optimization of multivariable function with nonlinear constraints

I am trying to find the optimum value of a parameter (alpha) for the following expression

function = -x(1)*y(1)*c + x(2)*y(2)*c + alpha*x(1)*x(2)

which x(1), y(1), x(2), y(2) and c have a wide range values

x(1) = x(2) = linspace(0,10,10)
y(1) = y(2) = linspace(0,2,10)
c = linspace(0,100,10)

and constraints are

-x(1)*y(1) + 2*y(1)*c - alpha*x(1)*x(2) < 0
x(2)*y(2) + 2*y(2)*c - alpha*x(1)*x(2) < 0

the goal is to find the alpha values for each set of variables: x(1), y(1), x(2), y(2) and c such that the function has the minimum values.

For example:

what would be the value of alpha for x(1) = 5, x(2) = 3, y(1) =2,y(2)=1 and c=70 which the function is minimum. Basically the code should calculate all different alpha values for all x(1), y(1), x(2), y(2) and c variables.

[Matt J]

The title of your post inaccurately describes the constraints as nonlinear. They are in fact linear in alpha (which is what matters).

Answer 1

[Matt J]

Because the objective function is a positive-sloped line in alpha for all x1,x2 combinations, the minimizing alpha is simply the smallest value that satifies the constraints. This can be obtained, for all combinations as,

[X1,X2,Y1,Y2,C]=ndgrid(linspace(0,10,10),linspace(0,10,10),...
                       linspace(0,2,10),  linspace(0,2,10),linspace(0,100,10));

X1X2 = X1.*X2;

infeasible = X1X2==0 & ( (2*Y1.*C-X1.*Y1)>0 | (2*Y2+X2.*Y2)>0 );
invariant = X1X2==0 & ~infeasible;

minAlpha = max( (2*Y1.*C-X1.*Y1)./X1X2 , (2*Y2+X2.*Y2)./X1X2   ); %combinations of solutions
  minAlpha(infeasible)=nan;
  minAlpha(invariant)=0;

[Boy]

@Matt J Thank you for the code. I think the code you wrote is optimizing the conditions instead to find the optimum value of alpha to minimize the objective fucntion that I wrote above. My goal is to find the optimum value for alpha to minimize this function

function = -x(1)*y(1)*c + x(2)*y(2)*c + alpha*x(1)*x(2)

and the boundery conditions are

-x(1)*y(1) + 2*y(1)*c - alpha*x(1)*x(2) < 0
x(2)*y(2) + 2*y(2)*c - alpha*x(1)*x(2) < 0

[Matt J]

No, my code is finding the optimum value(s) of alpha.

[Boy]

@Matt JThanks!

is there a way that I can export the alpha vales and corresponding values: x(1), x(2), y(1), y(2) and c in an .csv file?

[Matt J]

@Matt J Thanks!

You're quite welcome, but please Accept-click the answer to indicate that your question is resolved.

is there a way that I can export the alpha vales and corresponding values: x(1), x(2), y(1), y(2) and c in an .csv file?

writematrix([X1(:),X2(:),Y1(:),Y2(:),C(:), minAlpha(:)] , 'FileName.csv')

[Boy]

Thanks! @Matt J