MATLAB: Optimization of a function with constraints containing 15 variables

active-setconstrained optimizationconstraintsequalitiesfmincongainequalitiesinterior-pointlinearminimizationminimum valuenonlinearnonlinear contraintsnonlinear opimizationoptimizationsqp

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)

Best Answer

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
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