Optimization of a function with constraints containing 15 variables

I am trying to solve an optimization problem of a function to find its minimum value that satisfies some constraints ( linear and nonlinear equalities; linear inequalities). So far, I have tried the following solvers:

• fmincon (algorithms interior-point, active-set and sqp). The function converges for some initial values and deteriorates for other initial values that are only slightly different.
• ga. Either the algorithm doesn't find a feasible point in the first iteration (slighty less frequent when using nonlinear feasible creation function) or when it does, the stall generations limit is exceeded without a solution beeing found.

My code for the objective function:

function val = objFun(x)            
    val = x(1)+x(2)+x(3)+x(4);
end

My code for the constraints:

function [c, ceq] = constraints(x)

          %inequalities
          c(1) = -x(1);
          c(2) = -x(2);
          c(3) = -x(3);
          c(4) = -x(4);
          c(5) = x(15)-0.3;

          %linear equalities
          ceq(1) = x(5)+x(6)+x(7)+x(8);
          ceq(2) = -500+x(9)+x(10)+x(11)+x(12);
          ceq(3) = -500*0.636-0.25*(x(4)-x(2));
          ceq(4) = -0.25*(x(3)-x(1));
          ceq(5) = -500*0.242-0.25*(x(8)-x(6))-a*(x(11)-x(9));
          ceq(6) = x(1)+x(3)-x(2)-x(4);
          ceq(7) = (x(9)/x(5))+((0.25-x(13))/(-x(14)));
          ceq(8) = (x(10)/x(6))+((-x(13))/(-0.25-x(14)));
          ceq(9) = (x(11)/x(7))+((-0.25-x(13))/(-x(14)));
          ceq(10) = (x(12)/x(8))+((-x(13))/(0.25-x(14)));

          %nonlinear equalities
          ceq(11) = x(15)^2*x(1)^2-x(5)^2-x(9)^2;
          ceq(12) = x(15)^2*x(2)^2-x(6)^2-x(10)^2;
          ceq(13) = x(15)^2*x(3)^2-x(7)^2-x(11)^2;
          ceq(14) = x(15)^2*x(4)^2-x(8)^2-x(12)^2;

end

My code for the optimization with fmincon:

options = optimoptions(@fmincon,'Display','iter','Algorithm','sqp');
x0 = [300 400 300 200 100 400 100 200 500 400 200 100  0.1 0.1 0.25]; %this initial value converges
% x0 = [300 400 300 200 110 400 100 200 500 400 200 100  0.1 0.1 0.25]; %this initial value deteriorates
[x,fval] = fmincon(@objFun,x0,[],[],[],[],[],[],@constraints,options)

My code for the optimization with ga:

options = gaoptimset('Display','iter','CreationFcn',@gacreationnonlinearfeasible);
[x,fval] = ga(@objFun,15,[],[],[],[],[],[],@constraints,options)

[Dhanu Naik]

How can we tell Constraint function,The number of decision Variables in case of Generic Programming. Here you have taken 15 which was known to you,if my number of decision variables varies,how can I send that extra information to constraint function.

[Walter Roberson]

You can take the length() of the input to the constraint function. If somehow you have a different number of decision variables, then use http://www.mathworks.com/help/matlab/math/parameterizing-functions.html

Answer 1

[Alan Weiss]

Do not use nonlinear constraint functions to effect linear constraints and bounds. Your first five nonlinear inequality constraints should be represented as bounds: x(1) through x(4) have lower bounds of 0, and x(15) has an upper bound of 0.3.

Similarly, your first six nonlinear equality constraints should be written as linear equality constraints.

Don't bother trying to optimize a smooth function with smooth constraints using ga. The solver of choice for such problems is fmincon.

Do x(13) and x(14) have to be between -0.25 and 0.25? If so, give these as bounds.

I suggest that you examine your problem carefully to see if you can bound each component, and then include the tightest bounds you can in the problem.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation

[Lucas Giering]

Hey Alan,

thanks a lot for your fast answer. The advice about the bounds and the solver will help me a lot.

However, I am not entirel sure about how i should turn my nonlinear equality constraints into linear equality constraints. Right now i have 10 linear equalities and 4 nonlinear equalities.

Cheers, Lucas

[Alan Weiss]

The way I see it, you have 6 linear equalities and 8 nonlinear equalities. Take the first six equalities you call linear and represent them as linear equalities using the Aeq, beq syntax that I linked to before. The next 4 equalities that you call linear are, in fact, nonlinear equalities, and you can leave them in the form that you gave. The four constraints you call nonlinear equalities are indeed nonlinear equalities, and you can leave them in the form you gave.

Alan Weiss

MATLAB mathematical toolbox documentation

[Lucas Giering]

Yes, after writing my first comment i realized my mistake in that regard. Of course equalities 7-10 are indeed nonlinear.

I applied the changes you advised, but the behaviour is still the same: few initial values converge, others deteriorate to very high function values.

[Alan Weiss]

Maybe you can follow the suggestions in When the Solver Fails. In particular, I believe that you could easily come up with analytic gradients and Hessian for the 'interior-point' algorithm, either using Symbolic Math Toolbox as the easiest way, or calculate it by hand.

Good luck,

Alan Weiss

MATLAB mathematical toolbox documentation