Genetic Algorithms MATLAB: differenze tra le versioni

Versione attuale delle 12:11, 30 nov 2019

Nicola Altini nicola.altini@poliba.it

Vitoantonio Bevilacqua vitoantonio.bevilacqua@poliba.it

Mono-objective Function Optimization

Optimization Function

```
%%% Optimization Function 
```

% Define and plot the function to optimize

```
xi = linspace(-6,6,300);
```
```
yi = linspace(-6,6,300);
```
```
[X,Y] = meshgrid(xi,yi);
```
```
Z = ps_example([X(:),Y(:)]);
```

% Note that ps_example is a non-smooth function

```
Z = reshape(Z,size(X));
```
```
surf(X,Y,Z,'MeshStyle','none')
```
```
colormap 'jet'
```
```
view(-26,43)
```
```
xlabel('x(1)')
```
```
ylabel('x(2)')
```
```
title('Teaching Example')
```
```
hold on 
```
```
dim_area = 200;
```

Unconstrained Optimization

```
%% Optimization with Genetic Algorithm
```
```
rng default % For reproducibility
```
```
x = ga(@ps_example,2);
```

fprintf('The minimum found using GA is: ');

```
disp(x);
```

fprintf('f(x) = %.4f\n', ps_example(x));

scatter3(x(1), x(2), ps_example(x),dim_area,'g','filled');

Optimization with Linear Inequality Constraints

%% Nonsmooth Function with Linear Constraints

```
% Linear Constraints
```

% Inequalities to the matrix form A*x <= b

```
% -x(1) -x(2) <= -1
```
```
% -x(1) +x(2) <= 5
```
```
A = [-1, -1;
```
```
     -1,  1];
```
```
b = [-1; 5];
```
```
 
```
```
rng default % For reproducibility
```
```
fun = @ps_example;
```
```
x = ga(fun,2,A,b);
```

fprintf('The minimum found using GA (with Linear Constraints) is: ');

```
disp(x);
```

fprintf('f(x) = %.4f\n', ps_example(x));

```
fprintf('Check constraint:\n');
```
```
fprintf('A * x - b = ');
```
```
disp(A*x' - b);
```

scatter3(x(1), x(2), ps_example(x),dim_area,'r','filled');

Optimization with Linear Equality and Inequality Constraints

%% Linear Equality and Inequality Constraints

% Inequalities to the matrix form A*x <= b

% Equalities to the matrix form   Aeq*x = beq

```
A = [-1 -1];
```
```
b = -1;
```
```
Aeq = [-1 1];
```
```
beq = 5;
```
```
 
```
```
% Optimization
```
```
rng default % For reproducibility
```
```
fun = @ps_example;
```
```
x = ga(fun,2,A,b,Aeq,beq);
```

fprintf('The minimum found using GA (with Linear Equality and Inequality Constraints) is: ');

```
disp(x);
```

fprintf('f(x) = %.4f\n', ps_example(x));

```
fprintf('Check constraints:\n');
```
```
fprintf('A * x - b = ');
```
```
disp(A*x' - b);
```
```
fprintf('Aeq * x - beq = ');
```
```
disp(Aeq*x' - beq);
```

scatter3(x(1), x(2), ps_example(x),dim_area,'b','filled');

Excercise 3

%% Linear Equality and Inequality Constraints - Excercise 3

```
% Constraints
```
```
A = [-2 -4; 3 -1; -1 -1];
```
```
b = [-1; -2; 0];
```
```
Aeq = [2 1];
```
```
beq = 5;
```
```
 
```
```
% Optimization
```
```
rng default % For reproducibility
```
```
fun = @ps_example;
```
```
x = ga(fun,2,A,b,Aeq,beq);
```

fprintf('The minimum found using GA (with Linear Equality and Inequality Constraints) is: ');

```
disp(x);
```

fprintf('f(x) = %.4f\n', ps_example(x));

```
fprintf('Check constraints:\n');
```
```
fprintf('A * x - b = ');
```
```
disp(A*x' - b);
```
```
fprintf('Aeq * x - beq = ');
```
```
disp(Aeq*x' - beq);
```

scatter3(x(1), x(2), ps_example(x),dim_area,'b','filled');

Optimization with Linear Constraints and Bounds

```
%% Linear Constraints and Bounds
```
```
% -x(1) -x(2) <= -1
```
```
% -x(1) +x(2) == 5
```
```
A = [-1 -1];
```
```
b = -1;
```
```
Aeq = [-1 1];
```
```
beq = 5;
```
```
%  1 <= x(1) <= 6
```
```
% -3 <= x(2) <= 8
```
```
lb = [1 -3];
```
```
ub = [6  8];
```
```
 
```
```
% Optimization
```
```
rng default % For reproducibility
```
```
fun = @ps_example;
```
```
x = ga(fun,2,A,b,Aeq,beq,lb,ub);
```

fprintf('The minimum found using GA (with Linear Constraints and Bounds) is: ');

```
disp(x);
```

fprintf('f(x) = %.4f\n', ps_example(x));

```
fprintf('Check constraints:\n');
```
```
fprintf('A * x - b = ');
```
```
disp(A*x' - b);
```
```
fprintf('Aeq * x - beq = ');
```
```
disp(Aeq*x' - beq);
```

scatter3(x(1), x(2), ps_example(x),dim_area,'p','filled');

Optimization with Nonlinear Constraints

Example 1

```
%% Nonlinear Constraints 
```

% minimize ps_example function on the region

```
% 2x(1).^2 + x(2).^2 <= 3 
```
```
% (x(1) + 1).^2 = (x(2) / 2).^4
```
```
nonlcon = @ellipsecons;
```
```
 
```
```
% Optimization
```
```
fun = @ps_example;
```
```
rng default % For reproducibility
```

x = ga(fun,2,[],[],[],[],[],[],nonlcon);

```
disp(x);
```

fprintf('f(x) = %.4f\n', ps_example(x));

```
[c,ceq] = nonlcon(x);
```
```
fprintf('Check constraints:\n');
```

fprintf('nonlcon(x) -> c = %.3f, ceq = %.3f\n', c, ceq);

```
% Constraints satisfied when
```
```
% c   <= 0
```
```
% ceq == 0
```

scatter3(x(1), x(2), ps_example(x),dim_area,'y','filled');

Example 2

%% Genetic Algorithm with Non-Linear constraints

```
use_circle_equality_constraint = 1;
```
```
if use_circle_equality_constraint
```
```
    nonlcon = @circlecons_eq;
```
```
else
```
```
    nonlcon = @circlecons_lt;
```
```
end
```
```
 
```
```
rng default % For reproducibility
```
```
fun = @squarednorm;
```
```
A = [];
```
```
b = [];
```
```
Aeq = [];
```
```
beq = [];
```
```
lb = [2, 2];
```
```
ub = [7, 8];
```
```
IntCon = [];
```

[x,fval,exitflag,output,population,scores] = ga(fun,2,A,b,Aeq,beq,lb,ub,nonlcon,IntCon);

Optimization with Integer Constraints

```
%% Integer Constraints
```
```
IntCon = 1;
```
```
rng default % For reproducibility
```
```
fun = @ps_example;
```
```
A = [];
```
```
b = [];
```
```
Aeq = [];
```
```
beq = [];
```
```
lb = [];
```
```
ub = [];
```
```
nonlcon = [];
```

x = ga(fun,2,A,b,Aeq,beq,lb,ub,nonlcon,IntCon);

```
disp(x);
```

fprintf('f(x) = %.4f\n', ps_example(x));

scatter3(x(1), x(2), ps_example(x),dim_area,'c','filled');

Obtaining Diagnostic Information

Example 1

```
%%% Example 1
```
```
rng default % For reproducibility
```
```
fun = @ps_example;
```
```
A = [];
```
```
b = [];
```
```
Aeq = [];
```
```
beq = [];
```
```
lb = [];
```
```
ub = [];
```
```
nonlcon = [];
```
```
IntCon = 1;
```

% options = optimoptions(SolverName,Name,Value)

options = optimoptions('ga','PlotFcn', @gaplotbestf);

[x,fval,exitflag,output,population,scores] = ga(fun,2,A,b,Aeq,beq,lb,ub,nonlcon,IntCon,options);

```
% Final Population and Scores
```
```
disp(population(1:10,:));
```
```
disp(scores(1:10));
```

Example 2

```
%%% Example 2
```

options = optimoptions('ga','ConstraintTolerance',1e-6,'PlotFcn', @gaplotbestf);

[x,fval,exitflag,output,population,scores] = ga(fun,2,A,b,Aeq,beq,lb,ub,nonlcon,options);

```
% Final Population and Scores
```
```
disp(population(1:10,:));
```
```
disp(scores(1:10));
```

Helper Functions

```
%% Helper Functions
```
```
function [c,ceq] = circlecons_eq(x)
```
```
c = [];
```
```
ceq = (x(1)-8)^2 +(x(2)-6)^2 -9;
```
```
end
```
```
 
```
```
function [c,ceq] = circlecons_lt(x)
```
```
c = (x(1)-8)^2 +(x(2)-6)^2 -9;
```
```
ceq = [];
```
```
end
```
```
 
```
```
function [c, ceq] = ellipsecons(x)
```
```
c = 2*x(1)^2 + x(2)^2 - 3;
```
```
ceq = (x(1)+1)^2 - (x(2)/2)^4;
```
```
end
```
```
 
```
```
function f = squarednorm(x)
```
```
f = x(1)^2 + x(2)^2;
```
```
end
```

Multi-objective Function Optimization

Unconstrained Optimization

%%  Multi-objective Optimization Example

```
%%% No Constraints
```
```
% Defining fitness function
```

% Find the Pareto front for a simple multiobjective problem. There are two objectives and two decision variables x.

fitnessfcn = @(x)[norm(x)^2  ,  0.5*norm(x(:)-[2;-1])^2+2];

% Find the Pareto front for this objective function.

```
% Optimization
```
```
rng default % For reproducibility
```
```
x = gamultiobj(fitnessfcn,2);
```

% Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance.

```
% Plot the solution points.
```
```
plot(x(:,1),x(:,2),'ko')
```
```
xlabel('x(1)')
```
```
ylabel('x(2)')
```

title('Pareto Points in Parameter Space')

Optimization with Linear Constraints

```
%% Linear Inequality Constraint
```
```
A = [1,1];
```
```
b = 1/2;
```
```
% Find the Pareto front.
```
```
 
```
```
rng default % For reproducibility
```

[x, fval] = gamultiobj(fitnessfcn,2,A,b);

% Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance.

% Plot the constrained solution and the linear constraint.

```
 
```
```
plot(x(:,1),x(:,2),'ko')
```
```
t = linspace(-1/2,2);
```
```
y = 1/2 - t;
```
```
hold on
```
```
plot(t,y,'b--')
```
```
xlabel('x(1)')
```
```
ylabel('x(2)')
```

title('Pareto Points in Parameter Space')

```
hold off
```

Optimization with Bound Constraints

Example 1

```
%% Bound Constraints
```
```
fitnessfcn = @(x)[sin(x),cos(x)];
```
```
nvars = 1;
```
```
lb = 0;
```
```
ub = 2*pi;
```
```
rng default % for reproducibility
```

[x, fval] = gamultiobj(fitnessfcn,nvars,[],[],[],[],lb,ub);

```
 
```
```
plot(sin(x),cos(x),'r*')
```
```
xlabel('sin(x)')
```
```
ylabel('cos(x)')
```
```
title('Pareto Front')
```
```
legend('Pareto front')
```

Example 2

%% Multi-objective Optimization Example 2

% Defining and plotting function to optimize

```
x = -1:0.1:8;
```
```
y = schaffer2(x);
```
```
plot(x,y(:,1),'r',x,y(:,2),'b');
```
```
xlabel x
```
```
ylabel 'schaffer2(x)'
```
```
legend('Objective 1','Objective 2')
```
```
 
```
```
% Bounds
```
```
lb = -5;
```
```
ub = 10;
```
```
 
```

% Set options to view the Pareto front as gamultiobj runs.

options = optimoptions('gamultiobj','PlotFcn',@gaplotpareto);

```
 
```
```
% Optimization - call gamultiobj.
```
```
rng default % For reproducibility
```

[x,fval,exitflag,output] = gamultiobj(@schaffer2,1,[],[],[],[],lb,ub,options);

Helper Functions

```
%% Helper Functions
```
```
function f = my_multi(x)
```

f(1) = x(1)^4 - 10*x(1)^2+x(1)*x(2) + x(2)^4 -(x(1)^2)*(x(2)^2);

f(2) = x(2)^4 - (x(1)^2)*(x(2)^2) + x(1)^4 + x(1)*x(2);

```
end
```
```
 
```
```
function f = pareto_example(x)
```

f = [norm(x)^2  ,  0.5*norm(x(:)-[2;-1])^2+2];

```
end
```
```
 
```

function y = schaffer2(x) % y has two columns

```
 
```

% Initialize y for two objectives and for all x

```
y = zeros(length(x),2);
```
```
 
```
```
% Evaluate first objective. 
```

% This objective is piecewise continuous.

```
for i = 1:length(x)
```
```
    if x(i) <= 1
```
```
        y(i,1) = -x(i);
```
```
    elseif x(i) <= 3 
```
```
        y(i,1) = x(i) -2; 
```
```
    elseif x(i) <= 4 
```
```
        y(i,1) = 4 - x(i);
```
```
    else 
```
```
        y(i,1) = x(i) - 4;
```
```
    end
```
```
end
```
```
 
```
```
% Evaluate second objective
```
```
y(:,2) = (x -5).^2;
```
```
end
```

Genetic Algorithms MATLAB: differenze tra le versioni

Versione attuale delle 12:11, 30 nov 2019

Indice

Mono-objective Function Optimization

Optimization Function

Unconstrained Optimization

Optimization with Linear Inequality Constraints

Optimization with Linear Equality and Inequality Constraints

Excercise 3

Optimization with Linear Constraints and Bounds

Optimization with Nonlinear Constraints

Example 1

Example 2

Optimization with Integer Constraints

Obtaining Diagnostic Information

Example 1

Example 2

Helper Functions

Multi-objective Function Optimization

Unconstrained Optimization

Optimization with Linear Constraints

Optimization with Bound Constraints

Example 1

Example 2

Helper Functions

Menu di navigazione

Strumenti personali

Namespace

Varianti

Visite

Azioni

Ricerca

Navigazione

Strumenti

@@ Riga 1: / Riga 1: @@
-== Genetic Algorithms with MATLAB ==
+'''Nicola Altini''' [mailto:nicola.altini@poliba.it nicola.altini@poliba.it]
-[http://www.vitoantoniobevilacqua.it/wiki/images/8/88/GA.pdf Slides]
 '''Vitoantonio Bevilacqua''' [mailto:vitoantonio.bevilacqua@poliba.it vitoantonio.bevilacqua@poliba.it]
-'''Nicola Altini''' [mailto:nicola.altini@poliba.it nicola.altini@poliba.it]
+[http://www.vitoantoniobevilacqua.it/wiki/images/8/88/GA.pdf Slides]
 == Mono-objective Function Optimization ==
@@ Riga 236: / Riga 235: @@
 </syntaxhighlight>
-== Helper Functions ==
+=== Helper Functions ===
 <syntaxhighlight lang="matlab" line>
 %% Helper Functions
@@ Riga 256: / Riga 255: @@
 function f = squarednorm(x)
 f = x(1)^2 + x(2)^2;
+end
+</syntaxhighlight>
+== Multi-objective Function Optimization ==
+=== Unconstrained Optimization ===
+<syntaxhighlight lang="matlab" line>
+%%  Multi-objective Optimization Example
+%%% No Constraints
+% Defining fitness function
+% Find the Pareto front for a simple multiobjective problem. There are two objectives and two decision variables x.
+fitnessfcn = @(x)[norm(x)^2  ,  0.5*norm(x(:)-[2;-1])^2+2];
+% Find the Pareto front for this objective function.
+% Optimization
+rng default % For reproducibility
+x = gamultiobj(fitnessfcn,2);
+% Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance.
+% Plot the solution points.
+plot(x(:,1),x(:,2),'ko')
+xlabel('x(1)')
+ylabel('x(2)')
+title('Pareto Points in Parameter Space')
+</syntaxhighlight>
+=== Optimization with Linear Constraints ===
+<syntaxhighlight lang="matlab" line>
+%% Linear Inequality Constraint
+A = [1,1];
+b = 1/2;
+% Find the Pareto front.
+rng default % For reproducibility
+[x, fval] = gamultiobj(fitnessfcn,2,A,b);
+% Optimization terminated: average change in the spread of Pareto solutions less than options.FunctionTolerance.
+% Plot the constrained solution and the linear constraint.
+plot(x(:,1),x(:,2),'ko')
+t = linspace(-1/2,2);
+y = 1/2 - t;
+hold on
+plot(t,y,'b--')
+xlabel('x(1)')
+ylabel('x(2)')
+title('Pareto Points in Parameter Space')
+hold off
+</syntaxhighlight>
+=== Optimization with Bound Constraints ===
+==== Example 1 ====
+<syntaxhighlight lang="matlab" line>
+%% Bound Constraints
+fitnessfcn = @(x)[sin(x),cos(x)];
+nvars = 1;
+lb = 0;
+ub = 2*pi;
+rng default % for reproducibility
+[x, fval] = gamultiobj(fitnessfcn,nvars,[],[],[],[],lb,ub);
+plot(sin(x),cos(x),'r*')
+xlabel('sin(x)')
+ylabel('cos(x)')
+title('Pareto Front')
+legend('Pareto front')
+</syntaxhighlight>
+==== Example 2 ====
+<syntaxhighlight lang="matlab" line>
+%% Multi-objective Optimization Example 2
+% Defining and plotting function to optimize
+x = -1:0.1:8;
+y = schaffer2(x);
+plot(x,y(:,1),'r',x,y(:,2),'b');
+xlabel x
+ylabel 'schaffer2(x)'
+legend('Objective 1','Objective 2')
+% Bounds
+lb = -5;
+ub = 10;
+% Set options to view the Pareto front as gamultiobj runs.
+options = optimoptions('gamultiobj','PlotFcn',@gaplotpareto);
+% Optimization - call gamultiobj.
+rng default % For reproducibility
+[x,fval,exitflag,output] = gamultiobj(@schaffer2,1,[],[],[],[],lb,ub,options);
+</syntaxhighlight>
+=== Helper Functions ===
+<syntaxhighlight lang="matlab" line>
+%% Helper Functions
+function f = my_multi(x)
+f(1) = x(1)^4 - 10*x(1)^2+x(1)*x(2) + x(2)^4 -(x(1)^2)*(x(2)^2);
+f(2) = x(2)^4 - (x(1)^2)*(x(2)^2) + x(1)^4 + x(1)*x(2);
+end
+function f = pareto_example(x)
+f = [norm(x)^2  ,  0.5*norm(x(:)-[2;-1])^2+2];
+end
+function y = schaffer2(x) % y has two columns
+% Initialize y for two objectives and for all x
+y = zeros(length(x),2);
+% Evaluate first objective.
+% This objective is piecewise continuous.
+for i = 1:length(x)
+    if x(i) <= 1
+        y(i,1) = -x(i);
+    elseif x(i) <= 3
+        y(i,1) = x(i) -2;
+    elseif x(i) <= 4
+        y(i,1) = 4 - x(i);
+    else
+        y(i,1) = x(i) - 4;
+    end
+end
+% Evaluate second objective
+y(:,2) = (x -5).^2;
 end
 </syntaxhighlight>