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# Global Optimization Toolbox

## Optimization of Stochastic Objective Function

This example shows how to find a minimum of a stochastic objective function using PATTERNSEARCH function in the Global Optimization Toolbox. We also show why the Optimization Toolbox™ functions are not suitable for this kind of problems. A simple 2-dimensional optimization problem is selected for this example to help visualize the objective function.

Initialization

```X0 = [2.5 -2.5];   % Starting point.
LB = [-5 -5];      % Lower bound
UB = [5 5];        % Upper bound
range = [LB(1) UB(1); LB(2) UB(2)];
Objfcn = @smoothFcn; % Handle to the objective function.
% Plot the smooth objective function
fig = figure('Color','w');
showSmoothFcn(Objfcn,range);
hold on;
title('Smooth objective function');
ph = [];
ph(1) = plot3(X0(1),X0(2),Objfcn(X0)+30,'or','MarkerSize',10,'MarkerFaceColor','r');
hold off;
ax = gca;
ax.CameraPosition = [-31.0391  -85.2792 -281.4265];
ax.CameraTarget = [0 0 -50];
ax.CameraViewAngle = 6.7937;
legendLabels = {'Start point'};
lh = legend(ph,legendLabels,'Location','SouthEast');
lp = lh.Position;
lh.Position = [1-lp(3)-0.005 0.005 lp(3) lp(4)];
```

Run FMINCON on a Smooth Objective Function

The objective function is smooth (twice continuously differentiable). We will solve the optimization problem using FMINCON function from the Optimization Toolbox. FMINCON finds a constrained minimum of a function of several variables. This function has a unique minimum at the point x* = (-5.0,-5) where it has a function value f(x*) = -250.

```% Set options to display iterative results.
options = optimoptions(@fmincon,'Algorithm','active-set','Display','iter');
[Xop,Fop] = fmincon(Objfcn,X0,[],[],[],[],LB,UB,[],options)
figure(fig);
hold on;
% Plot the final point
ph(2) = plot3(Xop(1),Xop(2),Fop,'dm','MarkerSize',10,'MarkerFaceColor','m');
legendLabels = [legendLabels, 'FMINCON solution'];
lh = legend(ph,legendLabels,'Location','SouthEast');
lp = lh.Position;
lh.Position = [1-lp(3)-0.005 0.005 lp(3) lp(4)];
hold off;
```
```                                Max     Line search  Directional  First-order
Iter F-count        f(x)   constraint   steplength   derivative   optimality Procedure
0      3      -10.625         -2.5
1      6         -250            0            1        -23.4         82.1

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in
feasible directions, to within the default value of the function tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.

Active inequalities (to within options.TolCon = 1e-06):
lower      upper     ineqlin   ineqnonlin
1
2

Xop =

-5    -5

Fop =

-250

```

Stochastic Objective Function

The objective function we use now is the same as the previous example but with some random noise added to it. This is done by adding a random component to the function value.

```rng(0,'twister') % Reset the global random number generator
peaknoise = 4.5;
Objfcn = @(x) smoothFcn(x,peaknoise); % Handle to the objective function.
% Plot the objective function (non-smooth)
fig = figure('Color','w');
showSmoothFcn(Objfcn,range);
title('Stochastic objective function')
ax = gca;
ax.CameraPosition = [-31.0391  -85.2792 -281.4265];
ax.CameraTarget = [0 0 -50];
ax.CameraViewAngle = 6.7937;
```

Run FMINCON on a Stochastic Objective Function

The objective function is stochastic and not smooth. FMINCON is a general constrained optimization solver which finds a local minima using first derivative of the objective function. If derivative of the objective function is not provided, FMINCON uses finite difference to approximate first derivative of the objective function. In this example, the objective function have some random noise in it. The derivatives hence could be highly unreliable. FMINCON can potentially stop at a point which is not a minimum. This may happen because the optimal conditions seems to be satisfied at the final point because of noise or it could not make any progress.

```options = optimoptions(@fmincon,'Algorithm','active-set','Display','iter');
[Xop,Fop] = fmincon(Objfcn,X0,[],[],[],[],LB,UB,[],options)
figure(fig);
hold on;
ph = [];
ph(1) = plot3(X0(1),X0(2),Objfcn(X0)+30,'or','MarkerSize',10,'MarkerFaceColor','r');
ph(2) = plot3(Xop(1),Xop(2),Fop,'dm','MarkerSize',10,'MarkerFaceColor','m');
legendLabels = {'Start point','FMINCON solution'};
lh = legend(ph,legendLabels,'Location','SouthEast');
lp = lh.Position;
lh.Position = [1-lp(3)-0.005 0.005 lp(3) lp(4)];
hold off;
```
```                                Max     Line search  Directional  First-order
Iter F-count        f(x)   constraint   steplength   derivative   optimality Procedure
0      3     -19.2577         -2.5
1     12     -19.6222       -2.617       0.0156    -2.51e+08     1.59e+09
2     15     -53.5746            0            1    -5.47e+08     4.63e+08  Hessian modified twice

Local minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing in
feasible directions, to within the default value of the function tolerance,
and constraints are satisfied to within the default value of the constraint tolerance.

Active inequalities (to within options.TolCon = 1e-06):
lower      upper     ineqlin   ineqnonlin
1          2

Xop =

-5     5

Fop =

-53.5746

```

Run PATTERNSEARCH

We will now use PATTERNSEARCH from the Global Optimization Toolbox. Pattern search optimization techniques are a class of direct search methods for optimization. A pattern search algorithm does not require any derivative information of the objective function to find an optimal point.

```PSoptions = psoptimset('Display','iter');
[Xps,Fps] = patternsearch(Objfcn,X0,[],[],[],[],LB,UB,PSoptions)
figure(fig);
hold on;
ph(3) = plot3(Xps(1),Xps(2),Fps,'dc','MarkerSize',10,'MarkerFaceColor','c');
legendLabels = [legendLabels, 'Pattern Search solution'];
lh = legend(ph,legendLabels,'Location','SouthEast');
lp = lh.Position;
lh.Position = [1-lp(3)-0.005 0.005 lp(3) lp(4)];
hold off
```
```
Iter     f-count          f(x)      MeshSize     Method
0        1       -9.82132             1
1        3       -45.8992             2     Successful Poll
2        3       -45.8992             1     Refine Mesh
3        5       -45.8992           0.5     Refine Mesh
4        8       -88.4064             1     Successful Poll
5       10       -88.4064           0.5     Refine Mesh
6       13       -136.777             1     Successful Poll
7       15       -136.777           0.5     Refine Mesh
8       17       -136.777          0.25     Refine Mesh
9       20       -190.615           0.5     Successful Poll
10       22       -190.615          0.25     Refine Mesh
11       24       -190.615         0.125     Refine Mesh
12       27       -233.545          0.25     Successful Poll
13       29       -241.605           0.5     Successful Poll
14       31       -241.605          0.25     Refine Mesh
15       33        -250.17           0.5     Successful Poll
16       35        -250.17          0.25     Refine Mesh
17       37        -250.17         0.125     Refine Mesh
18       39        -250.17        0.0625     Refine Mesh
19       41        -250.17       0.03125     Refine Mesh
20       43        -250.17       0.01562     Refine Mesh
21       44       -254.925       0.03125     Successful Poll
22       46       -254.925       0.01562     Refine Mesh
23       47        -256.57       0.03125     Successful Poll
24       49        -261.07        0.0625     Successful Poll
25       51        -261.07       0.03125     Refine Mesh
26       53        -261.07       0.01562     Refine Mesh
27       55        -261.07      0.007812     Refine Mesh
28       57        -261.07      0.003906     Refine Mesh
29       59        -261.07      0.001953     Refine Mesh
30       61        -261.07     0.0009766     Refine Mesh

Iter     f-count        f(x)       MeshSize      Method
31       63        -261.07     0.0004883     Refine Mesh
32       65        -261.07     0.0002441     Refine Mesh
33       66       -262.065     0.0004883     Successful Poll
34       68       -262.065     0.0002441     Refine Mesh
35       71       -262.065     0.0001221     Refine Mesh
36       74       -262.065     6.104e-05     Refine Mesh
37       77       -262.065     3.052e-05     Refine Mesh
38       80       -262.065     1.526e-05     Refine Mesh
39       83       -262.065     7.629e-06     Refine Mesh
40       86       -262.065     3.815e-06     Refine Mesh
41       89       -262.065     1.907e-06     Refine Mesh
42       92       -262.065     9.537e-07     Refine Mesh
Optimization terminated: mesh size less than options.TolMesh.

Xps =

-4.9998   -5.0000

Fps =

-262.0651

```

Pattern search algorithm is not affected by random noise in the objective functions. Pattern search requires only function value and not the derivatives, hence noise (of some uniform kind) may not affect it. However, pattern search requires more function evaluation to find the true minimum than derivative based algorithms, a cost for not using the derivatives.