constraint optimization with alternatives under specific conditions

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I have the following optimization problem.

I'm using matlab to optimize a and b using fmincon function. I can define a and b as bound constraints but I'm confused where should I define the function c and d.

Thanks in advance for any help and/or clue. PS: I'm not sure if fmincon is okay for this kind of problem.

Answer 1

[Torsten]

Use this objective function for fmincon:

function f = objective(vec,x,y,z)

a=vec(1);
b=vec(2);

if 2*x*a-y*b-z >=4
  c=1;
elseif 2*x*a-y*b-z < 0
  c=0;
else
  c=(2*x*a-y*b-z)/2;
end

if x*c-y*b-z >=2
  d=1;
elseif x*c-y*b-z < 0
  d=0;
else
  d=(x*c-y*b-z)/2;
end 

f = -(a*c*x+b*d*y+z);

end

and call fmincon as

x=...;
y=...;
z=...;
X=fmincon(@(vec)objective(vec,x,y,z),[0.5 0.5],[],[],[],[],[0 0],[1 1]);

Best wishes

Torsten.

[Vizimax]

Thanks a lot. The problem is solved now with your answer.

Regards, Viz

Answer 2

[Alan Weiss]

Your problem looks straightforward enough for fmincon. You simply have to write your nonlinear constraints appropriately.

You need to define your variables as a single vector, usually called x. For example,

 a = x(1);
 b = x(2);
 c = x(3);
 d = x(4);
 xx = x(5); % I write xx for your original x
 y = x(6);
 z = x(7);

Then your objective function becomes

 @(x)-(x(1)*x(3)*x(5)+x(2)*x(4)*x(6)+x(7));

I write the negative of what you wrote as an objective function because you want to maximize the objective.

It is straightforward to write your nonlinear equality constraints in terms of x as well.

Start fmincon at a reasonable initial point and I believe that you will quickly arrive at an answer.

Alan Weiss

MATLAB mathematical toolbox documentation

[Vizimax]

Hi Alan, Thanks for your reply. My concern is how to define c and d. You wrote c = x(3) and d = x(4) but in the problem there are no explicit values of x(3) and x(4). These are defined by x(1), x(2), x(5), x(6) and x(7). I'm confused about where to define them ? Thanks for your time and help. Viz

[Torsten]

x,y and z are given constants ?

Best wishes

Torsten.

[Vizimax]

hi Torsten, Yes x, y, z are given constants.