Finding x & y-values that maximize an expression!!!!

Hi all,

I want to find the x and y values that maximize the following expression.

Delta_i's are numeric values and R_i's are in terms of x and y.

Thank you!

[Selen]

Actually, I am looking for x and y values that maximize this expression. All r's and BS's are known. Is there any equivalent function of fminsearch for finding max values?

Thanks a lot!

Answer 1

[Roger Wohlwend]

You can use the fminsearch function because maximizing J is the same as minimizing -J. That is why there is no optimizing function in MATLAB that finds maximas, there are only functions that find minimas.

Answer 2

[Selen]

I already tried that but it gives me the error: Maximum number of function evaluations has been exceeded!

Should the argument of fminsearch be something else than -F?

Thanks.

Answer 3

[George Papazafeiropoulos]

The expression for J does not have any maxima. This can be seen easily by geometric interpretation of the formula. J can be minimized according to he following:

BS=rand(7,2);
r=rand(7,1);
F=@(xy)(2*r(1).*hypot(xy(1)-BS(1,1),xy(2)-BS(1,2))+(hypot(xy(1)-BS(1,1),xy(2)-BS(1,2))).^2 ...
    +2*r(2).*hypot(xy(1)-BS(2,1),xy(2)-BS(2,2))+(hypot(xy(1)-BS(2,1),xy(2)-BS(2,2))).^2 ...
    +2*r(3).*hypot(xy(1)-BS(3,1),xy(2)-BS(3,2))+(hypot(xy(1)-BS(3,1),xy(2)-BS(3,2))).^2 ...
    +2*r(4).*hypot(xy(1)-BS(4,1),xy(2)-BS(4,2))+(hypot(xy(1)-BS(4,1),xy(2)-BS(4,2))).^2 ...
    +2*r(5).*hypot(xy(1)-BS(5,1),xy(2)-BS(5,2))+(hypot(xy(1)-BS(5,1),xy(2)-BS(5,2))).^2 ...
    +2*r(6).*hypot(xy(1)-BS(6,1),xy(2)-BS(6,2))+(hypot(xy(1)-BS(6,1),xy(2)-BS(6,2))).^2 ...
    +2*r(7).*hypot(xy(1)-BS(7,1),xy(2)-BS(7,2))+(hypot(xy(1)-BS(7,1),xy(2)-BS(7,2))).^2);
out=fmincon(F,[1,1],[1,1],inf)

[Selen]

Thank you.

Does this give me the minimizing x&y values or the maximizing ones?

[George Papazafeiropoulos]

It gives the minimizing values....

Answer 4

[Roger Stafford]

Revised statement: J has no maximum value. It could have local maxima points, depending on the locations of the points (xi,yi) and the delta values. However it is obvious that it can always be made arbitrarily large by moving the point (x,y) sufficiently far from the origin in any direction in the x-y plane.

For example, if there is only a single point (x1,y1), by locating (x,y) at that point, J would be at a local maximum - that is, it is greater there than at any point in its immediate neighborhood. If we draw a circle around the point (x1,y1) of radius delta1, J continues to decrease as (x,y) is moved away from (x1,y1) until reaching the circle where it would be zero. Beyond that, J increases again and it is clear that it can be made arbitrarily large by moving sufficiently far outside the circle. This also remains true with many points rather than one. There can be many local maxima points, but no absolute maximum is possible.