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`patternsearch`

This section presents an example of performing a pattern search on a constrained minimization problem. The example minimizes the function

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

where

$$\begin{array}{c}H=\left[\begin{array}{cccccc}36& 17& 19& 12& 8& 15\\ 17& 33& 18& 11& 7& 14\\ 19& 18& 43& 13& 8& 16\\ 12& 11& 13& 18& 6& 11\\ 8& 7& 8& 6& 9& 8\\ 15& 14& 16& 11& 8& 29\end{array}\right],\\ f=[\begin{array}{cccccc}20& 15& 21& 18& 29& 24\end{array}],\end{array}$$

subject to the constraints

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

where

$$\begin{array}{c}A=[\begin{array}{cccccc}-8& 7& 3& -4& 9& 0\end{array}],\\ b=7,\\ Aeq=\left[\begin{array}{cccccc}7& 1& 8& 3& 3& 3\\ 5& 0& -5& 1& -5& 8\\ -2& -6& 7& 1& 1& 9\\ 1& -1& 2& -2& 3& -3\end{array}\right],\\ beq=[\begin{array}{cccc}84& 62& 65& 1\end{array}].\end{array}$$

To perform a pattern search on the example, first enter

optimtool('patternsearch')

`optimtool`

and then choose
`patternsearch`

from the
@lincontest7

`lincontest7`

is a file included in Global
Optimization Toolbox software that computes the objective function for the example.
Because the matrices and vectors defining the starting point and constraints are
large, it is more convenient to set their values as variables in the MATLABx0 = [2 1 0 9 1 0]; Aineq = [-8 7 3 -4 9 0]; bineq = 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];

Then, enter the following in the Optimization app:

Set

**Start point**to`x0`

.Set the following

**Linear inequalities**:Set

**A**to`Aineq`

.Set

**b**to`bineq`

.Set

**Aeq**to`Aeq`

.Set

**beq**to`beq`

.

The following figure shows these settings in the Optimization app.

Since this is a linearly constrained problem, set the **Poll
method** to `GSS Positive basis 2N`

. For
more information about the efficiency of the GSS search methods for linearly
constrained problems, see Compare the Efficiency of Poll Options.

Then click **Start** to run the pattern search. When the
search is finished, the results are displayed in **Run solver and
view results** pane, as shown in the following figure.

To run this problem using command-line functions:

x0 = [2 1 0 9 1 0]; Aineq = [-8 7 3 -4 9 0]; bineq = 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]; options = optimoptions('patternsearch',... 'PollMethod','GSSPositiveBasis2N'); [x,fval,exitflag,output] = patternsearch(@lincontest7,x0,... Aineq,bineq,Aeq,beq,[],[],[],options);

View the solution, objective function value, and number of function evaluations during the solution process.

x,fval,output.funccount

x = 8.5165 -6.1094 4.0989 1.2877 -4.2348 2.1812 fval = 1.9195e+03 ans = 758

Suppose you want to minimize the simple objective function of
two variables `x1`

and `x2`

,

$$\underset{x}{\mathrm{min}}f(x)=\left(4-2.1{x}_{1}{}^{2}-{x}_{1}{}^{4/3}\right){x}_{1}{}^{2}+{x}_{1}{x}_{2}+\left(-4+4{x}_{2}{}^{2}\right){x}_{2}{}^{2}$$

subject to the following nonlinear inequality constraints and bounds

$$\begin{array}{ll}{x}_{1}{x}_{2}+{x}_{1}-{x}_{2}+1.5\le 0\hfill & \text{(nonlinearconstraint)}\hfill \\ 10-{x}_{1}{x}_{2}\le \text{0}\hfill & \text{(nonlinearconstraint)}\hfill \\ 0\le \text{}{x}_{1}\le \text{}1\hfill & \text{(bound)}\hfill \\ 0\le \text{}{x}_{2}\le 13\hfill & \text{(bound)}\hfill \end{array}$$

Begin by creating the objective and constraint functions. First, create a file named
`simple_objective.m`

as
follows:

function y = simple_objective(x) y = (4 - 2.1*x(1)^2 + x(1)^4/3)*x(1)^2 + x(1)*x(2) + (-4 + 4*x(2)^2)*x(2)^2;

The pattern search solver assumes the objective function will
take one input `x`

where `x`

has
as many elements as number of variables in the problem. The objective
function computes the value of the function and returns that scalar
value in its one return argument `y`

.

Then create a file named `simple_constraint.m`

containing
the constraints:

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

The pattern search solver assumes the constraint function will
take one input `x`

, where `x`

has
as many elements as the number of variables in the problem. The constraint
function computes the values of all the inequality and equality constraints
and returns two vectors, `c`

and `ceq`

,
respectively.

Next, to minimize the objective function using the `patternsearch`

function,
you need to pass in a function handle to the objective function as
well as specifying a start point as the second argument. Lower and
upper bounds are provided as `LB`

and `UB`

respectively.
In addition, you also need to pass a function handle to the nonlinear
constraint function.

ObjectiveFunction = @simple_objective; X0 = [0 0]; % Starting point LB = [0 0]; % Lower bound UB = [1 13]; % Upper bound ConstraintFunction = @simple_constraint; [x,fval] = patternsearch(ObjectiveFunction,X0,[],[],[],[],... LB,UB,ConstraintFunction)

Optimization terminated: mesh size less than options.MeshTolerance and constraint violation is less than options.ConstraintTolerance. x = 0.8122 12.3122 fval = 9.1324e+004

Next, plot the results. Create options using `optimoptions`

that
selects two plot functions. The first plot function `psplotbestf`

plots
the best objective function value at every iteration. The second plot
function `psplotmaxconstr`

plots the maximum constraint
violation at every iteration.

You can also visualize the progress of the algorithm by displaying
information to the Command Window using the `'Display'`

option.

options = optimoptions('patternsearch','PlotFcn',{@psplotbestf,@psplotmaxconstr},'Display','iter'); [x,fval] = patternsearch(ObjectiveFunction,X0,[],[],[],[],LB,UB,ConstraintFunction,options)

Max Iter Func-count f(x) Constraint MeshSize Method 0 1 0 10 0.8919 1 28 113580 0 0.001 Increase penalty 2 105 91324 1.776e-07 1e-05 Increase penalty 3 192 91324 1.185e-11 1e-07 Increase penalty Optimization terminated: mesh size less than options.MeshTolerance and constraint violation is less than options.ConstraintTolerance. x = 0.8122 12.3122 fval = 9.1324e+04

**Best Objective Function Value and Maximum Constraint Violation
at Each Iteration**