MATLAB: Unique variable values in ga

constrained optimizationgaoptimization

I am trying to write a code to optimise a stack of materials, based on certain parameters.

Currently I have an integer constrained ga optimisation, basically evaluating the objective function for a population of different stacks. However I would like to constrain this optimisation so that it doesn't look at stacks with two identical layers next to each other. Currently I am doing this by putting a big penalty on such structures in the objective function.

I was wondering if there is a way to constrain the ga function such that nearest neighbour variables cannot have the same value, i.e. x1~=x2, but x1 may equal x3.

Best Answer

Expressing x1 ~= x2 could in theory be done as ((x1 - x2) < 0 or (x2 - x1) < 0) . However, you cannot express "or" in inequality constraints or equality constraints: inequality constraints and equality constraints are "and"'d together. ga() does not offer strict inequality either, so the closest you could get would be ((x1 - x2) <= 0 and (x2 - x1) <= 0) and that simplifies to x1 == x2 which is not what you want .

ga() also cannot use equality constraints when there are integer constraints.

So... you are going to have to program this as nonlinear inequality constraints.

all(abs(diff(x)) ~= 1)

would correspond to the condition being met. But that returns logical true, and nonlinear constraints are considered to be satisfied if the output of the expression is <= 0 . So you need that the expression returns false (or negative) when the condition is satisfied, and true (or positive) when the condition is violated. One way of expressing that would be:

nonlcon = @(x) deal(any(abs(diff(x)) == 1), [])

that would return true, a positive number, if there was a constraint violation, which happens to work out; the output of the expression with be 0 if the constraint is not violated, and that is the condition for _continuing. It is confusing, but self-consistent.

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MATLAB: Specifying constraints for optimization using GA

As documented, when using a custom population type, "ga ignores all constraints, including bounds, linear constraints, and nonlinear constraints." So your custom creation, crossover, and mutation functions must enforce your constraints.

Alan Weiss

MATLAB mathematical toolbox documentation

MATLAB: Please Help Optimization using PSO

%{


x1+x2+x3+x4+x5 =1000;
x1^2-4x2+5x3^3+x4^4-3.5x5^2 = 300
%}

Notice that x2 is the only variable that appears linearly in the second equation. So rewrite the first equation as

%{
x2 = 1000 - x1 - x3 - x4 - x5
%}

Now substitute that into the second equation

%{
x1^2-4*(1000 - x1 - x3 - x4 - x5)+5x3^3+x4^4-3.5x5^2 = 300
%}

This is a quadratic in x1 (or x5), so solve it for x1

syms x1 x2 x3 x4 x5
eqn = x1^2-4*(1000 - x1 - x3 - x4 - x5)+5*x3^3+x4^4-3.5*x5^2 == 300
eqn = 
X1 = solve(eqn, x1)
X1 =

Now you run two different PSO runs, one substituting 1000 - x1 - x3 - x4 - x5 for x2 and the first of those X1 values for x1; the other run substituting 1000 - x1 - x3 - x4 - x5 for x2 and the second of those X1 values for x1. These runs will not require equality or inequality constraints (unless there are more constraints you did not tell us about.)

The runs you would do would be over 3 variables only, x3, x4, x5, with you having substituted for x1 and x2 in your objective function.

Best Answer

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MATLAB: Specifying constraints for optimization using GA

MATLAB: Please Help Optimization using PSO

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