Note
Go to the end to download the full example code.
A simple optimization problem: Rosenbrock function#
This example illustrates the basic usage of csnlp.Nlp in solving a simple
optimization problem, taken from
this CasADi blog post
Consider the parametric (in \(r\)) constrained Rosenbrock function
\[\min_{x}{ (1 - x_1)^2 + (x_2 - x_1^2)^2 } \text{ s.t. } x_1^2 + x_2^2 \leq r.\]
Creating and solving the problem#
The imports are pretty standard, as if we were using casadi alone. We just need to
additionally import the csnlp.Nlp class.
import casadi as cs
import matplotlib.pyplot as plt
import numpy as np
from csnlp import Nlp
In order to build the optimization problem, we first create an instance of
csnlp.Nlp (instructed to use casadi.MX under the hood) and
primal variable \(x\) and parameter \(r\).
nlp = Nlp[cs.MX](sym_type="MX")
x = nlp.variable("x", (2, 1))[0]
r = nlp.parameter("r")
Then, formulate the objective and the constraint, and initialize the solver (this is mandatory before any solving run). Under the hood, by default, the solver is set to IPOPT.
Note that for the constraint we save the corresponding Lagrange multiplier \(\lambda\) in the variable lam to be used later.
def rosenbrock(x):
return (1 - x[0]) ** 2 + (x[1] - x[0] ** 2) ** 2
nlp.minimize(rosenbrock(x))
_, lam = nlp.constraint("con1", cs.sumsqr(x), "<=", r)
nlp.init_solver()
For a given value of \(r\), e.g., \(r=1\), we can solve the problem as follows
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 7.0171713e-02 0.00e+00 1.63e-01 -1.7 5.11e-01 - 1.00e+00 1.00e+00h 1
3 4.0186889e-02 6.78e-03 5.35e-03 -1.7 4.77e-01 - 1.00e+00 1.00e+00h 1
4 4.1076918e-02 0.00e+00 2.46e-06 -3.8 1.05e-02 - 1.00e+00 1.00e+00h 1
5 4.0921291e-02 0.00e+00 7.37e-07 -5.7 1.45e-03 - 1.00e+00 1.00e+00h 1
6 4.0919039e-02 0.00e+00 1.46e-10 -8.6 2.13e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 6
(scaled) (unscaled)
Objective...............: 4.0919038671969346e-02 4.0919038671969346e-02
Dual infeasibility......: 1.4614634502585488e-10 1.4614634502585488e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5995835817288543e-09 2.5995835817288543e-09
Overall NLP error.......: 2.5995835817288543e-09 2.5995835817288543e-09
Number of objective function evaluations = 7
Number of objective gradient evaluations = 7
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 7
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 7
Number of Lagrangian Hessian evaluations = 6
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 15.00us ( 2.14us) 15.14us ( 2.16us) 7
nlp_g | 23.00us ( 3.29us) 19.92us ( 2.85us) 7
nlp_grad_f | 24.00us ( 3.00us) 21.86us ( 2.73us) 8
nlp_hess_l | 22.00us ( 3.67us) 21.00us ( 3.50us) 6
nlp_jac_g | 16.00us ( 2.00us) 15.68us ( 1.96us) 8
total | 3.51ms ( 3.51ms) 3.51ms ( 3.51ms) 1
And we can also visualize the solution
_, ax = plt.subplots(constrained_layout=True)
X, Y = np.meshgrid(np.linspace(0, 1.5, 100), np.linspace(-0.5, 1.5, 100))
F = rosenbrock((X, Y))
contour = ax.contour(X, Y, F, levels=100, cmap="viridis")
theta = np.linspace(0, 2 * np.pi, 100)
x_circle = np.cos(theta)
y_circle = np.sin(theta)
ax.plot(x_circle, y_circle, "k-")
x_opt = sol.value(x)
ax.plot(x_opt[0], x_opt[1], "r*", markersize=20)
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_aspect("equal")
ax.set_xlim(0, 1.5)
ax.set_ylim(-0.5, 1.5)
plt.show()
Computing multipliers and sensitivities#
What’s cooler, we can set the value of \(r\) at runtime to any valid value, run the solver, and compute the sensitivity of the objective with respect to \(r\) as the value of the mutiplier \(\lambda\) at the optimal point.
r_values = np.linspace(1, 3, 25)
f_values = []
lam_values = []
for r_value in r_values:
sol = nlp.solve(pars={"r": r_value})
f_values.append(sol.f)
lam_values.append(sol.value(lam))
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 7.0171713e-02 0.00e+00 1.63e-01 -1.7 5.11e-01 - 1.00e+00 1.00e+00h 1
3 4.0186889e-02 6.78e-03 5.35e-03 -1.7 4.77e-01 - 1.00e+00 1.00e+00h 1
4 4.1076918e-02 0.00e+00 2.46e-06 -3.8 1.05e-02 - 1.00e+00 1.00e+00h 1
5 4.0921291e-02 0.00e+00 7.37e-07 -5.7 1.45e-03 - 1.00e+00 1.00e+00h 1
6 4.0919039e-02 0.00e+00 1.46e-10 -8.6 2.13e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 6
(scaled) (unscaled)
Objective...............: 4.0919038671969346e-02 4.0919038671969346e-02
Dual infeasibility......: 1.4614634502585488e-10 1.4614634502585488e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5995835817288543e-09 2.5995835817288543e-09
Overall NLP error.......: 2.5995835817288543e-09 2.5995835817288543e-09
Number of objective function evaluations = 7
Number of objective gradient evaluations = 7
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 7
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 7
Number of Lagrangian Hessian evaluations = 6
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 18.00us ( 2.57us) 16.85us ( 2.41us) 7
nlp_g | 26.00us ( 3.71us) 21.29us ( 3.04us) 7
nlp_grad_f | 29.00us ( 3.62us) 26.88us ( 3.36us) 8
nlp_hess_l | 29.00us ( 4.83us) 28.69us ( 4.78us) 6
nlp_jac_g | 22.00us ( 2.75us) 20.37us ( 2.55us) 8
total | 3.93ms ( 3.93ms) 3.92ms ( 3.92ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 6.7526488e-02 0.00e+00 1.71e-01 -1.7 5.17e-01 - 1.00e+00 1.00e+00h 1
3 3.3194215e-02 0.00e+00 2.42e-03 -1.7 5.33e-01 - 1.00e+00 1.00e+00h 1
4 3.2811235e-02 0.00e+00 1.08e-05 -3.8 3.13e-02 - 1.00e+00 1.00e+00h 1
5 3.2593999e-02 0.00e+00 1.68e-06 -5.7 2.37e-03 - 1.00e+00 1.00e+00h 1
6 3.2591222e-02 0.00e+00 2.54e-10 -8.6 3.20e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 6
(scaled) (unscaled)
Objective...............: 3.2591222205591665e-02 3.2591222205591665e-02
Dual infeasibility......: 2.5355909039070923e-10 2.5355909039070923e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.6695320873635210e-09 2.6695320873635210e-09
Overall NLP error.......: 2.6695320873635210e-09 2.6695320873635210e-09
Number of objective function evaluations = 7
Number of objective gradient evaluations = 7
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 7
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 7
Number of Lagrangian Hessian evaluations = 6
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 16.00us ( 2.29us) 14.78us ( 2.11us) 7
nlp_g | 23.00us ( 3.29us) 18.31us ( 2.62us) 7
nlp_grad_f | 18.00us ( 2.25us) 18.78us ( 2.35us) 8
nlp_hess_l | 29.00us ( 4.83us) 28.94us ( 4.82us) 6
nlp_jac_g | 17.00us ( 2.12us) 15.51us ( 1.94us) 8
total | 3.32ms ( 3.32ms) 3.32ms ( 3.32ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 6.5303478e-02 0.00e+00 1.77e-01 -1.7 5.22e-01 - 1.00e+00 1.00e+00h 1
3 2.7779959e-02 0.00e+00 1.58e-03 -1.7 5.82e-01 - 1.00e+00 1.00e+00h 1
4 2.8306757e-02 0.00e+00 1.21e-05 -2.5 2.91e-02 - 1.00e+00 1.00e+00h 1
5 2.5901594e-02 0.00e+00 2.44e-04 -3.8 3.00e-02 - 1.00e+00 1.00e+00h 1
6 2.5616635e-02 0.00e+00 3.13e-06 -5.7 3.95e-03 - 1.00e+00 1.00e+00h 1
7 2.5613061e-02 0.00e+00 4.92e-10 -8.6 5.04e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 7
(scaled) (unscaled)
Objective...............: 2.5613061133718768e-02 2.5613061133718768e-02
Dual infeasibility......: 4.9234530341557559e-10 4.9234530341557559e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.8262707475320204e-09 2.8262707475320204e-09
Overall NLP error.......: 2.8262707475320204e-09 2.8262707475320204e-09
Number of objective function evaluations = 8
Number of objective gradient evaluations = 8
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 8
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 8
Number of Lagrangian Hessian evaluations = 7
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 19.00us ( 2.38us) 18.43us ( 2.30us) 8
nlp_g | 30.00us ( 3.75us) 23.42us ( 2.93us) 8
nlp_grad_f | 24.00us ( 2.67us) 21.73us ( 2.41us) 9
nlp_hess_l | 26.00us ( 3.71us) 23.79us ( 3.40us) 7
nlp_jac_g | 18.00us ( 2.00us) 17.48us ( 1.94us) 9
total | 3.82ms ( 3.82ms) 3.82ms ( 3.82ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 6.3410205e-02 0.00e+00 1.82e-01 -1.7 5.26e-01 - 1.00e+00 1.00e+00h 1
3 2.3549513e-02 0.00e+00 4.39e-03 -1.7 6.25e-01 - 1.00e+00 1.00e+00h 1
4 2.2524747e-02 0.00e+00 2.60e-05 -2.5 5.81e-02 - 1.00e+00 1.00e+00h 1
5 2.0107070e-02 0.00e+00 3.00e-04 -3.8 3.58e-02 - 1.00e+00 1.00e+00h 1
6 1.9788468e-02 0.00e+00 4.83e-06 -5.7 5.29e-03 - 1.00e+00 1.00e+00h 1
7 1.9783909e-02 0.00e+00 9.79e-10 -8.6 7.76e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 7
(scaled) (unscaled)
Objective...............: 1.9783909425558625e-02 1.9783909425558625e-02
Dual infeasibility......: 9.7912045138315307e-10 9.7912045138315307e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 3.1416546642908773e-09 3.1416546642908773e-09
Overall NLP error.......: 3.1416546642908773e-09 3.1416546642908773e-09
Number of objective function evaluations = 8
Number of objective gradient evaluations = 8
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 8
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 8
Number of Lagrangian Hessian evaluations = 7
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 17.00us ( 2.12us) 16.76us ( 2.10us) 8
nlp_g | 24.00us ( 3.00us) 21.97us ( 2.75us) 8
nlp_grad_f | 24.00us ( 2.67us) 20.63us ( 2.29us) 9
nlp_hess_l | 22.00us ( 3.14us) 23.03us ( 3.29us) 7
nlp_jac_g | 18.00us ( 2.00us) 17.29us ( 1.92us) 9
total | 3.70ms ( 3.70ms) 3.71ms ( 3.71ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 6.1779089e-02 0.00e+00 1.86e-01 -1.7 5.30e-01 - 1.00e+00 1.00e+00h 1
3 2.0210403e-02 0.00e+00 7.71e-03 -1.7 6.62e-01 - 1.00e+00 1.00e+00h 1
4 1.7814298e-02 0.00e+00 2.09e-04 -2.5 8.97e-02 - 1.00e+00 1.00e+00h 1
5 1.5322705e-02 0.00e+00 3.91e-04 -3.8 4.41e-02 - 1.00e+00 1.00e+00h 1
6 1.4950015e-02 0.00e+00 8.36e-06 -5.7 7.45e-03 - 1.00e+00 1.00e+00h 1
7 1.4943401e-02 0.00e+00 2.59e-09 -8.6 1.37e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 7
(scaled) (unscaled)
Objective...............: 1.4943401174661818e-02 1.4943401174661818e-02
Dual infeasibility......: 2.5936949393789988e-09 2.5936949393789988e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 4.1817302141429776e-09 4.1817302141429776e-09
Overall NLP error.......: 4.1817302141429776e-09 4.1817302141429776e-09
Number of objective function evaluations = 8
Number of objective gradient evaluations = 8
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 8
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 8
Number of Lagrangian Hessian evaluations = 7
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 19.00us ( 2.38us) 18.40us ( 2.30us) 8
nlp_g | 28.00us ( 3.50us) 24.68us ( 3.09us) 8
nlp_grad_f | 25.00us ( 2.78us) 23.04us ( 2.56us) 9
nlp_hess_l | 25.00us ( 3.57us) 24.94us ( 3.56us) 7
nlp_jac_g | 18.00us ( 2.00us) 18.47us ( 2.05us) 9
total | 3.79ms ( 3.79ms) 3.79ms ( 3.79ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 6.0359672e-02 0.00e+00 1.90e-01 -1.7 5.33e-01 - 1.00e+00 1.00e+00h 1
3 1.7546815e-02 0.00e+00 1.08e-02 -1.7 6.95e-01 - 1.00e+00 1.00e+00h 1
4 1.4029877e-02 0.00e+00 5.59e-04 -2.5 1.22e-01 - 1.00e+00 1.00e+00h 1
5 1.1426418e-02 0.00e+00 5.32e-04 -3.8 5.53e-02 - 1.00e+00 1.00e+00h 1
6 1.0972613e-02 0.00e+00 1.62e-05 -5.7 1.10e-02 - 1.00e+00 1.00e+00h 1
7 1.0963256e-02 0.00e+00 6.61e-09 -5.7 2.40e-04 - 1.00e+00 1.00e+00h 1
8 1.0961410e-02 0.00e+00 3.05e-10 -8.6 4.29e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 1.0961410459303357e-02 1.0961410459303357e-02
Dual infeasibility......: 3.0470378997726755e-10 3.0470378997726755e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.7041299620319639e-09 2.7041299620319639e-09
Overall NLP error.......: 2.7041299620319639e-09 2.7041299620319639e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 17.00us ( 1.89us) 17.99us ( 2.00us) 9
nlp_g | 31.00us ( 3.44us) 24.02us ( 2.67us) 9
nlp_grad_f | 26.00us ( 2.60us) 23.95us ( 2.40us) 10
nlp_hess_l | 44.00us ( 5.50us) 28.02us ( 3.50us) 8
nlp_jac_g | 20.00us ( 2.00us) 18.60us ( 1.86us) 10
total | 3.91ms ( 3.91ms) 3.91ms ( 3.91ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.9113564e-02 0.00e+00 1.93e-01 -1.7 5.36e-01 - 1.00e+00 1.00e+00h 1
3 1.5399389e-02 0.00e+00 1.36e-02 -1.7 7.24e-01 - 1.00e+00 1.00e+00h 1
4 1.1027756e-02 0.00e+00 1.04e-03 -2.5 1.53e-01 - 1.00e+00 1.00e+00h 1
5 8.3154312e-03 0.00e+00 7.31e-04 -3.8 6.92e-02 - 1.00e+00 1.00e+00h 1
6 7.7525665e-03 0.00e+00 3.38e-05 -5.7 1.68e-02 - 1.00e+00 1.00e+00h 1
7 7.7328231e-03 0.00e+00 4.14e-08 -5.7 6.20e-04 - 1.00e+00 1.00e+00h 1
8 7.7309570e-03 0.00e+00 4.25e-10 -8.6 5.40e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 7.7309569713546727e-03 7.7309569713546727e-03
Dual infeasibility......: 4.2531619198760140e-10 4.2531619198760140e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.7820333367420236e-09 2.7820333367420236e-09
Overall NLP error.......: 2.7820333367420236e-09 2.7820333367420236e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 20.00us ( 2.22us) 18.20us ( 2.02us) 9
nlp_g | 29.00us ( 3.22us) 24.14us ( 2.68us) 9
nlp_grad_f | 24.00us ( 2.40us) 23.15us ( 2.32us) 10
nlp_hess_l | 27.00us ( 3.37us) 25.98us ( 3.25us) 8
nlp_jac_g | 19.00us ( 1.90us) 18.24us ( 1.82us) 10
total | 3.92ms ( 3.92ms) 3.92ms ( 3.92ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.8011063e-02 0.00e+00 1.96e-01 -1.7 5.38e-01 - 1.00e+00 1.00e+00h 1
3 1.3649962e-02 0.00e+00 1.62e-02 -1.7 7.50e-01 - 1.00e+00 1.00e+00h 1
4 8.6707087e-03 0.00e+00 1.60e-03 -2.5 1.83e-01 - 1.00e+00 1.00e+00h 1
5 5.8954198e-03 0.00e+00 9.85e-04 -3.8 8.51e-02 - 1.00e+00 1.00e+00h 1
6 5.2072471e-03 0.00e+00 7.16e-05 -5.7 2.56e-02 - 1.00e+00 1.00e+00h 1
7 5.1650623e-03 0.00e+00 2.81e-07 -5.7 1.66e-03 - 1.00e+00 1.00e+00h 1
8 5.1630545e-03 0.00e+00 7.08e-10 -8.6 7.44e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 5.1630545058838810e-03 5.1630545058838810e-03
Dual infeasibility......: 7.0799513474106845e-10 7.0799513474106845e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.9638576191297612e-09 2.9638576191297612e-09
Overall NLP error.......: 2.9638576191297612e-09 2.9638576191297612e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 36.00us ( 4.00us) 35.51us ( 3.95us) 9
nlp_g | 30.00us ( 3.33us) 23.40us ( 2.60us) 9
nlp_grad_f | 24.00us ( 2.40us) 23.41us ( 2.34us) 10
nlp_hess_l | 29.00us ( 3.62us) 26.77us ( 3.35us) 8
nlp_jac_g | 18.00us ( 1.80us) 18.59us ( 1.86us) 10
total | 3.93ms ( 3.93ms) 3.92ms ( 3.92ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.7028844e-02 0.00e+00 1.98e-01 -1.7 5.41e-01 - 1.00e+00 1.00e+00h 1
3 1.2210370e-02 0.00e+00 1.84e-02 -1.7 7.72e-01 - 1.00e+00 1.00e+00h 1
4 6.8338682e-03 0.00e+00 2.19e-03 -2.5 2.10e-01 - 1.00e+00 1.00e+00h 1
5 4.0714589e-03 0.00e+00 1.27e-03 -3.8 1.02e-01 - 1.00e+00 1.00e+00h 1
6 3.3899608e-03 0.00e+00 1.01e-04 -3.8 3.21e-02 - 1.00e+00 1.00e+00h 1
7 3.1911382e-03 0.00e+00 1.04e-05 -5.7 9.67e-03 - 1.00e+00 1.00e+00h 1
8 3.1829352e-03 0.00e+00 1.76e-08 -8.6 4.16e-04 - 1.00e+00 1.00e+00h 1
9 3.1829246e-03 0.00e+00 2.87e-14 -8.6 5.45e-07 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: 3.1829246345669135e-03 3.1829246345669135e-03
Dual infeasibility......: 2.8664570717040760e-14 2.8664570717040760e-14
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5059217572868037e-09 2.5059217572868037e-09
Overall NLP error.......: 2.5059217572868037e-09 2.5059217572868037e-09
Number of objective function evaluations = 10
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 10
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 10
Number of Lagrangian Hessian evaluations = 9
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 30.00us ( 3.00us) 29.96us ( 3.00us) 10
nlp_g | 33.00us ( 3.30us) 27.43us ( 2.74us) 10
nlp_grad_f | 25.00us ( 2.27us) 24.66us ( 2.24us) 11
nlp_hess_l | 30.00us ( 3.33us) 28.92us ( 3.21us) 9
nlp_jac_g | 21.00us ( 1.91us) 20.40us ( 1.85us) 11
total | 4.22ms ( 4.22ms) 4.22ms ( 4.22ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.6148346e-02 0.00e+00 2.00e-01 -1.7 5.43e-01 - 1.00e+00 1.00e+00h 1
3 1.1014320e-02 0.00e+00 2.04e-02 -1.7 7.92e-01 - 1.00e+00 1.00e+00h 1
4 5.4088330e-03 0.00e+00 2.76e-03 -2.5 2.34e-01 - 1.00e+00 1.00e+00h 1
5 2.7440080e-03 0.00e+00 1.57e-03 -3.8 1.17e-01 - 1.00e+00 1.00e+00h 1
6 1.9864582e-03 0.00e+00 1.96e-04 -3.8 4.59e-02 - 1.00e+00 1.00e+00h 1
7 1.7451048e-03 0.00e+00 2.63e-05 -5.7 1.62e-02 - 1.00e+00 1.00e+00h 1
8 1.7290667e-03 0.00e+00 1.20e-07 -5.7 1.14e-03 - 1.00e+00 1.00e+00h 1
9 1.7271532e-03 0.00e+00 1.82e-09 -8.6 1.32e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: 1.7271531695755196e-03 1.7271531695755196e-03
Dual infeasibility......: 1.8226307303692568e-09 1.8226307303692568e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 3.6720751174653786e-09 3.6720751174653786e-09
Overall NLP error.......: 3.6720751174653786e-09 3.6720751174653786e-09
Number of objective function evaluations = 10
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 10
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 10
Number of Lagrangian Hessian evaluations = 9
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 21.00us ( 2.10us) 20.01us ( 2.00us) 10
nlp_g | 33.00us ( 3.30us) 27.65us ( 2.77us) 10
nlp_grad_f | 25.00us ( 2.27us) 24.84us ( 2.26us) 11
nlp_hess_l | 28.00us ( 3.11us) 28.66us ( 3.18us) 9
nlp_jac_g | 21.00us ( 1.91us) 20.26us ( 1.84us) 11
total | 4.21ms ( 4.21ms) 4.21ms ( 4.21ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.5354621e-02 0.00e+00 2.02e-01 -1.7 5.45e-01 - 1.00e+00 1.00e+00h 1
3 1.0011545e-02 0.00e+00 2.22e-02 -1.7 8.10e-01 - 1.00e+00 1.00e+00h 1
4 4.3051888e-03 0.00e+00 3.28e-03 -2.5 2.55e-01 - 1.00e+00 1.00e+00h 1
5 1.8114983e-03 0.00e+00 1.82e-03 -3.8 1.31e-01 - 1.00e+00 1.00e+00h 1
6 1.0586031e-03 0.00e+00 3.20e-04 -3.8 6.02e-02 - 1.00e+00 1.00e+00h 1
7 7.8447246e-04 0.00e+00 6.75e-05 -5.7 2.70e-02 - 1.00e+00 1.00e+00h 1
8 7.4439194e-04 0.00e+00 1.69e-06 -5.7 4.40e-03 - 1.00e+00 1.00e+00h 1
9 7.4151795e-04 0.00e+00 9.20e-09 -8.6 3.15e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: 7.4151795194904750e-04 7.4151795194904750e-04
Dual infeasibility......: 9.2008462446169226e-09 9.2008462446169226e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 8.3392495297081416e-09 8.3392495297081416e-09
Overall NLP error.......: 9.2008462446169226e-09 9.2008462446169226e-09
Number of objective function evaluations = 10
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 10
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 10
Number of Lagrangian Hessian evaluations = 9
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 21.00us ( 2.10us) 20.52us ( 2.05us) 10
nlp_g | 35.00us ( 3.50us) 27.95us ( 2.79us) 10
nlp_grad_f | 42.00us ( 3.82us) 43.13us ( 3.92us) 11
nlp_hess_l | 29.00us ( 3.22us) 28.80us ( 3.20us) 9
nlp_jac_g | 20.00us ( 1.82us) 19.86us ( 1.81us) 11
total | 4.20ms ( 4.20ms) 4.20ms ( 4.20ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.4635500e-02 0.00e+00 2.04e-01 -1.7 5.47e-01 - 1.00e+00 1.00e+00h 1
3 9.1635916e-03 0.00e+00 2.38e-02 -1.7 8.26e-01 - 1.00e+00 1.00e+00h 1
4 3.4498835e-03 0.00e+00 3.74e-03 -2.5 2.74e-01 - 1.00e+00 1.00e+00h 1
5 1.1774797e-03 0.00e+00 2.00e-03 -3.8 1.42e-01 - 1.00e+00 1.00e+00h 1
6 5.1088991e-04 0.00e+00 4.33e-04 -3.8 7.16e-02 - 1.00e+00 1.00e+00h 1
7 2.5921931e-04 0.00e+00 1.37e-04 -5.7 3.98e-02 - 1.00e+00 1.00e+00h 1
8 1.9120913e-04 0.00e+00 1.66e-05 -5.7 1.41e-02 - 1.00e+00 1.00e+00h 1
9 1.8139089e-04 0.00e+00 4.15e-07 -5.7 2.23e-03 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 1.7930937e-04 0.00e+00 1.96e-08 -8.6 4.75e-04 - 1.00e+00 1.00e+00h 1
11 1.7929715e-04 0.00e+00 6.69e-13 -8.6 2.83e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 11
(scaled) (unscaled)
Objective...............: 1.7929714680027842e-04 1.7929714680027842e-04
Dual infeasibility......: 6.6888682093146912e-13 6.6888682093146912e-13
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5063234157341978e-09 2.5063234157341978e-09
Overall NLP error.......: 2.5063234157341978e-09 2.5063234157341978e-09
Number of objective function evaluations = 12
Number of objective gradient evaluations = 12
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 12
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 12
Number of Lagrangian Hessian evaluations = 11
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 27.00us ( 2.25us) 25.58us ( 2.13us) 12
nlp_g | 36.00us ( 3.00us) 32.26us ( 2.69us) 12
nlp_grad_f | 29.00us ( 2.23us) 28.76us ( 2.21us) 13
nlp_hess_l | 36.00us ( 3.27us) 34.01us ( 3.09us) 11
nlp_jac_g | 24.00us ( 1.85us) 22.89us ( 1.76us) 13
total | 4.87ms ( 4.87ms) 4.86ms ( 4.86ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.3980978e-02 0.00e+00 2.05e-01 -1.7 5.48e-01 - 1.00e+00 1.00e+00h 1
3 8.4407793e-03 0.00e+00 2.52e-02 -1.7 8.41e-01 - 1.00e+00 1.00e+00h 1
4 2.7852896e-03 0.00e+00 4.13e-03 -2.5 2.90e-01 - 1.00e+00 1.00e+00h 1
5 7.5806761e-04 0.00e+00 2.11e-03 -3.8 1.49e-01 - 1.00e+00 1.00e+00h 1
6 2.2625261e-04 0.00e+00 4.87e-04 -3.8 7.76e-02 - 1.00e+00 1.00e+00h 1
7 5.7246642e-05 0.00e+00 1.79e-04 -5.7 4.66e-02 - 1.00e+00 1.00e+00h 1
8 1.4739390e-05 0.00e+00 4.42e-05 -5.7 2.35e-02 - 1.00e+00 1.00e+00h 1
9 4.1552852e-06 0.00e+00 1.03e-05 -5.7 1.14e-02 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 1.5499396e-06 0.00e+00 2.01e-06 -5.7 5.01e-03 - 1.00e+00 1.00e+00h 1
11 3.8848094e-07 0.00e+00 1.24e-06 -8.6 3.92e-03 - 1.00e+00 1.00e+00h 1
12 9.7729804e-08 0.00e+00 3.09e-07 -8.6 1.96e-03 - 1.00e+00 1.00e+00h 1
13 2.5060537e-08 0.00e+00 7.62e-08 -8.6 9.76e-04 - 1.00e+00 1.00e+00h 1
14 6.9068683e-09 0.00e+00 1.81e-08 -8.6 4.76e-04 - 1.00e+00 1.00e+00h 1
15 2.4098945e-09 0.00e+00 3.70e-09 -8.6 2.15e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 15
(scaled) (unscaled)
Objective...............: 2.4098945059785058e-09 2.4098945059785058e-09
Dual infeasibility......: 3.7025984853523812e-09 3.7025984853523812e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 4.8200273946044563e-09 4.8200273946044563e-09
Overall NLP error.......: 4.8200273946044563e-09 4.8200273946044563e-09
Number of objective function evaluations = 16
Number of objective gradient evaluations = 16
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 16
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 16
Number of Lagrangian Hessian evaluations = 15
Total seconds in IPOPT = 0.006
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 31.00us ( 1.94us) 30.90us ( 1.93us) 16
nlp_g | 53.00us ( 3.31us) 44.22us ( 2.76us) 16
nlp_grad_f | 40.00us ( 2.35us) 38.14us ( 2.24us) 17
nlp_hess_l | 47.00us ( 3.13us) 46.74us ( 3.12us) 15
nlp_jac_g | 32.00us ( 1.88us) 31.08us ( 1.83us) 17
total | 6.12ms ( 6.12ms) 6.12ms ( 6.12ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.3382755e-02 0.00e+00 2.06e-01 -1.7 5.50e-01 - 1.00e+00 1.00e+00h 1
3 7.8199926e-03 0.00e+00 2.64e-02 -1.7 8.54e-01 - 1.00e+00 1.00e+00h 1
4 2.2667217e-03 0.00e+00 4.45e-03 -2.5 3.04e-01 - 1.00e+00 1.00e+00h 1
5 4.8629617e-04 0.00e+00 2.13e-03 -3.8 1.54e-01 - 1.00e+00 1.00e+00h 1
6 9.4861390e-05 0.00e+00 4.67e-04 -3.8 7.77e-02 - 1.00e+00 1.00e+00h 1
7 8.7855125e-06 0.00e+00 1.45e-04 -5.7 4.28e-02 - 1.00e+00 1.00e+00h 1
8 2.6103864e-07 0.00e+00 1.89e-05 -5.7 1.57e-02 - 1.00e+00 1.00e+00h 1
9 6.9704935e-09 0.00e+00 5.74e-07 -5.7 2.74e-03 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 4.0026777e-13 0.00e+00 2.19e-08 -8.6 5.25e-04 - 1.00e+00 1.00e+00h 1
11 9.0466243e-15 0.00e+00 9.08e-13 -8.6 3.44e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 11
(scaled) (unscaled)
Objective...............: 9.0466243246771345e-15 9.0466243246771345e-15
Dual infeasibility......: 9.0786374495489244e-13 9.0786374495489244e-13
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5064681925437888e-09 2.5064681925437888e-09
Overall NLP error.......: 2.5064681925437888e-09 2.5064681925437888e-09
Number of objective function evaluations = 12
Number of objective gradient evaluations = 12
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 12
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 12
Number of Lagrangian Hessian evaluations = 11
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 25.00us ( 2.08us) 23.86us ( 1.99us) 12
nlp_g | 40.00us ( 3.33us) 32.37us ( 2.70us) 12
nlp_grad_f | 30.00us ( 2.31us) 27.75us ( 2.13us) 13
nlp_hess_l | 34.00us ( 3.09us) 34.00us ( 3.09us) 11
nlp_jac_g | 24.00us ( 1.85us) 23.15us ( 1.78us) 13
total | 4.89ms ( 4.89ms) 4.89ms ( 4.89ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.2833890e-02 0.00e+00 2.08e-01 -1.7 5.51e-01 - 1.00e+00 1.00e+00h 1
3 7.2830726e-03 0.00e+00 2.75e-02 -1.7 8.65e-01 - 1.00e+00 1.00e+00h 1
4 1.8599287e-03 0.00e+00 4.70e-03 -2.5 3.15e-01 - 1.00e+00 1.00e+00h 1
5 3.1254511e-04 0.00e+00 2.08e-03 -3.8 1.55e-01 - 1.00e+00 1.00e+00h 1
6 3.9273057e-05 0.00e+00 3.96e-04 -3.8 7.30e-02 - 1.00e+00 1.00e+00h 1
7 1.1323346e-06 0.00e+00 8.48e-05 -5.7 3.32e-02 - 1.00e+00 1.00e+00h 1
8 5.3143263e-09 0.00e+00 3.03e-06 -5.7 6.41e-03 - 1.00e+00 1.00e+00h 1
9 6.6123263e-14 0.00e+00 1.66e-08 -8.6 4.65e-04 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 2.2607847e-15 0.00e+00 1.35e-13 -8.6 1.36e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 10
(scaled) (unscaled)
Objective...............: 2.2607846751301537e-15 2.2607846751301537e-15
Dual infeasibility......: 1.3533781753801507e-13 1.3533781753801507e-13
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5059874535616747e-09 2.5059874535616747e-09
Overall NLP error.......: 2.5059874535616747e-09 2.5059874535616747e-09
Number of objective function evaluations = 11
Number of objective gradient evaluations = 11
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 11
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 11
Number of Lagrangian Hessian evaluations = 10
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 25.00us ( 2.27us) 24.01us ( 2.18us) 11
nlp_g | 37.00us ( 3.36us) 30.84us ( 2.80us) 11
nlp_grad_f | 28.00us ( 2.33us) 25.96us ( 2.16us) 12
nlp_hess_l | 32.00us ( 3.20us) 31.38us ( 3.14us) 10
nlp_jac_g | 23.00us ( 1.92us) 21.16us ( 1.76us) 12
total | 4.51ms ( 4.51ms) 4.51ms ( 4.51ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.2328535e-02 0.00e+00 2.09e-01 -1.7 5.52e-01 - 1.00e+00 1.00e+00h 1
3 6.8156322e-03 0.00e+00 2.84e-02 -1.7 8.76e-01 - 1.00e+00 1.00e+00h 1
4 1.5388433e-03 0.00e+00 4.89e-03 -2.5 3.25e-01 - 1.00e+00 1.00e+00h 1
5 2.0218974e-04 0.00e+00 1.98e-03 -3.8 1.55e-01 - 1.00e+00 1.00e+00h 1
6 1.6754549e-05 0.00e+00 3.08e-04 -3.8 6.55e-02 - 1.00e+00 1.00e+00h 1
7 1.5219076e-07 0.00e+00 4.26e-05 -5.7 2.38e-02 - 1.00e+00 1.00e+00h 1
8 7.1653837e-10 0.00e+00 4.00e-07 -5.7 2.37e-03 - 1.00e+00 1.00e+00h 1
9 2.6541777e-15 0.00e+00 2.27e-09 -8.6 1.70e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: 2.6541777156111496e-15 2.6541777156111496e-15
Dual infeasibility......: 2.2689809148682812e-09 2.2689809148682812e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 3.9219928248582801e-09 3.9219928248582801e-09
Overall NLP error.......: 3.9219928248582801e-09 3.9219928248582801e-09
Number of objective function evaluations = 10
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 10
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 10
Number of Lagrangian Hessian evaluations = 9
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 20.00us ( 2.00us) 20.47us ( 2.05us) 10
nlp_g | 33.00us ( 3.30us) 26.64us ( 2.66us) 10
nlp_grad_f | 24.00us ( 2.18us) 25.19us ( 2.29us) 11
nlp_hess_l | 29.00us ( 3.22us) 28.46us ( 3.16us) 9
nlp_jac_g | 20.00us ( 1.82us) 19.27us ( 1.75us) 11
total | 4.21ms ( 4.21ms) 4.21ms ( 4.21ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.1861727e-02 0.00e+00 2.10e-01 -1.7 5.53e-01 - 1.00e+00 1.00e+00h 1
3 6.4061795e-03 0.00e+00 2.93e-02 -1.7 8.86e-01 - 1.00e+00 1.00e+00h 1
4 1.2836965e-03 0.00e+00 5.03e-03 -2.5 3.34e-01 - 1.00e+00 1.00e+00h 1
5 1.3213997e-04 0.00e+00 1.85e-03 -3.8 1.52e-01 - 1.00e+00 1.00e+00h 1
6 7.6262603e-06 0.00e+00 2.26e-04 -3.8 5.71e-02 - 1.00e+00 1.00e+00h 1
7 2.4998766e-08 0.00e+00 2.10e-05 -5.7 1.68e-02 - 1.00e+00 1.00e+00h 1
8 3.1987638e-10 0.00e+00 5.87e-08 -5.7 9.25e-04 - 1.00e+00 1.00e+00h 1
9 9.4051883e-16 0.00e+00 1.02e-09 -8.6 1.13e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: 9.4051882503475303e-16 9.4051882503475303e-16
Dual infeasibility......: 1.0186468605667904e-09 1.0186468605667904e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 3.1423839562517952e-09 3.1423839562517952e-09
Overall NLP error.......: 3.1423839562517952e-09 3.1423839562517952e-09
Number of objective function evaluations = 10
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 10
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 10
Number of Lagrangian Hessian evaluations = 9
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 21.00us ( 2.10us) 20.38us ( 2.04us) 10
nlp_g | 36.00us ( 3.60us) 27.36us ( 2.74us) 10
nlp_grad_f | 25.00us ( 2.27us) 26.59us ( 2.42us) 11
nlp_hess_l | 29.00us ( 3.22us) 28.22us ( 3.14us) 9
nlp_jac_g | 21.00us ( 1.91us) 18.91us ( 1.72us) 11
total | 4.18ms ( 4.18ms) 4.18ms ( 4.18ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.1429229e-02 0.00e+00 2.10e-01 -1.7 5.54e-01 - 1.00e+00 1.00e+00h 1
3 6.0454626e-03 0.00e+00 3.00e-02 -1.7 8.95e-01 - 1.00e+00 1.00e+00h 1
4 1.0795087e-03 0.00e+00 5.13e-03 -2.5 3.41e-01 - 1.00e+00 1.00e+00h 1
5 8.7482132e-05 0.00e+00 1.71e-03 -3.8 1.49e-01 - 1.00e+00 1.00e+00h 1
6 3.7871010e-06 0.00e+00 1.60e-04 -3.8 4.90e-02 - 1.00e+00 1.00e+00h 1
7 5.4814828e-09 0.00e+00 1.08e-05 -5.7 1.21e-02 - 1.00e+00 1.00e+00h 1
8 1.9757079e-10 0.00e+00 1.06e-08 -5.7 3.99e-04 - 1.00e+00 1.00e+00h 1
9 5.0788853e-16 0.00e+00 6.30e-10 -8.6 8.88e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: 5.0788852844357805e-16 5.0788852844357805e-16
Dual infeasibility......: 6.3001010769284050e-10 6.3001010769284050e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.8996408826546640e-09 2.8996408826546640e-09
Overall NLP error.......: 2.8996408826546640e-09 2.8996408826546640e-09
Number of objective function evaluations = 10
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 10
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 10
Number of Lagrangian Hessian evaluations = 9
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 21.00us ( 2.10us) 20.34us ( 2.03us) 10
nlp_g | 31.00us ( 3.10us) 26.72us ( 2.67us) 10
nlp_grad_f | 27.00us ( 2.45us) 25.54us ( 2.32us) 11
nlp_hess_l | 33.00us ( 3.67us) 30.62us ( 3.40us) 9
nlp_jac_g | 23.00us ( 2.09us) 19.91us ( 1.81us) 11
total | 4.22ms ( 4.22ms) 4.22ms ( 4.22ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.1027401e-02 0.00e+00 2.11e-01 -1.7 5.55e-01 - 1.00e+00 1.00e+00h 1
3 5.7259771e-03 0.00e+00 3.07e-02 -1.7 9.03e-01 - 1.00e+00 1.00e+00h 1
4 9.1491524e-04 0.00e+00 5.19e-03 -2.5 3.46e-01 - 1.00e+00 1.00e+00h 1
5 5.8779922e-05 0.00e+00 1.56e-03 -3.8 1.45e-01 - 1.00e+00 1.00e+00h 1
6 2.0698707e-06 0.00e+00 1.12e-04 -3.8 4.16e-02 - 1.00e+00 1.00e+00h 1
7 1.6411495e-09 0.00e+00 6.05e-06 -5.7 9.04e-03 - 1.00e+00 1.00e+00h 1
8 1.5104441e-15 0.00e+00 4.89e-09 -8.6 2.67e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 1.5104441121569591e-15 1.5104441121569591e-15
Dual infeasibility......: 4.8852512565929027e-09 4.8852512565929027e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 5.4929464869459015e-09 5.4929464869459015e-09
Overall NLP error.......: 5.4929464869459015e-09 5.4929464869459015e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 20.00us ( 2.22us) 18.32us ( 2.04us) 9
nlp_g | 28.00us ( 3.11us) 24.25us ( 2.69us) 9
nlp_grad_f | 23.00us ( 2.30us) 22.80us ( 2.28us) 10
nlp_hess_l | 27.00us ( 3.38us) 25.84us ( 3.23us) 8
nlp_jac_g | 20.00us ( 2.00us) 17.06us ( 1.71us) 10
total | 3.90ms ( 3.90ms) 3.91ms ( 3.91ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.0653100e-02 0.00e+00 2.12e-01 -1.7 5.56e-01 - 1.00e+00 1.00e+00h 1
3 5.4415923e-03 0.00e+00 3.13e-02 -1.7 9.10e-01 - 1.00e+00 1.00e+00h 1
4 7.8126920e-04 0.00e+00 5.22e-03 -2.5 3.51e-01 - 1.00e+00 1.00e+00h 1
5 4.0130268e-05 0.00e+00 1.41e-03 -3.8 1.40e-01 - 1.00e+00 1.00e+00h 1
6 1.2419257e-06 0.00e+00 7.73e-05 -3.8 3.52e-02 - 1.00e+00 1.00e+00h 1
7 6.5078116e-10 0.00e+00 3.68e-06 -5.7 7.03e-03 - 1.00e+00 1.00e+00h 1
8 4.6942794e-16 0.00e+00 1.94e-09 -8.6 1.68e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 4.6942793688164477e-16 4.6942793688164477e-16
Dual infeasibility......: 1.9424394977504803e-09 1.9424394977504803e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 3.6946208020933589e-09 3.6946208020933589e-09
Overall NLP error.......: 3.6946208020933589e-09 3.6946208020933589e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 20.00us ( 2.22us) 19.46us ( 2.16us) 9
nlp_g | 31.00us ( 3.44us) 23.75us ( 2.64us) 9
nlp_grad_f | 23.00us ( 2.30us) 22.39us ( 2.24us) 10
nlp_hess_l | 27.00us ( 3.38us) 26.23us ( 3.28us) 8
nlp_jac_g | 19.00us ( 1.90us) 17.91us ( 1.79us) 10
total | 3.95ms ( 3.95ms) 3.95ms ( 3.95ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 5.0303596e-02 0.00e+00 2.12e-01 -1.7 5.57e-01 - 1.00e+00 1.00e+00h 1
3 5.1872652e-03 0.00e+00 3.19e-02 -1.7 9.17e-01 - 1.00e+00 1.00e+00h 1
4 6.7196566e-04 0.00e+00 5.23e-03 -2.5 3.55e-01 - 1.00e+00 1.00e+00h 1
5 2.7855751e-05 0.00e+00 1.27e-03 -3.8 1.35e-01 - 1.00e+00 1.00e+00h 1
6 8.0983086e-07 0.00e+00 5.36e-05 -3.8 2.97e-02 - 1.00e+00 1.00e+00h 1
7 3.2246843e-10 0.00e+00 2.43e-06 -5.7 5.68e-03 - 1.00e+00 1.00e+00h 1
8 2.4062498e-16 0.00e+00 9.67e-10 -8.6 1.18e-04 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 2.4062498476545355e-16 2.4062498476545355e-16
Dual infeasibility......: 9.6688372355952227e-10 9.6688372355952227e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 3.0985707245007653e-09 3.0985707245007653e-09
Overall NLP error.......: 3.0985707245007653e-09 3.0985707245007653e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 21.00us ( 2.33us) 18.81us ( 2.09us) 9
nlp_g | 31.00us ( 3.44us) 24.25us ( 2.69us) 9
nlp_grad_f | 24.00us ( 2.40us) 22.15us ( 2.22us) 10
nlp_hess_l | 26.00us ( 3.25us) 26.03us ( 3.25us) 8
nlp_jac_g | 18.00us ( 1.80us) 17.43us ( 1.74us) 10
total | 3.85ms ( 3.85ms) 3.85ms ( 3.85ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 4.9976509e-02 0.00e+00 2.13e-01 -1.7 5.58e-01 - 1.00e+00 1.00e+00h 1
3 4.9588208e-03 0.00e+00 3.23e-02 -1.7 9.23e-01 - 1.00e+00 1.00e+00h 1
4 5.8193506e-04 0.00e+00 5.22e-03 -2.5 3.59e-01 - 1.00e+00 1.00e+00h 1
5 1.9662481e-05 0.00e+00 1.15e-03 -3.8 1.30e-01 - 1.00e+00 1.00e+00h 1
6 5.6628358e-07 0.00e+00 3.74e-05 -3.8 2.52e-02 - 1.00e+00 1.00e+00h 1
7 1.8801878e-10 0.00e+00 1.71e-06 -5.7 4.74e-03 - 1.00e+00 1.00e+00h 1
8 1.5648994e-16 0.00e+00 5.67e-10 -8.6 8.98e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 1.5648994401963879e-16 1.5648994401963879e-16
Dual infeasibility......: 5.6658446296596025e-10 5.6658446296596025e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.8538211966995104e-09 2.8538211966995104e-09
Overall NLP error.......: 2.8538211966995104e-09 2.8538211966995104e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 15.00us ( 1.67us) 17.73us ( 1.97us) 9
nlp_g | 29.00us ( 3.22us) 23.01us ( 2.56us) 9
nlp_grad_f | 23.00us ( 2.30us) 22.68us ( 2.27us) 10
nlp_hess_l | 25.00us ( 3.12us) 25.24us ( 3.15us) 8
nlp_jac_g | 19.00us ( 1.90us) 16.95us ( 1.70us) 10
total | 3.88ms ( 3.88ms) 3.88ms ( 3.88ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 4.9669752e-02 0.00e+00 2.14e-01 -1.7 5.59e-01 - 1.00e+00 1.00e+00h 1
3 4.7527812e-03 0.00e+00 3.28e-02 -1.7 9.29e-01 - 1.00e+00 1.00e+00h 1
4 5.0726523e-04 0.00e+00 5.19e-03 -2.5 3.62e-01 - 1.00e+00 1.00e+00h 1
5 1.4111911e-05 0.00e+00 1.03e-03 -3.8 1.25e-01 - 1.00e+00 1.00e+00h 1
6 4.1893115e-07 0.00e+00 2.63e-05 -3.8 2.15e-02 - 1.00e+00 1.00e+00h 1
7 1.2290102e-10 0.00e+00 1.27e-06 -5.7 4.07e-03 - 1.00e+00 1.00e+00h 1
8 1.1429223e-16 0.00e+00 3.72e-10 -8.6 7.24e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 1.1429222975431541e-16 1.1429222975431541e-16
Dual infeasibility......: 3.7210581262012894e-10 3.7210581262012894e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.7347766265729349e-09 2.7347766265729349e-09
Overall NLP error.......: 2.7347766265729349e-09 2.7347766265729349e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 21.00us ( 2.33us) 19.32us ( 2.15us) 9
nlp_g | 47.00us ( 5.22us) 41.55us ( 4.62us) 9
nlp_grad_f | 24.00us ( 2.40us) 23.27us ( 2.33us) 10
nlp_hess_l | 29.00us ( 3.62us) 27.27us ( 3.41us) 8
nlp_jac_g | 17.00us ( 1.70us) 17.83us ( 1.78us) 10
total | 3.89ms ( 3.89ms) 3.89ms ( 3.89ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 4.9381490e-02 0.00e+00 2.14e-01 -1.7 5.59e-01 - 1.00e+00 1.00e+00h 1
3 4.5662323e-03 0.00e+00 3.32e-02 -1.7 9.35e-01 - 1.00e+00 1.00e+00h 1
4 4.4491969e-04 0.00e+00 5.15e-03 -2.5 3.64e-01 - 1.00e+00 1.00e+00h 1
5 1.0294433e-05 0.00e+00 9.28e-04 -3.8 1.20e-01 - 1.00e+00 1.00e+00h 1
6 3.2399690e-07 0.00e+00 1.87e-05 -3.8 1.84e-02 - 1.00e+00 1.00e+00h 1
7 8.7014412e-11 0.00e+00 9.89e-07 -5.7 3.58e-03 - 1.00e+00 1.00e+00h 1
8 8.8912522e-17 0.00e+00 2.65e-10 -8.6 6.07e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 8.8912522344570490e-17 8.8912522344570490e-17
Dual infeasibility......: 2.6454817415251347e-10 2.6454817415251347e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.6688523639630481e-09 2.6688523639630481e-09
Overall NLP error.......: 2.6688523639630481e-09 2.6688523639630481e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 20.00us ( 2.22us) 18.68us ( 2.08us) 9
nlp_g | 29.00us ( 3.22us) 24.59us ( 2.73us) 9
nlp_grad_f | 24.00us ( 2.40us) 22.92us ( 2.29us) 10
nlp_hess_l | 27.00us ( 3.37us) 27.16us ( 3.40us) 8
nlp_jac_g | 19.00us ( 1.90us) 17.86us ( 1.79us) 10
total | 3.90ms ( 3.90ms) 3.89ms ( 3.89ms) 1
This is Ipopt version 3.14.11, running with linear solver MUMPS 5.4.1.
Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 2
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 0
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 1
inequality constraints with only lower bounds: 0
inequality constraints with lower and upper bounds: 0
inequality constraints with only upper bounds: 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
0 1.0000000e+00 0.00e+00 2.00e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 3.1250000e-01 0.00e+00 5.00e-01 -1.0 5.00e-01 - 1.00e+00 1.00e+00f 1
2 4.9110102e-02 0.00e+00 2.14e-01 -1.7 5.60e-01 - 1.00e+00 1.00e+00h 1
3 4.3967182e-03 0.00e+00 3.35e-02 -1.7 9.40e-01 - 1.00e+00 1.00e+00h 1
4 3.9252745e-04 0.00e+00 5.11e-03 -2.5 3.66e-01 - 1.00e+00 1.00e+00h 1
5 7.6289532e-06 0.00e+00 8.35e-04 -3.8 1.15e-01 - 1.00e+00 1.00e+00h 1
6 2.5942225e-07 0.00e+00 1.35e-05 -3.8 1.58e-02 - 1.00e+00 1.00e+00h 1
7 6.5197585e-11 0.00e+00 7.95e-07 -5.7 3.20e-03 - 1.00e+00 1.00e+00h 1
8 7.1902281e-17 0.00e+00 1.99e-10 -8.6 5.25e-05 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 8
(scaled) (unscaled)
Objective...............: 7.1902281402064665e-17 7.1902281402064665e-17
Dual infeasibility......: 1.9892612672036513e-10 1.9892612672036513e-10
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.6285802469938070e-09 2.6285802469938070e-09
Overall NLP error.......: 2.6285802469938070e-09 2.6285802469938070e-09
Number of objective function evaluations = 9
Number of objective gradient evaluations = 9
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 9
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 9
Number of Lagrangian Hessian evaluations = 8
Total seconds in IPOPT = 0.003
EXIT: Optimal Solution Found.
solver_ipopt_Nlp1 : t_proc (avg) t_wall (avg) n_eval
nlp_f | 19.00us ( 2.11us) 19.47us ( 2.16us) 9
nlp_g | 30.00us ( 3.33us) 24.70us ( 2.74us) 9
nlp_grad_f | 24.00us ( 2.40us) 23.04us ( 2.30us) 10
nlp_hess_l | 27.00us ( 3.37us) 27.03us ( 3.38us) 8
nlp_jac_g | 18.00us ( 1.80us) 18.01us ( 1.80us) 10
total | 3.89ms ( 3.89ms) 3.89ms ( 3.89ms) 1
In the figure we plot the value of the optimal value \(f(x^\star; r)\) as a function of the parameter \(r\), and the tangent line at each point is computed via the corresponding optimal multiplier \(\lambda^\star(r)\).
ts = np.linspace(-0.02, 0.02)
fig, ax = plt.subplots(constrained_layout=True)
ax.plot(r_values, f_values, "o")
for r_value, f_value, lam_value in zip(r_values, f_values, lam_values):
ax.plot(r_value + ts, -lam_value * ts + f_value, "r-")
ax.set_xlabel("Value of r")
ax.set_ylabel("Objective value at solution")
plt.show()
Total running time of the script: (0 minutes 0.377 seconds)