Optimization with custom constraint

I am trying to formulate a linear optimization problem with a simple cost function of the form

Cost = X*Price_X + Y*Price_Y

But additionally, I have a parameter Z that I had obtained before through an iterative process. Z is a 2x2 matrix, where each value index is the value of X and Y.

Example: ParameterValue = Z(X,Y)

In my optimization problem, on top of having to minimize the cost of X and Y, the ParameterValue has to be within certain limits, or under a certain threshold

My Question is: is it possible to pass that parameter as a function handle, although it is a function of the other variables? Or should I formulate a multi-objective algorithm where ParameterValue is another objective function.

What I have tried was to have the following code:

if true
  Value= @(X,Y) Z(X,Y) %To have value as an anonymous function
ObjFunc = PriceX * X + PriceY * Y + 0*Value(X,Y)
end

Then adding the threshold as a constraint for Value. But this has not worked at all.

Is my formulation, procedure right? or doable in any case? or should I try to do some multi-objective linear optimization if possible. Please note that the values of Z (2x2 matrix)are obtained before, so I also need to call them somehow to the optimization function.

Thank you in advance.

Answer 1

[Alan Weiss]

I suggest that you read the documentation on how to specify an objective function and how to specify a nonlinear constraint.

Alan Weiss

MATLAB mathematical toolbox documentation

[Ali Chamseddine]

Thank you for your answer. I have tried specifying the parameter value as a non linear constraint but I am having a problem.

Following the "Specify a nonlinear constraint" document I did the following

if true
  Z = @(x,y) input(x,y);   %input is the value I had computed before
  c = @(x) Z(x(2),x(1)) - 0.3; % so I want Z for the values of x1 and x2 to be less than 0.3
ceq=[];
nonlinconstraint = @(x) deal(c(x),ceq(x));
obj = @(x) Price1*x(1) + Price2*x(2); %Price1 and Price2 are already introduced variables
opts = optimoptions(@fmincon,'Algorithm','sqp');
z = fmincon(obj,[lb;ub],[],[],[],[],lb,ub,nonlinconstraint,opts);
end

But this doesn't seem to be working, I get the "index exceeds matrix dimensions" error.

Is this lookup function handle possible? The reason I am doing this is because every combination of X and Y has a value parameter Z that is not a direct function of X and Y.

The only alternative to this that I am seeing is to create a surface fit function and implement the coefficients of the function as a constraint instead. But I would prefer to just go to the look-up if possible.

Thank you. Ali

[Alan Weiss]

I think that your x0 input to fmincon has the wrong number of elements. Each of lb and ub should be the same size, and your x0 has twice as many elements as they do.

Alan Weiss

MATLAB mathematical toolbox documentation

[Ali Chamseddine]

Thanks, I will try it out!