# Constrained Minimization Using `ga`

, Problem-Based

This example shows how to minimize an objective function, subject to nonlinear inequality constraints and bounds, using `ga`

in the problem-based approach. For a solver-based version of this problem, see Constrained Minimization Using the Genetic Algorithm.

### Constrained Minimization Problem

For this problem, the fitness function to minimize is a simple function of 2-D variables `X`

and `Y`

.

camxy = @(X,Y)(4 - 2.1.*X.^2 + X.^4./3).*X.^2 + X.*Y + (-4 + 4.*Y.^2).*Y.^2;

This function is described in Dixon and Szego [1].

Additionally, the problem has nonlinear constraints and bounds.

x*y + x - y + 1.5 <= 0 (nonlinear constraint) 10 - x*y <= 0 (nonlinear constraint) 0 <= x <= 1 (bound) 0 <= y <= 13 (bound)

Plot the nonlinear constraint region on a surface plot of the fitness function. The constraints limit the solution to the small region above both red curves.

x1 = linspace(0,1); y1 = (-x1 - 1.5)./(x1 - 1); y2 = 10./x1; [X,Y] = meshgrid(x1,linspace(0,13)); Z = camxy(X,Y); surf(X,Y,Z,"LineStyle","none") hold on z1 = camxy(x1,y1); z2 = camxy(x1,y2); plot3(x1,y1,z1,'r-',x1,y2,z2,'r-') xlim([0 1]) ylim([0 13]) zlim([0,max(Z,[],"all")]) hold off

### Create Optimization Variables, Problem, and Constraints

To set up this problem, create optimization variables `x`

and `y`

. Set the bounds when you create the variables.

x = optimvar("x","LowerBound",0,"UpperBound",1); y = optimvar("y","LowerBound",0,"UpperBound",13);

Create the objective as an optimization expression.

cam = camxy(x,y);

Create an optimization problem with this objective function.

`prob = optimproblem("Objective",cam);`

Create the two nonlinear inequality constraints, and include them in the problem.

prob.Constraints.cons1 = x*y + x - y + 1.5 <= 0; prob.Constraints.cons2 = 10 - x*y <= 0;

Review the problem.

show(prob)

OptimizationProblem : Solve for: x, y minimize : (((((4 - (2.1 .* x.^2)) + (x.^4 ./ 3)) .* x.^2) + (x .* y)) + (((-4) + (4 .* y.^2)) .* y.^2)) subject to cons1: ((((x .* y) + x) - y) + 1.5) <= 0 subject to cons2: (10 - (x .* y)) <= 0 variable bounds: 0 <= x <= 1 0 <= y <= 13

### Solve Problem

Solve the problem, specifying the `ga`

solver.

[sol,fval] = solve(prob,"Solver","ga")

Solving problem using ga. Optimization finished: average change in the fitness value less than options.FunctionTolerance and constraint violation is less than options.ConstraintTolerance.

`sol = `*struct with fields:*
x: 0.8122
y: 12.3103

fval = 9.1268e+04

### Add Visualization

To observe the solver's progress, specify options that select two plot functions. The plot function `gaplotbestf`

plots the best objective function value at every iteration, and the plot function `gaplotmaxconstr`

plots the maximum constraint violation at every iteration. Set these two plot functions in a cell array. Also, display information about the solver's progress in the Command Window by setting the `Display`

option to `'iter'`

.

options = optimoptions(@ga,... 'PlotFcn',{@gaplotbestf,@gaplotmaxconstr},... 'Display','iter');

Run the solver, including the `options`

argument.

[sol,fval] = solve(prob,"Solver","ga","Options",options)

Solving problem using ga. Single objective optimization: 2 Variable(s) 2 Nonlinear inequality constraint(s) Options: CreationFcn: @gacreationuniform CrossoverFcn: @crossoverscattered SelectionFcn: @selectionstochunif MutationFcn: @mutationadaptfeasible Best Max Stall Generation Func-count f(x) Constraint Generations 1 2520 91357.8 0 0 2 4982 91324.1 4.55e-05 0 3 7914 97166.6 0 0 4 16145 91268.4 0.0009997 0 Optimization finished: average change in the fitness value less than options.FunctionTolerance and constraint violation is less than options.ConstraintTolerance.

`sol = `*struct with fields:*
x: 0.8123
y: 12.3103

fval = 9.1268e+04

Nonlinear constraints cause `ga`

to solve many subproblems at each iteration. As shown in both the plots and the iterative display, the solution process has few iterations. However, the `Func-count`

column in the iterative display shows many function evaluations per iteration.

### Unsupported Functions

If your objective or nonlinear constraint functions are not supported (see Supported Operations for Optimization Variables and Expressions), use `fcn2optimexpr`

to convert them to a form suitable for the problem-based approach. For example, suppose that instead of the constraint $$xy\ge 10$$, you have the constraint $${I}_{1}(x)+{I}_{1}(y)\ge 10$$, where $${I}_{1}(x)$$ is the modified Bessel function `besseli(1,x)`

. (The Bessel functions are not supported functions.) Create this constraint using `fcn2optimexpr`

. First, create an optimization expression for $${I}_{1}(x)+{I}_{1}(y)$$.

bfun = fcn2optimexpr(@(t,u)besseli(1,t) + besseli(1,u),x,y);

Next, replace the constraint `cons2`

with the constraint `bfun >= 10`

.

prob.Constraints.cons2 = bfun >= 10;

Solve the problem. The solution is different because the constraint region is different.

[sol2,fval2] = solve(prob,"Solver","ga","Options",options)

Solving problem using ga. Single objective optimization: 2 Variable(s) 2 Nonlinear inequality constraint(s) Options: CreationFcn: @gacreationuniform CrossoverFcn: @crossoverscattered SelectionFcn: @selectionstochunif MutationFcn: @mutationadaptfeasible Best Max Stall Generation Func-count f(x) Constraint Generations 1 2512 974.044 0 0 2 4974 960.998 0 0 3 7436 963.12 0 0 4 12001 960.83 0.0009335 0 Optimization finished: average change in the fitness value less than options.FunctionTolerance and constraint violation is less than options.ConstraintTolerance.

`sol2 = `*struct with fields:*
x: 0.4999
y: 3.9979

fval2 = 960.8300

### References

[1] Dixon, L. C. W., and G .P. Szego (eds.). *Towards Global Optimisation 2.* North-Holland: Elsevier Science Ltd., Amsterdam, 1978.

## See Also

`ga`

| `solve`

| `fcn2optimexpr`