MATLAB: How to add some special constraints in ga optimization

gaMATLABmixed-integer programmingnonlinear constraints

Hello everyone, I am currently solving an Integer optimization problem, which involves some special constraints that I have no idea how to deal with them:

Suppose I need to minimize a function f(X), where X is an integer vector only contains "0"s and "1"s, i.e. X=[x1, x2 … xn];

Constraint 1: for any (2 to n)-th entries of vector X, if the X(i-1) entry is 0 and the sum of (1-X(i-5:i-1))<5, then the X(i) must be 0 (of course, if i<=5, the X(i) must be 0).

Constraint 2: for any (2 to n)-th entries of vector X, if the X(i-1) entry is 1 and the sum of X(i-8:i-1)<8, then the X(i) must be 1 (of course, if i<=8, the X(i) must be 1).

There also is known vector S=[s1,s2 … sn] in the objective function f(X), each one depends on the corresponding entry of the vector X, and such that:

Constraint 3: suppose s1 is given, for any (2 to n)-th entries of vector S, if X(i-1) is 1, then S(i)=0. If X(i-N+1:i-1) are all '0's and X(i-N) is 1, if N<=6 S(i)=1000; if N>6 S(i)=2000.

P.S. Can I use if-else conditions in the nonlinear constraint function file?

Thank you very much!!

Ying

Best Answer

The answer to your specific question is yes, of course, you can use if-else or any other MATLAB statement in a nonlinear constraint function.

As far as your constraints, it seems to me that constraint 1 states that for any i >= 5, x(i) can be 1 only if all previous x(j) = 1 for j < i. Is this true? My statement depends on all x(j) = 0 or 1. So the first x(i-1) that = 0 causes the sum x(i-5:i-1) to be 4, and makes x(i) = 0, and this continues.

If this is true, then constraint 2 is superfluous.

In your constraint 3, it seems to me impossible that "the sum of X(i-6:i-1)>6" because all the x(i) are either 0 or 1, so the sum is bounded above by 6.

Alan Weiss

MATLAB mathematical toolbox documentation

Related Solutions

MATLAB: MINLP optimization using GA reaching different solutions every run

The results of ga can be random, because it uses random mutation of a population of test points as part of its search.

If I take your problem statement literally, it looks like it can be simplified to a common MILP and solved with intlinprog, which should make the results more reliable. In particular, totalrevenue6 looks like it is a monotonic function of income6 which is in turn monotonic as a function of x_1. So you can optimize with respect to x1 independently, reducing the problem to an objective and constraints that are linear in the binary variables x_2 and x_3.

However, your problem description looks like it has several holes:

First, it is unclear what totalrevenue is summing over. Are you summing over hours? Over generators? Both? If x1,x2,x3 have further hidden indices over generators/hours, you should unhide them for us.

Similarly, what is x_j in the "startup constraint"? Does j=1,2,3 or is j an index for something else that's been hideen? The constraint x_2-x_1-s_1<=0 looks very strange if x_1 still refers here to generator output and x_2 still refers to the binary on/off state of the generator. Since you mention (x3), I assume you really might mean x3_j where j s an index for .... something.

Finally, it looks like you are minimizing income6, which is also strange. If I expand your expression for totalrevenue6, rearranging the minus signs, I obtain

   totalrevenue6 = sum( SC_1*x_3 + x_2*(income6) )

In other words, your objective function improves if you decrease income6, which seems counter-intuitive.

MATLAB: Constraint with decision variable

You should use a couple of different kinds of constraints. But you are only constraining variable #i, so we have to initialize the constraints so the other variables are not affected

Nx = length(x);
%no equality constraints
Aeq = [];
beq = [];
%the constraint has to be for some _specific_ i
i = 13;
%now to construct x(i-1) - P <= x(i) and x(i) <= x(i-1) + D
A = zeros(2, Nx);
b = zeros(2, 1);
A(1, i-1) = 1; A(1, i) = -1; b(1) = P;   % (1)  * x(i-1) + (-1) * x(i) <= (P)
A(2, i-1) = -1; A(2, i) = 1; b(2) = D;   % (-1) * x(i-1) + (1)  * x(i) <= (D)
%lower bounds are -inf, upper bounds are +inf for all variables except #i
lb = -inf * ones(1, Nx);
ub = inf * ones(1, Nx);
%but for variable #i, the value cannot be less than 0 or more than C
lb(i) = 0;
ub(i) = C;
fmincon(@objective, x0, A, b, Aeq, beq, lb, ub)

Your constraints cannot be generalized to all of the variables because there is no x(0) to constrain x(1) against.

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