+06 -3.7 4.69e+01 - 1.73e-02 5.09e-03h 1
192 1.4986885e+01 7.99e-03 1.78e+06 2.8 3.25e+02 - 5.21e-04 3.31e-04f 3
193 1.5007602e+01 7.99e-03 1.54e+06 2.2 3.57e+02 - 5.63e-03 4.53e-04f 2
194 1.5190191e+01 7.95e-03 1.24e+06 1.5 2.06e+01 - 2.54e-03 4.52e-03h 2
195 1.5245114e+01 7.94e-03 3.86e+05 1.1 7.43e+01 - 5.19e-03 1.36e-03f 1
196 1.5541191e+01 7.87e-03 3.19e+06 2.0 2.71e+01 - 1.49e-01 9.00e-03h 1
197 1.5690437e+01 7.84e-03 8.15e+07 2.8 1.76e+01 - 1.39e-01 3.60e-03h 1
198 1.5688104e+01 7.82e-03 1.69e+08 2.8 6.14e+01 - 6.16e-03 1.89e-03h 1
199 1.5860011e+01 7.80e-03 2.46e+07 2.8 3.62e+01 - 1.89e-04 2.10e-03h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
200 1.5849339e+01 7.80e-03 7.22e+08 -3.5 5.25e+02 - 4.25e-04 2.72e-05f 1
Number of Iterations....: 200
(scaled) (unscaled)
Objective...............: 1.5849339467302581e+01 1.5849339467302581e+01
Dual infeasibility......: 7.2171132001666582e+08 7.2171132001666582e+08
Constraint violation....: 7.8037068786840358e-03 7.8037068786840358e-03
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.6670615089360967e+03 2.6670615089360967e+03
Overall NLP error.......: 1.7450866344943236e+02 7.2171132001666582e+08
Number of objective function evaluations = 431
Number of objective gradient evaluations = 200
Number of equality constraint evaluations = 431
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 206
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 200
Total seconds in IPOPT = 0.863
EXIT: Maximum Number of Iterations Exceeded.
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.8.2.
Number of nonzeros in equality constraint Jacobian...: 616
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 994
Total number of variables............................: 156
variables with only lower bounds: 0
variables with lower and upper bounds: 80
variables with only upper bounds: 0
Total number of equality constraints.................: 78
Total number of inequality constraints...............: 0
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 0
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 5.8500029e+01 1.39e-01 1.00e+01 0.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 2.0443350e+01 4.42e-02 9.15e+00 -0.7 2.12e-01 2.0 1.00e+00 6.57e-01f 1
2 1.9842494e+01 4.17e-02 5.50e+01 0.1 1.20e+00 1.5 1.00e+00 6.16e-02f 1
3 1.9773633e+01 4.17e-02 1.65e+02 0.5 2.02e+00 1.0 1.00e+00 6.42e-02h 1
4 2.9596750e+01 7.08e-02 3.13e+02 1.1 3.69e+00 - 1.00e+00 2.11e-01f 1
5 3.1545313e+01 6.32e-02 2.47e+02 0.1 1.04e+00 - 3.32e-01 2.40e-01h 1
6 3.8918800e+01 5.33e-02 5.27e+02 1.3 2.11e+00 - 8.14e-01 1.54e-01f 1
7 4.6601221e+01 4.80e-02 2.61e+03 1.9 2.66e+00 - 5.95e-01 9.86e-02h 1
8 5.1055134e+01 3.91e-02 8.55e+03 2.1 1.39e+00 - 1.00e+00 1.86e-01h 1
9 5.2184172e+01 3.39e-02 1.62e+04 2.6 1.27e+00 - 2.92e-01 1.35e-01h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 5.5534199e+01 3.22e-02 4.26e+04 2.8 5.43e+00 - 3.14e-01 4.98e-02h 1
11 6.0635061e+01 2.97e-02 3.47e+04 2.8 4.30e+00 - 6.30e-02 7.94e-02h 1
12 6.2274297e+01 2.78e-02 3.32e+04 -3.5 2.27e+00 - 6.85e-02 6.38e-02h 1
13 6.3210421e+01 2.64e-02 1.33e+05 2.3 3.43e+00 - 2.65e-01 5.09e-02h 1
14 6.4180088e+01 2.59e-02 1.98e+06 2.8 7.60e+00 4.2 9.93e-01 2.21e-02h 1
15 6.8265172e+01 2.48e-02 1.97e+06 2.8 9.56e+00 - 6.22e-02 4.14e-02h 1
16 6.8022794e+01 2.37e-02 3.19e+06 2.8 7.27e+00 - 1.61e-01 4.53e-02h 1
17 7.1058047e+01 2.37e-02 3.56e+06 -2.8 2.37e+02 3.7 2.70e-04 1.30e-03h 1
18 7.2109731e+01 2.35e-02 1.69e+07 2.4 2.02e+01 - 1.00e+00 5.65e-03h 1
19 6.7935271e+01 2.28e-02 3.51e+07 2.8 1.42e+01 - 1.14e-01 3.93e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 6.8226059e+01 2.26e-02 1.98e+07 -2.4 4.04e+00 - 3.67e-03 7.24e-03h 1
21 6.8599150e+01 2.26e-02 2.06e+07 2.8 4.81e+01 - 9.14e-04 1.13e-03h 1
22 6.9481201e+01 2.25e-02 2.99e+07 2.8 2.86e+01 - 2.03e-02 3.15e-03h 1
23 7.2398034e+01 2.23e-02 9.58e+08 1.6 2.98e+01 - 1.00e+00 1.00e-02h 1
24 7.2428250e+01 2.23e-02 1.02e+09 -1.9 1.70e+01 - 3.90e-03 1.76e-04h 1
25 7.4642032e+01 2.22e-02 1.00e+09 2.8 5.19e+01 - 1.75e-04 4.31e-03h 1
26 7.5712021e+01 2.21e-02 1.00e+09 2.8 4.24e+01 - 5.86e-03 2.59e-03h 1
27 7.7133638e+01 2.20e-02 1.42e+09 2.8 4.49e+01 - 3.64e-01 3.25e-03h 1
28 7.7147649e+01 2.20e-02 1.73e+09 2.8 4.28e+01 - 1.15e-02 3.41e-05h 1
29 7.7142807e+01 2.20e-02 2.06e+09 -1.5 5.07e+01 - 1.33e-03 2.30e-04h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 7.9133630e+01 2.20e-02 2.06e+09 2.8 9.36e+01 - 1.31e-04 2.25e-03h 1
31 8.0719918e+01 2.19e-02 2.05e+09 2.8 5.96e+01 - 3.85e-03 2.72e-03h 1
32 8.1744346e+01 2.19e-02 2.14e+09 2.8 6.56e+01 - 1.49e-02 1.59e-03h 1
33 8.1754150e+01 2.19e-02 2.75e+09 -1.4 6.06e+01 - 7.53e-03 1.73e-05h 1
34 8.3944861e+01 2.19e-02 2.65e+09 2.8 1.77e+02 - 2.41e-05 1.29e-03h 1
35 8.5023052e+01 2.18e-02 2.67e+09 2.8 8.84e+01 - 2.73e-03 1.25e-03h 1
36 8.6452690e+01 2.18e-02 2.87e+09 2.8 9.62e+01 - 9.72e-03 1.52e-03h 1
37 8.6466403e+01 2.18e-02 4.01e+09 2.8 9.57e+01 - 8.74e-03 1.53e-05h 1
38 8.6324158e+01 2.18e-02 1.59e+10 2.8 1.92e+01 - 1.01e-02 3.85e-04h 1
39 8.6970481e+01 2.18e-02 1.55e+10 -0.7 4.67e+02 - 1.16e-05 1.46e-04h 3
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 8.9942286e+01 2.18e-02 1.66e+10 2.8 4.11e+02 - 1.78e-04 7.07e-04f 1
41 9.1189873e+01 2.18e-02 1.64e+10 2.8 1.84e+02 - 3.99e-03 6.76e-04h 1
42 9.1181341e+01 2.18e-02 1.64e+10 2.1 2.09e+01 - 5.96e-05 3.77e-05h 1
43r 9.1181341e+01 2.18e-02 9.82e+02 2.1 0.00e+00 - 0.00e+00 2.96e-07R 6
44r 8.8691105e+01 3.08e-02 1.10e+01 1.2 3.14e-01 - 9.90e-01 1.00e+00f 1
45r 8.8778217e+01 5.34e-03 1.90e+01 -0.4 9.75e-02 - 9.96e-01 8.81e-01f 1
46 8.2272523e+01 5.13e-03 9.51e+00 -4.0 1.22e+00 - 9.17e-02 5.96e-02f 1
47 7.9910252e+01 5.10e-03 5.57e+01 0.1 1.73e+01 - 4.61e-02 3.23e-03f 1
48 8.6734619e+01 5.01e-03 3.96e+03 0.7 4.70e+00 - 7.62e-01 1.54e-02f 1
49 8.2927453e+01 4.89e-03 3.81e+04 1.0 2.97e+00 - 1.00e+00 4.51e-02h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 8.7082281e+01 4.79e-03 3.57e+04 -4.0 4.02e+00 - 2.13e-02 2.12e-02h 1
51 8.8503085e+01 4.78e-03 2.05e+06 2.2 8.13e+01 - 1.00e+00 1.72e-03f 2
52 8.8665375e+01 4.78e-03 2.65e+06 2.2 4.77e+01 - 8.34e-03 2.61e-04h 1
53 9.0101058e+01 4.95e-03 2.81e+06 -3.7 3.19e+01 - 4.84e-03 3.75e-03h 1
54 9.0817854e+01 4.95e-03 2.58e+06 2.8 6.37e+01 - 5.28e-05 1.01e-03f 1
55 9.2890326e+01 4.93e-03 9.48e+06 2.8 1.03e+02 - 6.08e-02 2.03e-03f 2
56 9.3608086e+01 5.00e-03 9.34e+06 2.5 9.57e+01 - 2.25e-04 5.25e-04h 1
57 9.4078557e+01 5.00e-03 9.30e+06 1.8 1.97e+02 - 1.38e-04 2.62e-04f 2
58 9.4680740e+01 4.92e-03 9.25e+06 1.1 2.85e+02 - 1.53e-05 3.50e-04h 1
59 9.4646762e+01 4.90e-03 9.26e+06 1.1 2.42e+03 - 1.51e-05 1.88e-05f 2
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 9.4660932e+01 4.87e-03 8.74e+06 1.1 7.54e+02 - 3.12e-03 6.75e-05h 2
61 9.5432836e+01 4.92e-03 8.47e+06 1.1 5.83e+01 - 1.03e-04 5.59e-04h 3
62 9.6435832e+01 4.94e-03 8.23e+06 1.1 7.89e+01 4.1 5.29e-04 8.35e-04h 1
63 9.7170260e+01 4.94e-03 2.78e+07 1.1 9.15e+01 3.7 2.56e-02 6.06e-04h 1
64 9.7537356e+01 4.93e-03 2.89e+07 1.1 1.87e+02 - 7.85e-04 2.28e-04h 4
65 9.7942426e+01 4.92e-03 4.37e+07 1.1 1.42e+02 - 6.30e-03 2.62e-04h 4
66 9.8491367e+01 4.95e-03 6.29e+07 1.1 8.31e+01 - 4.53e-03 3.22e-04h 3
67 9.8847814e+01 4.96e-03 9.58e+07 1.1 9.10e+01 - 4.52e-03 2.21e-04h 3
68 9.9390994e+01 5.00e-03 1.29e+08 1.1 8.40e+01 - 3.00e-03 3.61e-04h 2
69 9.9918649e+01 4.92e-03 1.21e+08 1.1 1.71e+02 - 2.46e-05 4.10e-04h 2
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 1.0030747e+02 4.89e-03 1.25e+08 1.1 2.70e+02 5.0 4.72e-04 2.14e-04h 1
71 1.0038198e+02 4.83e-03 1.25e+08 1.1 2.06e+03 - 4.80e-05 3.84e-05f 2
72 1.0040844e+02 4.83e-03 1.23e+08 1.1 1.39e+03 - 1.43e-04 1.91e-05f 4
73 1.0043727e+02 4.82e-03 1.05e+08 1.1 1.20e+03 - 2.03e-03 2.60e-05f 4
74 1.0113398e+02 4.84e-03 9.56e+07 1.1 2.16e+02 5.4 1.24e-03 5.47e-04h 1
75 1.0136515e+02 4.85e-03 1.16e+08 1.1 2.80e+02 - 3.16e-03 1.25e-04h 2
76 1.0144776e+02 4.85e-03 1.21e+08 1.1 4.32e+02 - 7.80e-04 8.25e-05h 2
77 1.0153054e+02 4.84e-03 1.28e+08 1.1 4.40e+02 - 8.82e-04 8.19e-05h 1
78 1.0157412e+02 4.84e-03 1.45e+08 1.1 2.31e+02 - 8.06e-04 1.71e-05h 1
79 1.0188747e+02 4.84e-03 1.83e+09 1.1 2.72e+02 - 6.87e-02 1.13e-04h 4
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 1.0408582e+02 4.84e-03 1.83e+09 1.1 8.32e+02 - 2.59e-04 2.59e-04s 11
81r 1.0408582e+02 4.84e-03 9.99e+02 1.1 0.00e+00 - 0.00e+00 0.00e+00R 1
82r 1.0361012e+02 4.83e-03 3.88e+01 -0.2 5.08e-02 - 9.88e-01 8.69e-01f 1
83r 1.0300655e+02 4.81e-03 1.57e+02 -1.1 5.26e-02 - 9.85e-01 7.32e-01f 1
84r 9.9293183e+01 6.01e-03 2.32e+01 -1.2 4.91e-01 - 8.75e-01 1.00e+00f 1
85r 9.9236253e+01 9.17e-03 6.71e+00 -2.1 1.74e-02 2.0 1.00e+00 8.56e-01f 1
86r 1.0015662e+02 1.13e-02 1.21e+01 -2.8 1.27e-01 1.5 2.35e-01 1.39e-01f 1
87r 1.0025888e+02 1.31e-02 4.38e+00 -2.6 1.79e-02 1.9 1.00e+00 7.21e-01f 1
88r 1.0058319e+02 1.67e-02 1.03e+01 -2.8 2.98e-02 1.5 1.00e+00 5.07e-01f 1
89r 1.0100867e+02 1.98e-02 2.85e+00 -3.6 9.53e-03 1.9 1.00e+00 7.88e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
90r 1.0129705e+02 2.23e-02 1.40e+01 -4.0 2.28e-02 1.4 1.00e+00 2.67e-01f 1
91r 1.0174425e+02 2.46e-02 2.00e+00 -4.0 1.00e-02 1.8 1.00e+00 8.04e-01f 1
92r 1.0225424e+02 2.76e-02 1.18e+01 -4.0 2.36e-02 1.4 1.00e+00 4.21e-01f 1
93r 1.0242299e+02 2.85e-02 4.72e+00 -4.0 9.08e-03 1.8 1.00e+00 3.63e-01f 1
94r 1.0274896e+02 3.06e-02 1.37e+01 -4.0 2.30e-02 1.3 1.00e+00 2.97e-01f 1
95r 1.0308055e+02 3.25e-02 1.77e+00 -4.0 9.35e-03 1.7 1.00e+00 7.72e-01f 1
96r 1.0355001e+02 3.24e-02 1.07e+01 -4.0 2.05e-02 1.3 1.00e+00 3.91e-01f 1
97r 1.0525910e+02 3.26e-02 1.91e+00 -4.0 4.16e-02 0.8 7.58e-01 8.23e-01f 1
98r 1.0563602e+02 3.24e-02 4.47e+00 -4.0 1.55e-02 1.2 1.00e+00 5.55e-01h 1
99r 1.0591704e+02 3.30e-02 1.53e+01 -3.2 2.95e-02 0.7 1.00e+00 2.64e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
100r 1.0690631e+02 3.29e-02 6.32e+00 -3.5 8.01e-02 0.3 5.98e-01 5.82e-01f 1
101r 1.0719337e+02 3.21e-02 5.27e+00 -3.2 2.97e-02 0.7 1.00e+00 5.56e-01f 1
102r 1.0780428e+02 3.13e-02 3.84e-01 -3.4 8.20e-02 0.2 1.00e+00 1.00e+00f 1
103r 1.0792934e+02 3.09e-02 3.67e+00 -3.6 1.85e-01 -0.3 1.00e+00 7.36e-01f 1
104r 1.0784030e+02 3.28e-02 3.61e+00 -4.0 3.46e-01 -0.7 6.37e-01 4.56e-01f 1
105r 1.0795634e+02 3.34e-02 8.78e-02 -4.0 2.28e-02 0.6 1.00e+00 1.00e+00h 1
106r 1.0793364e+02 3.45e-02 4.29e-01 -4.0 6.18e-02 0.1 1.00e+00 8.70e-01h 1
107r 1.0785021e+02 3.74e-02 4.13e-02 -4.0 1.28e-02 -0.4 1.00e+00 1.00e+00H 1
108r 1.0767873e+02 4.35e-02 7.60e-02 -4.0 1.88e-02 -0.8 1.00e+00 1.00e+00f 1
109r 1.0747085e+02 5.42e-02 7.94e-02 -4.1 3.03e-02 -1.3 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
110r 1.0730394e+02 5.30e-02 1.35e+01 -3.8 1.25e-01 - 7.87e-01 3.74e-01h 1
111r 1.0727991e+02 5.29e-02 6.76e-02 -4.1 8.34e-03 0.9 1.00e+00 1.00e+00h 1
112r 1.0712104e+02 6.20e-02 3.10e-01 -4.0 5.35e-02 - 1.00e+00 1.00e+00h 1
113r 1.0703956e+02 6.58e-02 9.67e-02 -4.1 1.95e-02 - 1.00e+00 1.00e+00h 1
114r 1.0732583e+02 6.08e-02 8.54e-02 -4.0 2.63e-02 - 1.00e+00 1.00e+00h 1
115r 1.0712209e+02 6.43e-02 6.75e-02 -4.1 2.11e-02 - 1.00e+00 1.00e+00h 1
116r 1.0728004e+02 6.09e-02 6.48e-02 -4.0 1.99e-02 - 1.00e+00 1.00e+00h 1
117r 1.0711244e+02 6.45e-02 6.56e-02 -4.1 2.02e-02 - 1.00e+00 1.00e+00h 1
118r 1.0728413e+02 6.08e-02 6.63e-02 -4.0 2.04e-02 - 1.00e+00 1.00e+00h 1
119r 1.0711341e+02 6.45e-02 6.61e-02 -4.1 2.03e-02 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
120r 1.0716434e+02 6.56e-02 4.14e-03 -4.1 3.93e-03 - 1.00e+00 1.00e+00h 1
121r 1.0728140e+02 6.07e-02 3.87e-02 -4.0 1.52e-02 - 1.00e+00 1.00e+00h 1
122r 1.0711278e+02 6.44e-02 6.14e-02 -4.1 1.91e-02 - 1.00e+00 1.00e+00h 1
123r 1.0728379e+02 6.08e-02 6.59e-02 -4.0 2.03e-02 - 1.00e+00 1.00e+00h 1
124r 1.0711332e+02 6.45e-02 6.61e-02 -4.1 2.03e-02 - 1.00e+00 1.00e+00h 1
125r 1.0716435e+02 6.56e-02 4.14e-03 -4.1 3.93e-03 - 1.00e+00 1.00e+00h 1
126r 1.0728140e+02 6.07e-02 3.87e-02 -4.0 1.52e-02 - 1.00e+00 1.00e+00h 1
127r 1.0711278e+02 6.44e-02 6.14e-02 -4.1 1.91e-02 - 1.00e+00 1.00e+00h 1
128r 1.0716434e+02 6.56e-02 4.16e-03 -4.1 3.93e-03 - 1.00e+00 1.00e+00h 1
129r 1.0728138e+02 6.07e-02 3.87e-02 -4.0 1.52e-02 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
130r 1.0711278e+02 6.44e-02 6.14e-02 -4.1 1.91e-02 - 1.00e+00 1.00e+00h 1
131r 1.0716434e+02 6.56e-02 4.16e-03 -4.1 3.93e-03 - 1.00e+00 1.00e+00h 1
132r 1.0716400e+02 6.56e-02 1.77e-06 -4.1 1.13e-04 - 1.00e+00 1.00e+00h 1
133r 1.0728160e+02 6.07e-02 3.87e-02 -4.0 1.53e-02 - 1.00e+00 1.00e+00f 1
134r 1.0711284e+02 6.44e-02 6.13e-02 -4.1 1.91e-02 - 1.00e+00 1.00e+00h 1
135r 1.0716434e+02 6.56e-02 4.16e-03 -4.1 3.93e-03 - 1.00e+00 1.00e+00h 1
136r 1.0716400e+02 6.56e-02 1.77e-06 -4.1 1.12e-04 - 1.00e+00 1.00e+00h 1
137r 1.0670385e+02 8.49e-02 5.58e+00 -6.1 6.10e-02 - 6.00e-01 9.82e-01f 1
138r 1.0754875e+02 7.37e-02 6.90e+01 -4.0 5.19e-01 - 1.00e+00 2.86e-01f 1
139r 1.0710617e+02 7.21e-02 4.27e+01 -4.3 7.11e-02 - 4.91e-01 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
140r 1.0792703e+02 6.17e-02 1.76e+00 -4.0 6.56e-02 - 1.00e+00 1.00e+00h 1
141r 1.0704510e+02 6.37e-02 6.38e-01 -4.1 4.75e-02 - 1.00e+00 1.00e+00h 1
142r 1.0717169e+02 6.53e-02 1.52e-02 -4.1 6.14e-03 - 1.00e+00 1.00e+00h 1
143r 1.0716630e+02 6.53e-02 1.72e-04 -4.1 1.22e-03 - 1.00e+00 1.00e+00h 1
144r 1.0670618e+02 8.47e-02 5.61e+00 -6.1 6.08e-02 - 6.00e-01 9.82e-01f 1
145r 1.0656640e+02 1.30e-01 2.39e+00 -6.1 5.29e-02 - 6.27e-01 8.56e-01h 1
146r 1.0765416e+02 9.60e-02 1.38e+02 -4.0 8.98e-01 - 1.00e+00 1.77e-01f 1
147r 1.0729965e+02 7.85e-02 7.85e+01 -4.4 6.74e-02 - 4.95e-01 1.00e+00h 1
148r 1.0729819e+02 7.98e-02 1.19e+01 -4.4 1.79e-01 - 1.00e+00 3.71e-01h 1
149r 1.0704678e+02 7.99e-02 3.42e-02 -4.4 1.30e-02 - 1.00e+00 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
150r 1.0769522e+02 5.71e-02 1.08e+00 -4.0 9.40e-02 - 1.00e+00 1.00e+00f 1
151r 1.0711575e+02 6.39e-02 5.63e-01 -4.1 6.14e-02 - 1.00e+00 1.00e+00h 1
152r 1.0717122e+02 6.54e-02 1.29e-02 -4.1 1.44e-02 - 1.00e+00 1.00e+00h 1
153r 1.0716679e+02 6.52e-02 4.10e-04 -4.1 3.54e-03 - 1.00e+00 1.00e+00h 1
154r 1.0670667e+02 8.46e-02 5.61e+00 -6.1 6.07e-02 - 6.00e-01 9.82e-01f 1
155r 1.0656639e+02 1.30e-01 2.39e+00 -6.1 5.27e-02 - 6.28e-01 8.60e-01h 1
156r 1.0647195e+02 1.35e-01 5.87e-02 -6.1 1.09e-02 - 9.85e-01 1.00e+00h 1
157r 1.0797022e+02 1.26e-01 6.22e+02 -4.0 1.79e+00 - 1.00e+00 5.03e-02f 1
158r 1.0803218e+02 1.22e-01 1.78e+01 -4.4 5.11e-02 - 9.72e-01 1.00e+00h 1
159r 1.0711283e+02 9.53e-02 7.69e-01 -4.4 7.38e-02 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
160r 1.0698279e+02 6.65e-02 2.37e-01 -4.4 3.82e-02 - 1.00e+00 1.00e+00h 1
161r 1.0704027e+02 7.91e-02 7.59e-03 -4.4 1.26e-02 - 1.00e+00 1.00e+00h 1
162r 1.0775314e+02 6.18e-02 1.23e+00 -4.0 9.44e-02 - 1.00e+00 1.00e+00f 1
163r 1.0710994e+02 6.48e-02 6.21e-01 -4.1 6.22e-02 - 1.00e+00 1.00e+00h 1
164r 1.0717370e+02 6.54e-02 1.53e-02 -4.1 1.58e-02 - 1.00e+00 1.00e+00h 1
165r 1.0716699e+02 6.52e-02 5.85e-04 -4.1 4.14e-03 - 1.00e+00 1.00e+00h 1
166r 1.0670688e+02 8.46e-02 5.60e+00 -6.1 6.07e-02 - 6.00e-01 9.82e-01f 1
167r 1.0656633e+02 1.30e-01 2.39e+00 -6.1 5.26e-02 - 6.29e-01 8.63e-01h 1
168r 1.0647200e+02 1.35e-01 5.88e-02 -6.1 1.09e-02 - 9.85e-01 1.00e+00h 1
169r 1.0618201e+02 1.35e-01 2.39e-01 -6.1 2.91e-02 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
170r 1.0794215e+02 1.29e-01 1.31e+02 -4.0 2.11e+00 - 1.56e-01 3.61e-02f 1
171r 1.0782851e+02 1.28e-01 2.81e-01 -5.1 4.53e-03 0.4 1.00e+00 9.14e-01h 1
172r 1.0719131e+02 1.27e-01 3.39e-01 -5.1 9.33e-02 - 2.57e-01 2.47e-01f 1
173r 1.0718771e+02 1.26e-01 4.04e-03 -5.1 4.37e-03 -0.0 1.00e+00 1.00e+00f 1
174r 1.0687564e+02 1.27e-01 1.49e+00 -5.1 5.43e-02 - 4.94e-01 4.36e-01H 1
175r 1.0687908e+02 1.26e-01 2.06e-03 -5.1 2.42e-03 -0.5 1.00e+00 1.00e+00f 1
176r 1.0798560e+02 1.00e-01 1.13e+02 -4.0 6.53e-01 - 1.00e+00 2.42e-01f 1
177r 1.0754170e+02 8.77e-02 5.42e+01 -4.3 7.63e-02 - 5.87e-01 1.00e+00h 1
178r 1.0742384e+02 7.94e-02 1.78e-01 -4.3 3.22e-02 - 1.00e+00 1.00e+00h 1
179r 1.0708144e+02 7.47e-02 2.55e-02 -4.3 5.49e-03 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
180r 1.0709052e+02 7.46e-02 2.50e-05 -4.3 8.55e-04 - 1.00e+00 1.00e+00h 1
181r 1.0750792e+02 5.97e-02 4.62e-01 -4.0 5.59e-02 - 1.00e+00 1.00e+00f 1
182r 1.0712220e+02 6.43e-02 2.76e-01 -4.1 4.18e-02 - 1.00e+00 1.00e+00h 1
183r 1.0716699e+02 6.54e-02 3.75e-03 -4.1 3.99e-03 - 1.00e+00 1.00e+00h 1
184r 1.0670687e+02 8.48e-02 5.57e+00 -6.1 6.07e-02 - 6.01e-01 9.82e-01f 1
185r 1.0657643e+02 1.29e-01 2.38e+00 -6.1 5.70e-02 - 6.09e-01 7.74e-01h 1
186r 1.0648478e+02 1.35e-01 1.17e-01 -6.1 1.09e-02 - 9.88e-01 9.42e-01h 1
187r 1.0618278e+02 1.35e-01 2.50e-01 -6.1 3.00e-02 - 1.00e+00 1.00e+00h 1
188r 1.0670279e+02 1.33e-01 4.69e+00 -4.0 2.13e+00 - 1.52e-02 1.06e-02f 1
189r 1.0655077e+02 1.31e-01 1.03e-01 -5.8 1.36e-02 -1.0 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
190r 1.0624107e+02 1.34e-01 6.72e-02 -5.8 1.65e-02 - 1.00e+00 1.00e+00h 1
191r 1.0626480e+02 1.33e-01 3.20e-02 -5.8 8.48e-03 - 1.00e+00 1.00e+00h 1
192r 1.0626493e+02 1.34e-01 2.39e-04 -5.8 9.24e-04 - 1.00e+00 1.00e+00h 1
193r 1.0790360e+02 1.24e-01 1.51e+02 -4.0 1.41e+00 - 2.92e-01 6.14e-02f 1
194r 1.0781451e+02 1.23e-01 9.02e+00 -4.8 2.72e-02 0.3 1.00e+00 1.44e-01h 1
195r 1.0567398e+02 1.18e-01 5.85e+00 -4.8 8.97e-02 - 8.03e-01 1.00e+00f 1
196r 1.0602229e+02 1.12e-01 5.54e-01 -4.8 6.71e-02 - 1.00e+00 1.00e+00h 1
197r 1.0654967e+02 1.12e-01 1.47e+01 -4.8 1.34e-01 - 1.00e+00 6.40e-01h 1
198r 1.0678856e+02 1.09e-01 2.34e+00 -4.8 7.30e-02 - 8.19e-01 7.81e-01f 1
199r 1.0674529e+02 1.10e-01 2.21e+00 -4.8 1.67e-01 - 4.24e-01 1.52e-01f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
200r 1.0673974e+02 1.10e-01 3.38e-01 -4.8 5.94e-02 0.8 1.00e+00 1.00e+00f 1
Number of Iterations....: 200
(scaled) (unscaled)
Objective...............: 1.0673973851813122e+02 1.0673973851813122e+02
Dual infeasibility......: 1.0005622575443891e+01 1.0005622575443891e+01
Constraint violation....: 1.0976829160236151e-01 1.0976829160236151e-01
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.0190448111747951e-02 2.0190448111747951e-02
Overall NLP error.......: 6.1119787118723234e+00 1.0005622575443891e+01
Number of objective function evaluations = 271
Number of objective gradient evaluations = 82
Number of equality constraint evaluations = 271
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 203
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 200
Total seconds in IPOPT = 0.683
EXIT: Maximum Number of Iterations Exceeded.
Time weight effects:
Weight | Total Time | ||u|| | Max |u|
---------------------------------------------
0.1 | 18.2480 s | 5.9416 | 0.9997
1.0 | 16.0657 s | 6.1587 | 1.0000
10.0 | 10.9262 s | 6.0077 | 1.0000
Observations:
- Lower weight → slower trajectory, gentler controls
- Higher weight → faster trajectory, more aggressive controlsStep 9: Time Step Adaptation
Δt_variation = std(Δt_sol_mintime)
println("\nTime step adaptation:")
println(" Std dev(Δt): ", Δt_variation)
println(
" Coefficient of variation: ",
@sprintf("%.3f", Δt_variation / mean(Δt_sol_mintime))
)
Time step adaptation:
Std dev(Δt): 0.18860465171849528
Coefficient of variation: 0.470Key Insights
- Free time variables: Setting
timestep=:Δtmakes time steps optimizable - Time bounds are crucial: Lower bound prevents Δt -> 0, upper bound prevents unrealistically large steps
- Time weight balances speed vs control: High weight -> fast but aggressive, low weight -> slow but gentle
- Control saturation: Time-optimal solutions often saturate control bounds (bang-bang behavior)
- Non-uniform time steps: Optimizer may choose variable Δt — larger steps where less control is needed
- Initial guess: Start with reasonable Δt to help convergence
Best Practices
Time Step Bounds
- Lower bound: ~0.01 to 0.05 (prevent numerical issues)
- Upper bound: 1/10 to 1/5 of expected total time
- Start conservative, relax if needed
Control Weights
- Usually small (1e-3 to 1e-2) for regularization
- Just enough to ensure well-conditioned problem
- Too large defeats the purpose of time minimization
Time Weights
- Start with ~1.0 and adjust
- Increase to prioritize speed more
- Decrease if controls become too aggressive
Number of Time Steps
- Fewer steps = less resolution, harder to satisfy dynamics
- More steps = more variables, slower solve
- Rule of thumb: 30-100 steps for most problems
Initialization
- Use solution from fixed-time problem as warm start
- Or solve with high control weight first, then reduce
Exercises
Exercise 1: Bang-Bang Control
Increase time weight to w_time=100.0. Do controls saturate more?
Exercise 2: Time Step Constraints
Try tighter bounds: Δt ∈ [0.05, 0.15]. How does total time change?
Exercise 3: Longer Distance
Change goal to x_goal = [2.0, 0.0]. How does optimal time scale?
Exercise 4: Multiple Objectives
Add terminal cost with soft goal:
obj = w_control * QuadraticRegularizer(:u, traj, 1.0) +
w_time * MinimumTimeObjective(traj, 1.0) +
100.0 * TerminalObjective(x -> norm(x - x_goal)^2, :x, traj)Exercise 5: Warm Starting
Solve fixed-time problem first, use as initial guess for free-time:
traj_warm = NamedTrajectory(
(x = prob_fixed.trajectory.x,
u = prob_fixed.trajectory.u,
Δt = Δt_guess);
# ... rest of setup
)Next Steps
- Smooth Controls Tutorial: Add derivative penalties while minimizing time
- How-To Guide: Tune the Solver: Improve convergence for difficult problems
- Advanced Topics: Performance: Optimize large-scale problems
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