Optimization of objective function with multiple constraints and variables

I need to minimize an objective function with four variables. The function is to be optimized under 6 nonlinear inequalities, as well as lower and upper bounds.

My basic script is this:

P = 6000; 
E = 30e6;  
G = 12e6; 
L = 14; 
fun=@(x) (1+0.10471)*x(1)*x(2)+0.04811*x(3)*x(4)*(L+1);
gs = GlobalSearch;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
problem = createOptimProblem('fmincon','x0',[0.1,0.1,0.1,0.1],'objective',fun,'lb',[0.125,0.1,0.1,0.1],'ub',[5,10,10,5],'nonlcon',nonlcon,'options',opts);

where

function [c,ceq] = nonlcon(~)
P = 6000; 
E = 30e6;  
G = 12e6; 
L = 14; 
c =@(x) [-13600 + sqrt((P/(sqrt(2)*x(1)*x(2)))^2+2*(P/(sqrt(2)*x(1)*x(2)))*((P*(L+x(2)/2))*(sqrt((x(2)^2)/4+...
((x(1)+x(3))/2)^2)))/(2*(sqrt(2)*x(1)*x(2)*((x(2)^2)/12+((x(1)+x(3))/2)^2))))*x(2)/(2*(sqrt((x(2)^2)/4+...
((x(1)+x(3))/2)^2)))+((P*(L+x(2)/2))*(sqrt((x(2)^2)/4+...
((x(1)+x(3))/2)^2)))/(2*(sqrt(2)*x(1)*x(2)*((x(2)^2)/12+((x(1)+x(3))/2)^2)))^2;
-30000+(6*P*L)/(x(4)*x(3)^2);x(1)-x(4);0.10471*x(1)^2+0.04811*x(3)*x(4)*(14+x(2))-5;...
0.125-x(1);(4*P*L^3)/(E*x(4)*x(3)^3)-0.25; P-4.013*E*sqrt((x(3)^2*x(4)^6)/36)*(1-x(3)*sqrt(E/(4*G))/(2*L))/(L^2)]; 
ceq = [];
end

Up until that point, my script runs with no issues. However when I try to insert a way to run the problem and display some results I get errors. One way I tried is

min=run(gs,problem);

and I get

Error using fmincon (line 626)
The constraint function must return two outputs; the nonlinear inequality constraints and the nonlinear equality constraints.

Error in globalsearchnlp

Error in GlobalSearch/run (line 340)
                globalsearchnlp(FUN,X0,A,B,Aeq,Beq,LB,UB,NONLCON,options,localOptions);

Error in confuneq1 (line 23)
 min=run(gs,problem);

Caused by:
    Failure in initial call to fmincon with user-supplied problem structure.

And another one I tried is

sol=solve(problem);

and I get

Error using char
Conversion to char from struct is not possible.

Error in solve>isOption (line 459)
b = ~isa(a, 'logical') && any(strcmpi(char(a), ...

Error in solve>getEqns (line 392)
while k <= numel(argv) && ~isOption(argv{k})

Error in solve (line 226)
[eqns,vars,options] = getEqns(varargin{:});

Error in confuneq1 (line 22)
 sol=solve(problem);

I would appreciate some help. I don't know if the problem creation is wrong, or the ways I try to run it, or both. Thanks

Answer 1

[Stephan]

Hi,

using an anonymus function works here:

P = 6000; 
E = 30e6;  
G = 12e6; 
L = 14; 
fun=@(x)(1+0.10471)*x(1)*x(2)+0.04811*x(3)*x(4)*(L+1);

c = @(x)[-13600 + sqrt((P/(sqrt(2)*x(1)*x(2)))^2+...
    2*(P/(sqrt(2)*x(1)*x(2)))*((P*(L+x(2)/2))*(sqrt((x(2)^2)/4+...
    ((x(1)+x(3))/2)^2)))/(2*(sqrt(2)*x(1)*x(2)*((x(2)^2)/12+((x(1)+...
    x(3))/2)^2))))*x(2)/(2*(sqrt((x(2)^2)/4+...
    ((x(1)+x(3))/2)^2)))+((P*(L+x(2)/2))*(sqrt((x(2)^2)/4+...
    ((x(1)+x(3))/2)^2)))/(2*(sqrt(2)*x(1)*x(2)*((x(2)^2)/12+((x(1)+...
    x(3))/2)^2)))^2;
-30000+(6*P*L)/(x(4)*x(3)^2);
x(1)-x(4);
0.10471*x(1)^2+0.04811*x(3)*x(4)*(14+x(2))-5;
0.125-x(1);(4*P*L^3)/(E*x(4)*x(3)^3)-0.25;
P-4.013*E*sqrt((x(3)^2*x(4)^6)/36)*(1-x(3)*sqrt(E/(4*G))/(2*L))/(L^2)]; 
ceq = @(x)[];
nonl_con = @(x)deal(c(x),ceq(x));

gs = GlobalSearch;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
problem = createOptimProblem('fmincon','x0',[0.1;0.1;0.1;0.1],'objective',fun,'lb',[0.125,0.1,0.1,0.1],'ub',[5,10,10,5],'nonlcon',nonl_con,'options',opts);

min=run(gs,problem);

this gives following result:

GlobalSearch stopped because it analyzed all the trial points.

10 out of 11 local solver runs converged with a positive local solver exit flag.

with min:

>> min

min =

      0.2057
      0.5305
      9.0366
      0.2057

Best regards

Stephan

[Stephan]

The nice thing is that you have a solution to your problem. The unsatisfactory thing is that I can not tell you why an anonmye function works for the nonlinear constraints, but not a "normal" function.

Does anyone have any idea why that is?

Best regards

Stephan

[Star Strider]

In order to pass a function reference as an argument to another function, it is necessary to pass it as a function handle.

I did not run the code to test this, however adding the ‘@’ to create a function handle from nonlcon, specifically: ‘...,'nonlcon',@nonlcon,...’ in the argument list, would probably work.

The anonymous function creates a function handle, eliminating that problem.

[Stephan]

Thank you.

[Nick Chatz]

Thanks a lot

[Star Strider]

My pleasure.

[Nick Chatz]

while we are on the subject. What if I have lower and upper bounds that depend on the variables? How do I introduce that in the fmincon problem?

For example I have lower bounds [0.01*x(1)*x(2), 28, 5] and upper bounds [0.04*x(1)*x(2), 40, 10] for three variables.

[Stephan]

I guess it is better to ask this in a new question.

[Walter Roberson]

You would use nonlinear constraints for that.