Maxima and Minima using fminsearch()?

I am trying to find the minima and maxima of this function. The first section of my code is the graph of a surface I had to create using this function. The second section of my code is my (failed) attempt at using fminsearch() to find the minimum of the function (though I also have to find the max as well). I require some assistance on how this fminsearch() function is formatted, and how it can be used to find the max too. The picture I linked is what my answers are supposed to look like. Here is my code

x=[-10:1:10];
y=x;

   subplot(2,2,1)  % separates figure window into 2 rows and 2 columns
  [xGrid yGrid]=meshgrid(x,y);
  z=(1./((xGrid+3).^2+(yGrid-1).^2+2))+((xGrid-yGrid)./((xGrid-1).^2+(yGrid-2).^2+4)); % function of x/y
  surf(x,y,z) % standard projection of surface is isometric
  title('Isometric View') % graph title
  xlabel('x'),ylabel('y'),zlabel('z') % graph labels

   NegFunction=@(x)(1./((x(1)+3).^2+(x(2)-1).^2+2))+((x(1)-x(2))./((x(1)-1).^2+(x(2)-2).^2+4)); 
  [xyMinVector,zMin]=fminsearch(NegFunction,[3.5,0]);
  xMin = xyMinVector(1); % value of x when z is a minimum
  yMin = xyMinVector(2); % value of y when z is a minimum
  fprintf('The minimum value was: z(%0.3f,%0.3f)=%2.0f\n',xMin,yMin,zMin)

Answer 1

[Walter Roberson]

You do not define NegFunction in what you posted.

[Karan Sandhu]

After running the code again, i get this: "Exiting: Maximum number of function evaluations has been exceeded - increase MaxFunEvals option. Current function value: 0.000000

The minimum value was: z(358139351549666070000000000000000000000000000.000,95234708364011678000000000000000000000000.0"

[Walter Roberson]

[xyMinVector,zMin]=fminsearch(NegFunction,[3.5,0], struct('MaxFunEvals', 2000, 'MaxIter', 2000));

[John BG]

out of bounds: Apply fminsearch within [0 0] centered square side 20.

When the figures show up as large as your results, with so many zeros, either you are encrypting something, lost precision many loops before MATLAB stops, or it's loose way to say Inf.

[Walter Roberson]

fminsearch() is not a global minimizer. It can get stuck in local minima. For this function, there is a minima of value 0 as x(1) approaches infinity and x(2) is something smaller. If you do not happen to enter into the catch-basin of negative values near the solution, then you can end up chasing the 0 out at infinity.

Judging by eye, along y = 0, it appears to me that the watershed for the catch-basin is x = 3.0, so [3.5,0] happens to be at risk of that behaviour. If the starting point had been, for example, [2.5,0] then it would have quickly gone to the minima.

Let this be a cautionary tale about using fminsearch! It is very useful, but it is a local minimizer that cannot always overcome bad starting points.

[Karan Sandhu]

Ok so I am trying to find the maximum of this function now. Here is my code. I basically negated the original function. The problem is, I keep getting 0.6... as my answer when john pointed out that it should be around 0.3. Any ideas?

 NegFunction=@(x)-1*(1./((x(1)+3).^2+(x(2)-1).^2+2))+((x(1)-x(2))./((x(1)-1).^2+(x(2)-2).^2+4)); 
[xyMaxVector,zMax] = fminsearch(NegFunction,[-0.3,0])
xMax = xyMaxVector(1); % value of x when z is a maximum
yMax = xyMaxVector(2); % value of y when z is a maximum
fprintf('\nThe maximum value was: z(%6.3f,%6.3f)=%6.3f\n',xMax,yMax,-zMax)

Answer 2

[John BG]

with x step 1 you get

min(min(z))
ans =  -0.433333333333333

max(max(z))
ans =   0.309523809523810

zmax_lin_index=find(z==max(max(z)))

=159

zmin_lin_index=find(z==min(min(z)))

=224

[xmax ymax]=ind2sub(size(z),zmax_lin_index)
xmax =
    12
ymax =
     8
[xmin ymin]=ind2sub(size(z),zmin_lin_index)
xmin =
    14
ymin =
    11

refining x step down to .01

min(min(z))
 =
  -0.436048111705817
 max(max(z))
 =
   0.314386299028360
 zmax_lin_index=find(z==max(max(z)))
zmax_lin_index =
     1393784
 zmin_lin_index=find(z==min(min(z)))
zmin_lin_index =
     2030325
 [xmax ymax]=ind2sub(size(z),zmax_lin_index)
xmax =
        1088
ymax =
   697
 [xmin ymin]=ind2sub(size(z),zmin_lin_index)
xmin =
        1311
ymin =
        1015

If you find this answer of any help solving this question, please click on the thumbs-up vote link,

thanks in advance

John

[e-mail]