Constraining Function in fmincon

I'm attempting to optimize the design of a structure using fmincon. I have x(1) and x(2) as the base and height of an I-Beam, and am trying to find the smallest dimensions to meet all constraints. The problem I'm running into is that there isn't a way to place a constraint on the function itself. I have calculated the smallest area needed to ensure the structure is able to withstand the loads applied to it, but am having trouble applying this constraint. Below is my code, I have tried applying the constraint to be in A, but due to the way the area of an Ibeam is calculated, I haven't been able to figure out placing that in there.

% Variables
%hI and wI (Design Goals) x1 and x2 respectively
x0 = [0, 0];
A_f  = [];
B_f  = [];
Aeq  = [];
Beq  = [];
lb = [0.14, 0.14];
ub = [4.5, 4.5];
xout = fmincon(@IBEAMOP,x0,A_f,B_f,Aeq,Beq,lb,ub)
weight = IBEAMOP(xout)

function [xsectionarea] = IBEAMOP(x)
    %% I-Beam Optimization
    syms P A
    sig      = P/A;

    % Variables
    %hI and wI (Design Goals) x1 and x2 respectively
    t =.140; %in
    P = 118371;
    %Material Parameters
    E_Ti  =  15.5*(10^6);  %pside
    sig_Y =  110*(10^3) ;  %psi
    sig_U =  115*(10^3) ;  %psi
    rho   =  .162;       %lb/(in^3)
    nu    =  .33;
    
    A     =(2*x(2)*t+(x(1)-2*t)*t);
    xsectionarea = (2*x(2)*t+(x(1)-2*t)*t);
    end
end

Answer 1

[Matt J]

It sounds like you need to apply a nonlinear constraint. If so, you need to use fmincon with 9 or more input arguments specified,

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)

[Kip Risch-Andrews]

I tried using fmincon, but couldn't figure out how to perscribe values to c and ceq, as my conditions are constants, rather than other non linear constraints.

[Matt J]

I can't tell what you mean, I'm afraid. I suggest you write down the problem for us in mathematical equation form, rather than trying to communicate it with code.

[Kip Risch-Andrews]

The problem is to optimize the dimensions for an I-Beam so that it can withstand a force without reaching the critical stress values.

Areq = P/sig_U

Since I have already calculated the load applied (P), and through the material specifications (sig_U), I know the minimum area required. The problem I have is that I cannot find a way to apply this constraint to my area, which is what @IBEAMOP produces, by optimizing two of the dimensions. I've tried multiplying x1 and x2 into the area through A in fmincon, but am unable to due to the fact it involves subtraction.

[Matt J]

It's still not very clear to me, but if you are trying to minimize IBEAMOP subject also to a lower bound minArea on the output of IBEAMOP, you would provide the nonlcon function as follows,

lb = [0.14, 0.14];
ub = [4.5, 4.5];
xout = fmincon(@IBEAMOP,x0,[],[],[],[],lb,ub,@nonlcon)

function [c,ceq]=nonlcon(x)

  ceq=[];
  
  c=minArea - IBEAMOP(x);

end


function [xsectionarea] = IBEAMOP(x)

%% I-Beam Optimization

    % Variables
    %hI and wI (Design Goals) x1 and x2 respectively
    t =.140; %in
    P = 118371;
    %Material Parameters
    E_Ti  =  15.5*(10^6);  %pside
    sig_Y =  110*(10^3) ;  %psi
    sig_U =  115*(10^3) ;  %psi
    rho   =  .162;       %lb/(in^3)
    nu    =  .33;

    xsectionarea = (2*x(2)*t+(x(1)-2*t)*t);
end

[Kip Risch-Andrews]

That worked perfectly, thank you for the help! Sorry for being a bit unclear with the problem statement.

Answer 2

[Matt J]

Now that the problem is clearer, it appears to me that you could have done the whole thing with linprog,

f=[2,1]*t;
Aineq  = -[2,1]*t;
Bineq  = -minArea-2*t^2;
lb = [0.14, 0.14];
ub = [4.5, 4.5];

xout=linprog(f,Aineq,Bineq,[],[],lb,ub)