How to define objective function that is not a direct function of decision variable?

I want to minimize "Cost" while "a, b, c" are the variables.

F1 = (1/a) + b

F2 = 2/F1

F3 = F2*3/c

Cost = F2 + F3

I was wondering how can I solve for minimum Cost, using a problem-based non-linear optimization approach? Any lead will be greatly appreciated.

[Matt J]

The best advice will depend on what constraints you have on a,b,c. Without constraints, your objective function is unbounded.

[Thaneer Malai Narayanan]

Thanks, lets assume the variables ae bounded and continuous. My question is how do i express the constaints and obj function of this problem?

Answer 1

[Ameer Hamza]

See this example to see how define the objective function and bound on the optimization variables.

F1 = @(a,b,c) (1./a) + b;
F2 = @(a,b,c) 2./F1(a,b,c);
F3 = @(a,b,c) F2(a,b,c).*3./c;
Cost = @(a,b,c) F2(a,b,c) + F3(a,b,c);

lb = [0 0 0]; % lower bounds on a, b, and c
ub = [1 1 1]; % lower bounds on a, b, and c
x0 = rand(1,3); % initial guess

x_sol = fmincon(@(x) Cost(x(1), x(2), x(3)), x0, [], [], [], [], lb, ub);
a_sol = x_sol(1);
b_sol = x_sol(2);
c_sol = x_sol(3);

[Thaneer Malai Narayanan]

Thank you Ameer. This worked. Now I want to add a constrant on F2. Could you please show how to do it?

For example,

F2 <= 10

I was wondering how to change include that as Ax = b? Really appreciate your help.

[Ameer Hamza]

Thaneer, You cannot include as Ax=b, because your constraint is not linear. You can add it like this.

F1 = @(a,b,c) (1./a) + b;
F2 = @(a,b,c) 2./F1(a,b,c);
F3 = @(a,b,c) F2(a,b,c).*3./c;
Cost = @(a,b,c) F2(a,b,c) + F3(a,b,c);

lb = [0 0 0]; % lower bounds on a, b, and c
ub = [1 1 1]; % lower bounds on a, b, and c
x0 = rand(1,3); % initial guess

x_sol = fmincon(@(x) Cost(x(1), x(2), x(3)), x0, [], [], [], [], lb, ub, @(x) cons(x, F2));
a_sol = x_sol(1);
b_sol = x_sol(2);
c_sol = x_sol(3);

function [c, ceq] = cons(x, F2)
    ceq = [];
    c = F2(x(1), x(2), x(3)) - 10;
end

Solution:

>> x_sol
x_sol =
    0.0000    0.7288    0.8549

Verify of constraint is met:

    >> F2(x_sol(1), x_sol(2), x_sol(3))
ans =
   4.4348e-09