Solving systems of nonlinear equations

This example demonstrates how to solve systems of nonlinear equations using a problem-oriented approach.

We will use the functions of the [JuMP.jl] library(https://github.com/jump-dev/JuMP .jl) for the formulation of the optimization problem and the library of nonlinear optimization Ipopt.jl.

Installing Libraries

If the latest version of the package is not installed in your environment JuMP, uncomment and run the cell below:

Pkg.add(["Ipopt", "JuMP"])

#Pkg.add("JuMP");

To launch a new version of the library after the installation is complete, click on the "Personal Account" button:

Then click on the "Stop" button:

Restart the session by clicking the "Start Engee" button:

Task description

We'll find solutions and for the system of nonlinear equations presented below, using a problem-oriented approach:

\begin{cases} exp(-exp(-(x_1+x_2)))=x_2(1+x_1^2) \ \x_1cos(x_2)+x_2sin(x_1)={\frac{{1}}{2}}\end

Connecting libraries

Connect the library JuMP:

using JuMP;

Connect the library of the nonlinear solver Ipopt:

using Ipopt;

Creating an optimization task

Create an optimization task using the function Model() and specify the name of the solver in parentheses.:

sys_prob = Model(Ipopt.Optimizer)

A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: EMPTY_OPTIMIZER
Solver name: Ipopt

Create a variable x containing two values – and :

@variable(sys_prob, x[1:2]);

Set the first non-linear condition using a macro @NLconstraint:

@NLconstraint(sys_prob, exp(-exp(-(x[1] + x[2]))) == x[2] * (1 + x[1]^2))

Set the second non-linear condition using a macro @NLconstraint:

@NLconstraint(sys_prob, x[1] * cos(x[2]) + x[2] * sin(x[1]) == 1/2)

Setting initial values for optimization variables can help to manage the optimization process more effectively. Solve the problem starting from the point [0,0]. To do this, use the function set_start_value() and set the initial values for $x_1$ and $x_2$ :

set_start_value(x[1], 0.0);
set_start_value(x[2], 0.0);

Solving the optimization problem

Solve the optimization problem:

optimize!(sys_prob)

This is Ipopt version 3.14.13, running with linear solver MUMPS 5.6.1.

Number of nonzeros in equality constraint Jacobian...:        4
Number of nonzeros in inequality constraint Jacobian.:        0
Number of nonzeros in Lagrangian Hessian.............:        6

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.................:        2
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  0.0000000e+00 5.00e-01 0.00e+00  -1.0 0.00e+00    -  0.00e+00 0.00e+00   0
   1  0.0000000e+00 3.15e-01 0.00e+00  -1.7 8.73e-01    -  1.00e+00 1.00e+00h  1
   2  0.0000000e+00 5.73e-02 0.00e+00  -1.7 1.98e-01    -  1.00e+00 1.00e+00h  1
   3  0.0000000e+00 8.66e-04 0.00e+00  -2.5 6.84e-02    -  1.00e+00 1.00e+00h  1
   4  0.0000000e+00 2.66e-07 0.00e+00  -5.7 6.23e-04    -  1.00e+00 1.00e+00h  1
   5  0.0000000e+00 2.28e-14 0.00e+00  -8.6 3.20e-07    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 5

                                   (scaled)                 (unscaled)
Objective...............:   0.0000000000000000e+00    0.0000000000000000e+00
Dual infeasibility......:   0.0000000000000000e+00    0.0000000000000000e+00
Constraint violation....:   2.2759572004815709e-14    2.2759572004815709e-14
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   0.0000000000000000e+00    0.0000000000000000e+00
Overall NLP error.......:   2.2759572004815709e-14    2.2759572004815709e-14


Number of objective function evaluations             = 6
Number of objective gradient evaluations             = 6
Number of equality constraint evaluations            = 6
Number of inequality constraint evaluations          = 0
Number of equality constraint Jacobian evaluations   = 6
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations             = 5
Total seconds in IPOPT                               = 0.003

EXIT: Optimal Solution Found.

Save the values x in the variable:

sol_x = value.(x);

Output the optimization results:

println("Solution: x = ", sol_x)

Solution: x = [0.35324661959670156, 0.6060817366414603]

The solutions we found for the system of nonlinear equations:

Conclusion

In this example, we have found solutions for a system of nonlinear equations using a problem-oriented approach. We used the library JuMP for the formulation of our problem and the library of the nonlinear solver Ipopt to find solutions to the problem.