1.3 Alternative Formulations:#
Consider the following problem with ini- tial values x=5, y=5.
\[min_{x,y} f(x,y) = (x-1.01)^{2} + y^{2}\]
\[s.t \;\;\;\; \frac{x-1}{y} = 1\]
Note that the solution to this problem is \(x=1.005\) and \(y=0.005\). There are several ways that the problem above can be reformulated. Some examples are shown below. Which ones do you expect to be better? Why? Note the number of iterations and quality of solutions. What can you learn about problem formulation from these examples?
(a)
\[min_{x,y} f(x,y) = (x-1.01)^{2} + y^{2}\]
\[s.t \;\;\;\; \frac{x-1}{y} = 1\]
import pyomo.environ as pyo
model = pyo.ConcreteModel()
model.x = pyo.Var(initialize=5.0)
model.y = pyo.Var(initialize=5.0)
def obj_rule(m):
return (m.x-1.01)**2 + m.y**2
model.obj = pyo.Objective(rule=obj_rule)
def con_rule(m):
return (m.x - 1.0) / m.y == 1.0
model.con = pyo.Constraint(rule=con_rule)
solver = pyo.SolverFactory('ipopt')
solver.solve(model, tee=True)
print(pyo.value(model.x))
print(pyo.value(model.y))
Ipopt 3.14.19:
******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
Ipopt is released as open source code under the Eclipse Public License (EPL).
For more information visit https://github.com/coin-or/Ipopt
******************************************************************************
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 2
Number of nonzeros in inequality constraint Jacobian.: 0
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.................: 1
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 4.0920100e+01 2.00e-01 9.99e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 5.9762286e-01 2.27e+00 1.52e+01 -1.0 5.48e+00 - 1.00e+00 1.00e+00f 1
2 1.9787064e-01 2.97e+00 2.40e+01 -1.0 8.02e+00 - 1.00e+00 1.25e-01f 4
3 6.5241867e+01 2.82e+00 2.98e+01 -1.0 6.68e+00 - 1.00e+00 1.00e+00h 1
4 9.5583161e+01 1.54e+00 2.31e+01 -1.0 4.70e+00 - 1.00e+00 1.00e+00h 1
5 1.8959813e+02 4.39e-01 2.77e+01 -1.0 1.14e+01 - 1.00e+00 1.00e+00h 1
6 2.2666991e+01 2.02e+00 1.24e+01 -1.0 1.54e+01 - 1.00e+00 1.00e+00f 1
7 3.7976779e+01 9.23e-01 1.21e+01 -1.0 3.88e+00 - 1.00e+00 1.00e+00h 1
8 2.0620942e+01 6.84e-02 1.64e+01 -1.0 5.65e+00 - 1.00e+00 5.00e-01f 2
9 9.8180623e-02 8.02e-01 1.24e+02 -1.0 3.63e+00 - 1.00e+00 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 2.3622224e+03 7.97e-01 1.42e+02 -1.0 4.73e+01 - 1.00e+00 1.00e+00h 1
11 3.6759509e+03 1.64e-01 2.37e+02 -1.0 3.63e+01 - 1.00e+00 1.00e+00h 1
12 7.0378669e+01 6.52e-01 5.20e+02 -1.0 4.32e+01 - 1.00e+00 1.00e+00f 1
13 2.5142436e+04 6.17e-01 5.94e+02 -1.0 1.40e+02 - 1.00e+00 1.00e+00h 1
14 2.6505724e+04 3.06e-01 1.02e+03 -1.7 7.26e+01 - 1.00e+00 1.00e+00h 1
15 3.7335004e+03 2.99e-01 5.43e+02 -1.7 9.42e+01 - 1.00e+00 1.00e+00f 1
16 3.5127061e+03 7.10e-02 3.16e+02 -1.7 9.60e+00 - 1.00e+00 1.00e+00f 1
17 3.2134630e+01 4.96e-01 1.50e+03 -1.7 4.08e+01 - 1.00e+00 1.00e+00f 1
18 1.0915648e+05 4.88e-01 1.76e+03 -1.7 2.89e+02 - 1.00e+00 1.00e+00h 1
19 1.1667899e+05 1.53e-01 1.76e+03 -1.7 1.07e+02 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 3.7449172e+03 4.78e-01 3.37e+03 -1.7 2.30e+02 - 1.00e+00 1.00e+00f 1
21 4.9463570e+05 4.36e-01 3.65e+03 -1.7 5.58e+02 - 1.00e+00 1.00e+00h 1
22 3.6988313e+05 2.75e-01 5.13e+03 -1.7 2.37e+02 - 1.00e+00 1.00e+00f 1
23 8.9309627e+04 1.73e-01 1.92e+03 -1.7 2.88e+02 - 1.00e+00 1.00e+00f 1
24 4.0590995e+04 3.57e-02 4.55e+02 -1.7 1.71e+02 - 1.00e+00 5.00e-01f 2
25 1.8907970e+01 5.44e+00 1.24e+05 -1.7 1.46e+02 - 1.00e+00 1.00e+00f 1
26 1.4670968e+01 5.44e+00 1.24e+05 -1.7 1.63e+04 - 1.00e+00 4.88e-04f 12
27 2.3540992e+01 5.44e+00 1.24e+05 -1.7 1.63e+04 - 1.00e+00 6.10e-05h 15
28 2.8758563e+01 5.44e+00 1.24e+05 -1.7 1.63e+04 - 1.00e+00 3.05e-05h 16
29 3.1562971e+01 5.44e+00 1.24e+05 -1.7 1.63e+04 - 1.00e+00 1.53e-05h 17
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 2.8032208e+08 5.44e+00 1.32e+05 -1.7 1.63e+04 - 1.00e+00 1.00e+00s 22
31 2.9975467e+08 2.93e+00 7.23e+04 -1.7 4.29e+03 - 1.00e+00 1.00e+00s 22
32 5.0495436e+08 1.67e+00 5.58e+04 -1.7 1.07e+04 - 1.00e+00 1.00e+00s 22
33r 5.0495436e+08 1.67e+00 9.99e+02 0.2 0.00e+00 - 0.00e+00 0.00e+00R 1
34r 1.7350836e+09 9.31e-01 7.02e+02 0.2 6.83e+06 - 1.00e+00 3.35e-03f 1
35r 9.2739578e+08 4.28e-01 8.63e+00 0.2 4.56e+04 - 1.00e+00 3.31e-01f 1
36 2.5595903e+08 9.68e-02 4.45e+04 -1.7 3.13e+04 - 1.00e+00 5.00e-01f 2
37 9.5896907e+05 9.69e-01 2.09e+05 -2.5 1.18e+04 - 1.00e+00 1.00e+00f 1
38 1.0215080e+10 9.60e-01 2.15e+05 -2.5 1.00e+05 - 1.00e+00 1.00e+00h 1
39 1.9190691e+10 5.97e-02 4.31e+05 -2.5 9.67e+04 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
40 3.6729262e+07 8.83e-01 3.34e+06 -2.5 1.00e+05 - 1.00e+00 1.00e+00f 1
41 2.1363532e+12 8.79e-01 3.67e+06 -2.5 1.45e+06 - 1.00e+00 1.00e+00h 1
42 3.7010171e+12 1.13e-01 6.64e+06 -2.5 1.26e+06 - 1.00e+00 1.00e+00h 1
43 2.8512383e+10 7.70e-01 2.44e+07 -2.5 1.39e+06 - 1.00e+00 1.00e+00f 1
44 8.3324853e+13 7.56e-01 2.81e+07 -2.5 8.70e+06 - 1.00e+00 1.00e+00h 1
45 1.1867571e+14 2.10e-01 4.78e+07 -2.5 6.23e+06 - 1.00e+00 1.00e+00h 1
46 4.3825285e+12 5.52e-01 6.94e+07 -2.5 7.54e+06 - 1.00e+00 1.00e+00f 1
47 2.9193897e+14 4.83e-01 7.29e+07 -2.5 1.33e+07 - 1.00e+00 1.00e+00h 1
48 1.9270328e+14 4.45e-01 1.55e+08 -2.5 7.28e+06 - 1.00e+00 1.00e+00f 1
49 1.5322309e+14 3.60e-02 4.23e+07 -2.5 2.51e+06 - 1.00e+00 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
50 4.0213041e+13 1.47e-03 1.01e+07 -2.5 8.84e+06 - 1.00e+00 5.00e-01f 2
51 2.1909501e+07 1.78e+00 1.36e+09 -5.7 4.49e+06 - 1.00e+00 1.00e+00f 1
52r 2.1909501e+07 1.78e+00 9.99e+02 0.2 0.00e+00 - 0.00e+00 4.77e-07R 22
53r 6.2264681e+07 9.52e-01 6.60e+02 0.2 1.43e+06 - 1.00e+00 2.94e-03f 1
54 2.9605877e+07 4.56e-02 2.39e+04 -5.7 8.24e+03 - 1.00e+00 5.00e-01f 2
55 7.5041437e+04 6.23e-01 2.12e+05 -5.7 3.84e+03 - 1.00e+00 1.00e+00f 1
56 3.8044524e+09 6.20e-01 2.54e+05 -5.7 5.74e+04 - 1.00e+00 1.00e+00h 1
57 4.8409848e+09 1.71e-01 3.17e+05 -5.7 3.10e+04 - 1.00e+00 1.00e+00h 1
58 1.4096798e+08 5.45e-01 5.83e+05 -5.7 4.80e+04 - 1.00e+00 1.00e+00f 1
59 2.0273680e+10 4.99e-01 6.42e+05 -5.7 1.16e+05 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
60 1.6357818e+10 3.26e-01 1.05e+06 -5.7 5.03e+04 - 1.00e+00 1.00e+00f 1
61 4.5791481e+09 1.59e-01 2.86e+05 -5.7 5.85e+04 - 1.00e+00 1.00e+00f 1
62 1.5986897e+09 1.44e-02 6.17e+04 -5.7 4.66e+04 - 1.00e+00 5.00e-01f 2
63 3.9919794e+08 1.14e-04 2.86e+04 -5.7 2.87e+04 - 1.00e+00 5.00e-01f 2
64 1.3061165e+00 1.96e+00 6.01e+06 -5.7 1.41e+04 - 1.00e+00 1.00e+00f 1
65r 1.3061165e+00 1.96e+00 9.99e+02 0.3 0.00e+00 - 0.00e+00 4.77e-07R 22
66r 2.6019454e+00 9.77e-01 4.87e+02 0.3 1.00e+03 - 1.00e+00 1.94e-03f 1
67 1.2554928e+00 2.21e-02 4.85e+00 -5.7 1.65e+00 - 1.00e+00 5.00e-01f 2
68 9.7999887e-04 5.40e-01 7.80e+01 -5.7 7.92e-01 - 1.00e+00 1.00e+00f 1
69 3.6572559e+02 5.39e-01 9.38e+01 -5.7 1.73e+01 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
70 4.3470356e+02 1.43e-01 9.63e+01 -5.7 7.69e+00 - 1.00e+00 1.00e+00h 1
71 1.4059476e+02 2.44e-02 2.93e+01 -5.7 1.44e+01 - 1.00e+00 5.00e-01f 2
72 3.5786237e+01 5.20e-04 8.68e+00 -5.7 8.51e+00 - 1.00e+00 5.00e-01f 2
73 7.4391281e-05 3.59e-01 1.54e+02 -5.7 4.23e+00 - 1.00e+00 1.00e+00f 1
74 5.3687560e+02 3.59e-01 1.78e+02 -5.7 1.95e+01 - 1.00e+00 1.00e+00h 1
75 5.7064266e+02 7.02e-02 1.09e+02 -5.7 4.95e+00 - 1.00e+00 1.00e+00h 1
76 1.6408381e+02 6.79e-03 2.64e+01 -5.7 1.67e+01 - 1.00e+00 5.00e-01f 2
77 4.1148273e+01 3.71e-05 9.08e+00 -5.7 9.10e+00 - 1.00e+00 5.00e-01f 2
78 5.1696817e-05 3.31e-02 1.12e+01 -5.7 4.54e+00 - 1.00e+00 1.00e+00f 1
79 1.7827388e-02 3.14e-02 1.08e+01 -5.7 9.57e-02 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
80 1.4428077e-02 3.91e-03 1.69e+00 -5.7 1.11e-02 - 1.00e+00 1.00e+00h 1
81 3.7713449e-03 2.69e-04 1.93e-01 -5.7 8.37e-02 - 1.00e+00 5.00e-01h 2
82 5.0116310e-05 2.32e-03 9.17e-01 -5.7 4.31e-02 - 1.00e+00 1.00e+00h 1
83 5.0578118e-05 2.18e-04 8.63e-02 -5.7 5.30e-04 - 1.00e+00 1.00e+00h 1
84 4.9998894e-05 2.30e-05 9.01e-03 -5.7 5.28e-04 - 1.00e+00 1.00e+00h 1
85 5.0000001e-05 2.08e-08 8.15e-06 -5.7 4.64e-06 - 1.00e+00 1.00e+00h 1
86 5.0000000e-05 2.07e-13 8.34e-11 -8.6 5.17e-08 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 86
(scaled) (unscaled)
Objective...............: 4.9999999999989723e-05 4.9999999999989723e-05
Dual infeasibility......: 8.3368130107674965e-11 8.3368130107674965e-11
Constraint violation....: 2.0738966099997924e-13 2.0738966099997924e-13
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 8.3368130107674965e-11 8.3368130107674965e-11
Number of objective function evaluations = 240
Number of objective gradient evaluations = 86
Number of equality constraint evaluations = 240
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 90
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 86
Total seconds in IPOPT = 0.032
EXIT: Optimal Solution Found.
1.0050000000000423
0.0050000000000412674
(b)
\[min_{x,y} f(x,y) = (x-1.01)^{2} + y^{2}\]
\[s.t \;\;\;\; \frac{x}{y + 1} = 1\]
import pyomo.environ as pyo
model = pyo.ConcreteModel()
model.x = pyo.Var(initialize=5.0)
model.y = pyo.Var(initialize=5.0)
def obj_rule(m):
return (m.x-1.01)**2 + m.y**2
model.obj = pyo.Objective(rule=obj_rule)
def con_rule(m):
return m.x / (m.y + 1.0) == 1.0
model.con = pyo.Constraint(rule=con_rule)
solver = pyo.SolverFactory('ipopt')
solver.solve(model, tee=True)
print(pyo.value(model.x))
print(pyo.value(model.y))
Ipopt 3.14.19:
******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
Ipopt is released as open source code under the Eclipse Public License (EPL).
For more information visit https://github.com/coin-or/Ipopt
******************************************************************************
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 2
Number of nonzeros in inequality constraint Jacobian.: 0
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.................: 1
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 4.0920100e+01 1.67e-01 9.83e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 4.0023722e-01 1.49e+00 2.82e+01 -1.0 5.39e+00 - 1.00e+00 1.00e+00f 1
2 1.0494629e+01 1.04e+00 2.24e+01 -1.0 2.58e+00 - 1.00e+00 1.00e+00h 1
3 5.4394796e+00 1.78e-01 8.48e+00 -1.0 1.23e+00 - 1.00e+00 1.00e+00f 1
4 1.0538196e-01 1.52e-01 3.90e+00 -1.0 1.75e+00 - 1.00e+00 1.00e+00f 1
5 4.6918851e-02 2.17e-02 1.20e+00 -1.0 1.63e-01 - 1.00e+00 1.00e+00h 1
6 1.3071752e-04 2.88e-03 1.19e-01 -1.0 1.62e-01 - 1.00e+00 1.00e+00h 1
7 4.9833353e-05 1.79e-05 1.02e-03 -2.5 6.25e-03 - 1.00e+00 1.00e+00h 1
8 5.0000013e-05 1.28e-09 6.30e-08 -5.7 9.00e-05 - 1.00e+00 1.00e+00h 1
9 5.0000000e-05 0.00e+00 3.30e-16 -8.6 4.58e-09 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 9
(scaled) (unscaled)
Objective...............: 4.9999999999999169e-05 4.9999999999999169e-05
Dual infeasibility......: 3.2959746043559335e-16 3.2959746043559335e-16
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 3.2959746043559335e-16 3.2959746043559335e-16
Number of objective function evaluations = 10
Number of objective gradient evaluations = 10
Number of equality constraint evaluations = 10
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 10
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 9
Total seconds in IPOPT = 0.004
EXIT: Optimal Solution Found.
1.0050000000000001
0.0050000000000000235
(c)
\[min_{x,y} f(x,y) = (x-1.01)^{2} + y^{2}\]
\[s.t \;\;\;\; y = x - 1\]
import pyomo.environ as pyo
model = pyo.ConcreteModel()
model.x = pyo.Var(initialize=5.0)
model.y = pyo.Var(initialize=5.0)
def obj_rule(m):
return (m.x-1.01)**2 + m.y**2
model.obj = pyo.Objective(rule=obj_rule)
def con_rule(m):
return m.y == m.x - 1.0
model.con = pyo.Constraint(rule=con_rule)
solver = pyo.SolverFactory('ipopt')
solver.solve(model, tee=True)
print(pyo.value(model.x))
print(pyo.value(model.y))
Ipopt 3.14.19:
******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
Ipopt is released as open source code under the Eclipse Public License (EPL).
For more information visit https://github.com/coin-or/Ipopt
******************************************************************************
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 2
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 2
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.................: 1
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 4.0920100e+01 1.00e+00 8.99e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 5.0000000e-05 0.00e+00 4.44e-16 -1.0 5.00e+00 - 1.00e+00 1.00e+00f 1
Number of Iterations....: 1
(scaled) (unscaled)
Objective...............: 5.0000000000000090e-05 5.0000000000000090e-05
Dual infeasibility......: 4.4408920985006262e-16 4.4408920985006262e-16
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 0.0000000000000000e+00 0.0000000000000000e+00
Overall NLP error.......: 4.4408920985006262e-16 4.4408920985006262e-16
Number of objective function evaluations = 2
Number of objective gradient evaluations = 2
Number of equality constraint evaluations = 2
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 2
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 1
Total seconds in IPOPT = 0.001
EXIT: Optimal Solution Found.
1.005
0.004999999999999893
(d) Bounds and initialization can be very helpful when solving nonlinear optimization problems. Starting with the code below, resolve the original problem, but add bounds, \(y \geq 0\). Note the number of iterations and quality of solution, and compare with what you found in 1.2 (a).
import pyomo.environ as pyo
model = pyo.ConcreteModel()
model.x = pyo.Var(initialize=5.0)
model.y = pyo.Var(initialize=5.0, bounds=(0,None))
def obj_rule(m):
return (m.x-1.01)**2 + m.y**2
model.obj = pyo.Objective(rule=obj_rule)
def con_rule(m):
return (m.x - 1.0) / m.y == 1.0
model.con = pyo.Constraint(rule=con_rule)
solver = pyo.SolverFactory('ipopt')
solver.solve(model, tee=True)
print(pyo.value(model.x))
print(pyo.value(model.y))
Ipopt 3.14.19:
******************************************************************************
This program contains Ipopt, a library for large-scale nonlinear optimization.
Ipopt is released as open source code under the Eclipse Public License (EPL).
For more information visit https://github.com/coin-or/Ipopt
******************************************************************************
This is Ipopt version 3.14.19, running with linear solver MUMPS 5.7.3.
Number of nonzeros in equality constraint Jacobian...: 2
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 3
Total number of variables............................: 2
variables with only lower bounds: 1
variables with lower and upper bounds: 0
variables with only upper bounds: 0
Total number of equality constraints.................: 1
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 4.0920100e+01 2.00e-01 9.38e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
1 9.7049410e-01 1.89e+01 2.29e+03 -1.0 5.19e+00 - 1.00e+00 9.54e-01f 1
2 2.8303563e+03 1.85e+01 2.25e+03 -1.0 5.21e+01 - 1.88e-02 1.00e+00h 1
3 2.9704435e+03 9.12e+00 1.14e+03 -1.0 2.64e+00 - 1.00e+00 1.00e+00h 1
4 2.5293245e+03 4.09e+00 5.51e+02 -1.0 4.89e+00 - 8.94e-01 1.00e+00f 1
5 1.7547126e+03 1.52e+00 2.52e+02 -1.0 1.04e+01 - 1.00e+00 1.00e+00f 1
6 6.8047428e+02 1.49e-01 8.49e+01 -1.0 1.93e+01 - 1.00e+00 1.00e+00f 1
7 7.4721920e+00 8.01e-01 2.32e+02 -1.0 1.91e+01 - 1.00e+00 1.00e+00f 1
8 8.1969347e+03 7.77e-01 2.62e+02 -1.0 8.57e+01 - 3.37e-02 1.00e+00h 1
9 1.1605265e+04 2.36e-01 4.89e+02 -1.0 6.40e+01 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
10 5.5411085e+02 5.19e-01 5.83e+02 -1.0 7.36e+01 - 1.00e+00 1.00e+00f 1
11 1.7377623e+04 4.22e-01 5.65e+02 -1.0 9.29e+01 - 3.00e-01 1.00e+00h 1
12 1.1573362e+04 1.25e-01 1.63e+02 -1.0 6.64e+01 - 1.00e+00 5.00e-01f 2
13 3.3128549e+03 8.55e-03 9.20e+01 -1.0 8.52e+01 - 1.00e+00 4.70e-01f 2
14 7.1443870e-01 8.34e-01 1.80e+03 -1.0 4.11e+01 - 1.00e+00 9.85e-01f 1
15 3.2462198e-05 4.75e-01 1.05e+03 -1.0 1.73e+02 - 1.00e+00 4.30e-03f 1
16 9.8963008e-05 2.74e-01 8.52e+02 -1.0 1.41e+00 - 1.00e+00 4.23e-03h 1
17 9.2105755e-05 2.40e-01 5.57e+02 -1.0 3.56e-04 - 1.00e+00 1.00e+00f 1
18 7.9772743e-05 1.58e-01 3.60e+02 -1.0 7.13e-04 - 1.00e+00 1.00e+00h 1
19 6.9741224e-05 6.86e-02 1.47e+02 -1.0 7.40e-04 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
20 4.9888137e-05 3.97e-02 6.88e+01 -1.0 2.39e-03 - 1.00e+00 1.00e+00h 1
21 3.3950490e-04 3.02e-02 4.89e+01 -1.0 1.32e-02 - 1.00e+00 1.00e+00f 1
22 3.7536170e-03 1.95e-02 3.11e+01 -1.0 3.12e-02 - 1.00e+00 1.00e+00h 1
23 1.2981580e-02 8.58e-03 1.36e+01 -1.0 3.74e-02 - 1.00e+00 1.00e+00h 1
24 3.4681398e-02 3.23e-03 5.01e+00 -1.0 5.13e-02 - 1.00e+00 1.00e+00h 1
25 4.7179466e-02 4.50e-04 7.03e-01 -1.0 2.21e-02 - 1.00e+00 1.00e+00h 1
26 1.7082443e-02 2.83e-04 3.55e-01 -1.7 6.13e-02 - 1.00e+00 1.00e+00f 1
27 1.0216180e-02 7.79e-05 1.09e-01 -1.7 2.10e-02 - 1.00e+00 1.00e+00h 1
28 3.1289054e-03 5.64e-05 3.27e-02 -2.5 3.21e-02 - 1.00e+00 1.00e+00h 1
29 1.4987215e-03 2.18e-05 2.21e-02 -2.5 1.23e-02 - 1.00e+00 1.00e+00h 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
30 3.8554063e-04 1.69e-05 4.50e-03 -3.8 1.40e-02 - 1.00e+00 1.00e+00h 1
31 1.3832051e-04 9.18e-06 4.39e-03 -3.8 6.31e-03 - 1.00e+00 1.00e+00h 1
32 8.9968997e-05 2.11e-06 9.99e-06 -3.8 2.18e-03 - 1.00e+00 1.00e+00h 1
33 5.4301800e-05 9.79e-07 2.80e-03 -5.7 3.00e-03 - 1.00e+00 1.00e+00h 1
34 5.0216846e-05 2.09e-07 1.11e-04 -5.7 1.14e-03 - 1.00e+00 1.00e+00h 1
35 5.0020268e-05 9.37e-09 2.55e-05 -5.7 2.29e-04 - 1.00e+00 1.00e+00h 1
36 5.0016425e-05 1.85e-11 1.06e-08 -5.7 1.00e-05 - 1.00e+00 1.00e+00h 1
37 5.0000006e-05 3.39e-13 3.16e-06 -8.6 8.89e-05 - 1.00e+00 1.00e+00h 1
38 5.0000000e-05 5.55e-15 4.02e-13 -8.6 1.58e-06 - 1.00e+00 1.00e+00h 1
Number of Iterations....: 38
(scaled) (unscaled)
Objective...............: 5.0000000031647193e-05 5.0000000031647193e-05
Dual infeasibility......: 4.0225288377992996e-13 4.0225288377992996e-13
Constraint violation....: 5.5511151231257827e-15 5.5511151231257827e-15
Variable bound violation: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 2.5158913589264014e-09 2.5158913589264014e-09
Overall NLP error.......: 2.5158913589264014e-09 2.5158913589264014e-09
Number of objective function evaluations = 44
Number of objective gradient evaluations = 39
Number of equality constraint evaluations = 44
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 39
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 38
Total seconds in IPOPT = 0.013
EXIT: Optimal Solution Found.
1.0050001257911454
0.005000125791145421