## Constrained Minimization Using `patternsearch`

and
`Optimize`

Live Editor Task

This example shows how to solve a constrained minimization problem using both the Optimize Live Editor task, which offers a visual approach, and the command line.

### Problem Description

The problem involves using linear and nonlinear constraints when minimizing a
nonlinear function with `patternsearch`

. The objective function
is

$$F(x)=\frac{1}{2}{x}^{T}Hx+{f}^{T}x,$$

where

H = [36 17 19 12 8 15; 17 33 18 11 7 14; 19 18 43 13 8 16; 12 11 13 18 6 11; 8 7 8 6 9 8; 15 14 16 11 8 29]; f = [ 20 15 21 18 29 24 ]'; F = @(x)0.5*x'*H*x + f'*x;

The linear constraints are

$$\begin{array}{c}A\cdot x\le b,\\ Aeq\cdot x=beq,\end{array}$$

where

A = [-8 7 3 -4 9 0]; b = 7; Aeq = [7 1 8 3 3 3; 5 0 -5 1 -5 8; -2 -6 7 1 1 9; 1 -1 2 -2 3 -3]; beq = [84 62 65 1]';

Enter the preceding code sections to get the problem variables into your workspace before proceeding.

### Solve Using `patternsearch`

in Optimize Live Editor Task

Create a new live script by clicking the

**New Live Script**button in the**File**section on the**Home**tab.Insert an Optimize Live Editor task. Click the

**Insert**tab and then, in the**Code**section, select**Task > Optimize**.For this example, choose the solver-based task.

**Specify Problem Type**In the

**Specify problem type**section of the task, click the**Objective > Nonlinear**button.Click the

**Constraints > Linear inequality**and**Linear equality**buttons.Select

**Solver > patternsearch - Pattern search**.**Select Problem Data**Enter the problem variables in the

**Select problem data**section of the task. To specify the objective function, select**Objective function > Function handle**and choose**F**.Set the inequality constraints to

`A`

and`b`

. Set the equality constraints to`Aeq`

and`beq`

.To set the initial point, you first need to create a new section above the task. To do so, click the

**Section Break**button on the**Insert**tab. In the new section above the task, enter the following code for the initial point.x0 = [2 1 0 9 1 0]';

Run the section to place

`x0`

into the workspace. To run the section, place the cursor in the section and press**Ctrl+Enter**or click the blue striped bar to the left of the line number.In the

**Select problem data**section of the task, set`x0`

as the initial point.**Specify Solver Options**Because this problem is linearly constrained, specify an additional solver option. Expand the

**Specify solver options**section of the task, and then click the**Add**button. Set the**Poll settings > Poll method**to`GSSPositiveBasis2N`

. For more information about the efficiency of the GSS poll methods for linearly constrained problems, see Compare the Efficiency of Poll Options.**Set Display Options**In the

**Display progress**section of the task, select the**Best value**and**Mesh size**plot functions.Your setup looks like this:

**Run Solver and Examine Results**To run the solver, click the options button

**⁝**at the top right of the task window, and select**Run Section**.The plots appear in a separate figure window and in the task output area.

To obtain the solution point and objective function value at the solution, look at the top of the task.

The

`Optimize`

Live Editor task returns the solution in a variable named`solution`

and returns the objective function value in a variable named`objectiveValue`

. View these values by entering the following code in the section below the task and then running the section, or by entering the code at the MATLAB^{®}command line.disp(solution)

8.5165 -6.1094 4.0989 1.2877 -4.2348 2.1812

disp(objectiveValue)

1.9195e+03

**Include Nonlinear Constraints**Add the following nonlinear constraints to the problem.

$$\begin{array}{c}-1.5+{x}_{1}{x}_{2}+{x}_{1}-{x}_{2}\le 0\\ -{x}_{1}{x}_{2}-10\le 0.\end{array}$$

To include these constraints, first click the

**Constraints > Nonlinear**button.In the

**Select problem data**section, under**Constraints**, select**Nonlinear > Local function**and then click the**New**button. The function appears in a new section below the task. Edit the resulting code to contain the following lines.function [c, ceq] = double_ineq(x) c = [-1.5 + x(1)*x(2) + x(1) - x(2); -x(1)*x(2) - 10]; ceq = []; end

In the

**Nonlinear**constraints section, select**double_ineq**.The nonlinear constraint algorithm causes

`patternsearch`

to take many function evaluations. In the**Specify solver options**section, click the plus sign to the right of the current options to display additional options. Then increase the maximum function evaluation limit to 5e4.Run the task again to rerun the optimization.

View the solution and objective function value.

disp(solution)

7.2083 -1.3873 4.9579 -3.1393 -3.1843 4.7457

disp(objectiveValue)

2.4018e+03

The objective function value is higher than the value in the problem without nonlinear constraints. The previous solution is not feasible with respect to the nonlinear constraints.

The plots show many fewer iterations than before because the nonlinear constraint
algorithm changes the `patternsearch`

algorithm to include
another outer loop to solve a modified problem. The outer loop reduces the
modification to the problem at each major iteration. In this case, the algorithm
makes only four outer iterations. For algorithm details, see Nonlinear Constraint Solver Algorithm for Pattern Search.

### Solve Using `patternsearch`

at the Command Line

To solve the original problem (only linear constraints) at the command line, execute the following code.

x0 = [2 1 0 9 1 0]'; options = optimoptions('patternsearch',... 'PollMethod','GSSPositiveBasis2N',... 'PlotFcn',{'psplotbestf','psplotmeshsize'}); lb = []; ub = []; nonlcon = []; [x,fval] = patternsearch(F,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

Optimization terminated: mesh size less than options.MeshTolerance. x = 8.5165 -6.1094 4.0989 1.2877 -4.2348 2.1812 fval = 1.9195e+03

`patternsearch`

generates the first pair of plots shown in the
Optimize Live Editor task example.

To include the nonlinear constraints, save the following code to a file named
`double_ineq.m`

on the MATLAB path.

function [c, ceq] = double_ineq(x) c = [-1.5 + x(1)*x(2) + x(1) - x(2); -x(1)*x(2) - 10]; ceq = []; end

To allow the solver to run to completion with nonlinear constraints, increase the allowed number of function evaluations.

options.MaxFunctionEvaluations = 5e4;

Solve the problem including nonlinear constraints.

nonlcon = @double_ineq; [x,fval] = patternsearch(F,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)

Optimization terminated: mesh size less than options.MeshTolerance and constraint violation is less than options.ConstraintTolerance. x = 7.2083 -1.3873 4.9579 -3.1393 -3.1843 4.7457 fval = 2.4018e+03

`patternsearch`

also generates the second pair of plots shown
in the Optimize Live Editor task example.

Both the Optimize Live Editor task and the command line allow you to formulate and solve problems, and they give identical results. The command line is more streamlined, but provides less help for choosing a solver, setting up the problem, and choosing options such as plot functions. You can also start a problem using Optimize, and then generate code for command line use, as in Constrained Nonlinear Problem Using Optimize Live Editor Task or Solver.