Julia JuMP + IpOpt

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VERSION

v"1.0.3"

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using JuMP
using Ipopt

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m = Model(solver = IpoptSolver())
@variable(m, x, start = 0.0)
@variable(m, y, start = 0.0)

@NLobjective(m, Min, (1-x)^2 + 100(y-x^2)^2)

solve(m)
println("x = ", getvalue(x), " y = ", getvalue(y))

******************************************************************************
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 http://projects.coin-or.org/Ipopt
******************************************************************************

This is Ipopt version 3.12.10, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

Number of nonzeros in equality constraint Jacobian...:        0
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.................:        0
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  1.0000000e+00 0.00e+00 2.00e+00  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  9.5312500e-01 0.00e+00 1.25e+01  -1.0 1.00e+00    -  1.00e+00 2.50e-01f  3
   2  4.8320569e-01 0.00e+00 1.01e+00  -1.0 9.03e-02    -  1.00e+00 1.00e+00f  1
   3  4.5708829e-01 0.00e+00 9.53e+00  -1.0 4.29e-01    -  1.00e+00 5.00e-01f  2
   4  1.8894205e-01 0.00e+00 4.15e-01  -1.0 9.51e-02    -  1.00e+00 1.00e+00f  1
   5  1.3918726e-01 0.00e+00 6.51e+00  -1.7 3.49e-01    -  1.00e+00 5.00e-01f  2
   6  5.4940990e-02 0.00e+00 4.51e-01  -1.7 9.29e-02    -  1.00e+00 1.00e+00f  1
   7  2.9144630e-02 0.00e+00 2.27e+00  -1.7 2.49e-01    -  1.00e+00 5.00e-01f  2
   8  9.8586451e-03 0.00e+00 1.15e+00  -1.7 1.10e-01    -  1.00e+00 1.00e+00f  1
   9  2.3237475e-03 0.00e+00 1.00e+00  -1.7 1.00e-01    -  1.00e+00 1.00e+00f  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  10  2.3797236e-04 0.00e+00 2.19e-01  -1.7 5.09e-02    -  1.00e+00 1.00e+00f  1
  11  4.9267371e-06 0.00e+00 5.95e-02  -1.7 2.53e-02    -  1.00e+00 1.00e+00f  1
  12  2.8189505e-09 0.00e+00 8.31e-04  -2.5 3.20e-03    -  1.00e+00 1.00e+00f  1
  13  1.0095040e-15 0.00e+00 8.68e-07  -5.7 9.78e-05    -  1.00e+00 1.00e+00f  1
  14  1.3288608e-28 0.00e+00 2.02e-13  -8.6 4.65e-08    -  1.00e+00 1.00e+00f  1

Number of Iterations....: 14

                                   (scaled)                 (unscaled)
Objective...............:   1.3288608467480825e-28    1.3288608467480825e-28
Dual infeasibility......:   2.0183854587685121e-13    2.0183854587685121e-13
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00
Overall NLP error.......:   2.0183854587685121e-13    2.0183854587685121e-13


Number of objective function evaluations             = 36
Number of objective gradient evaluations             = 15
Number of equality constraint evaluations            = 0
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 0
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 14
Total CPU secs in IPOPT (w/o function evaluations)   =      0.202
Total CPU secs in NLP function evaluations           =      0.051

EXIT: Optimal Solution Found.
x = 0.9999999999999899 y = 0.9999999999999792

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# adding a (linear) constraint
@constraint(m, x + y == 10)
solve(m)
println("x = ", getvalue(x), " y = ", getvalue(y))

This is Ipopt version 3.12.10, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

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  1.3288608e-28 8.00e+00 1.56e-13  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  5.0288811e+03 1.78e-15 1.04e+04  -1.0 5.34e+00    -  1.00e+00 1.00e+00h  1
   2  2.9288883e+02 0.00e+00 2.25e+03  -1.0 7.07e-01    -  1.00e+00 1.00e+00f  1
   3  5.6537755e+00 0.00e+00 2.04e+02  -1.0 2.30e-01    -  1.00e+00 1.00e+00f  1
   4  2.8949941e+00 0.00e+00 2.39e+00  -1.0 2.55e-02    -  1.00e+00 1.00e+00f  1
   5  2.8946076e+00 0.00e+00 3.43e-04  -1.0 3.07e-04    -  1.00e+00 1.00e+00f  1
   6  2.8946076e+00 0.00e+00 7.87e-12  -5.7 4.42e-08    -  1.00e+00 1.00e+00f  1

Number of Iterations....: 6

                                   (scaled)                 (unscaled)
Objective...............:   2.8946075504894604e+00    2.8946075504894604e+00
Dual infeasibility......:   7.8660411517716966e-12    7.8660411517716966e-12
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00
Overall NLP error.......:   7.8660411517716966e-12    7.8660411517716966e-12


Number of objective function evaluations             = 7
Number of objective gradient evaluations             = 7
Number of equality constraint evaluations            = 7
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 7
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 6
Total CPU secs in IPOPT (w/o function evaluations)   =      0.002
Total CPU secs in NLP function evaluations           =      0.005

EXIT: Optimal Solution Found.
x = 2.7011471240982194 y = 7.2988528759017814

IpOpt test

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using Ipopt

m = Model(solver=IpoptSolver())
@variable(m, 0 <= x <= 2 )
@variable(m, 0 <= y <= 30 )

@objective(m, Min, x*x+ 2x*y + y*y )
@constraint(m, x + y >= 1 )

print(m)

status = solve(m)

Min x² + 2 x*y + y²
Subject to
 x + y ≥ 1
 0 ≤ x ≤ 2
 0 ≤ y ≤ 30
This is Ipopt version 3.12.10, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

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:        2
                     variables with only upper bounds:        0
Total number of equality constraints.................:        0
Total number of inequality constraints...............:        1
        inequality constraints with only lower bounds:        1
   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  3.9999920e-04 9.80e-01 6.40e-01  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  3.2681447e-03 9.43e-01 7.36e-01  -1.0 3.92e-01    -  3.28e-02 4.75e-02h  1
   2  1.1040193e+00 0.00e+00 1.51e+00  -1.0 7.62e-01    -  6.37e-01 1.00e+00f  1
   3  1.1237215e+00 0.00e+00 2.18e-03  -1.0 2.05e-02    -  9.99e-01 1.00e+00f  1
   4  1.0015482e+00 0.00e+00 2.83e-08  -2.5 5.93e-02    -  1.00e+00 1.00e+00f  1
   5  1.0001417e+00 0.00e+00 1.50e-09  -3.8 3.92e-03    -  1.00e+00 1.00e+00f  1
   6  1.0000018e+00 0.00e+00 1.84e-11  -5.7 1.81e-03    -  1.00e+00 1.00e+00f  1
   7  9.9999998e-01 0.00e+00 2.49e-14  -8.6 2.00e-04    -  1.00e+00 1.00e+00f  1

Number of Iterations....: 7

                                   (scaled)                 (unscaled)
Objective...............:   9.9999998250465849e-01    9.9999998250465849e-01
Dual infeasibility......:   2.4868995751603507e-14    2.4868995751603507e-14
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   3.1573161178575155e-09    3.1573161178575155e-09
Overall NLP error.......:   3.1573161178575155e-09    3.1573161178575155e-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 CPU secs in IPOPT (w/o function evaluations)   =      0.157
Total CPU secs in NLP function evaluations           =      0.144

EXIT: Optimal Solution Found.

:Optimal

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print(status)

Optimal

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println("x = ", getvalue(x), "and y = ", getvalue(y))

x = 0.27864390938942063and y = 0.7213560818629086

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