I tried to show effect of uncertain parameters on bi-variate function with color map and aslo highlight changes to max and min,I am not sure the function I wrote is right!!!

for i=1:length(x)   
    x=lhsdesign(10,1,"iterations",2);
    for j=1:length(y)
        y=lhsnorm(50,10,10,"on");
        Z=[x,y];
        
        Z(i,j)=2.*x(i).^3+y(j).^3-3.*x(i).^2-12.*x(i)-3.*y(j);    
    end
end

options = optimset('Display','iter','PlotFcns',@optimplotfval);
x0=[x(1),y(1)]
[xmin,fval]=fminsearch(@(xy)Z(xy(1),xy(2)),x0,options); %min 
[xmax ,fval]=fminsearch(@(xy) -1 *Z(xy(1),xy(2)),x0,options); %max

[X,Y]=meshgrid(x,y);
surf(X,Y,Z(i,j),'EdgeColor',"interp","FaceAlpha",0.5);
hold on
plot(xmin(1),xmin(2), f(xmin(1),xmin(2)),'MarkerSize',6);
hold on
plot(xmax(1),xmax(2), f(xmax(1),xmax(2)),'MarkerSize',6);
hold off
xlabel("optimum valeus for x")
ylabel("optimum valeus for y")
colormap(parula(6));%default colormap with 6 colors
colorbar

can anyone help me for correcting this function?

[sogol bandekian]

@Star Strider

can you tell me here?

[Torsten]

You calculate a Z matrix here that is not referenced later on:

for i=1:size(x)   
    for j=1:size(y)
        Z(i,j)=2.*x(i).^3+y(j).^3-3.*x(i).^2-12.*x(i)-3.*y(j);    
    end
end

Do you want to search for the minimum of Z by using Zmin = min(min(Z)) ?

[sogol bandekian]

I tried to generate same uncertain points by x and y then implement them to Z ,for showing the effect of uncertain parameters on a bi-variate function(f(x,y)=z)and also detemine the maxima and minima

[Torsten]

x = 0:0.1:1;
y = 0:0.1:1;
[X,Y] = meshgrid(x,y);
Z = 2.*X.^3+Y.^3-3.*X.^2-12.*X-3.*Y;
Z = Z +(-0.1+0.2*rand(size(Z)));
fun = @(x)interp2(X,Y,Z,x(1),x(2));
options = optimset('Display','iter','PlotFcns',@optimplotfval);
x0=[x(1),y(1)]
[xmin,fval]=fminsearch(fun,x0,options); %min 

But since the function is not smooth, fminsearch is not the correct tool. You can just use min(min(Z)).

[sogol bandekian]

min(min(Z)) gives global min?how about max?

[Torsten]

min(min(Z)) gives the minimum element of the matrix Z.

Same for max(max(Z)).

[sogol bandekian]

it means find global min and max?

[Torsten]

It means finding min and max of Z in the points specified by (x(i),y(j)).

[Walter Roberson]

Some people find it easier to understand by using

min(Z(:))

and

max(Z(:))

instead of min(min(Z)) and max(max(Z))

[sogol bandekian]

but they are not optimization solvers like fminsearch.are not they?

[Torsten]

No. It's a search for a minimum or maximum of a functrion on a discrete set of (x/y) values. The function is evaluated on a (fine enough) grid and the point where this evaluation is minimal (or maximal) is taken as the minimum (or maximum) of the function. It's an alternative to fminsearch and other optimizers if the region where minimum (or maximum) is is approximately known and if the function is badly behaved (not differentiable etc).

Answer 1

[Walter Roberson]

y=lhsnorm(50,10,10,"on");

[xmin,fval]=fminsearch(@(x)Z(x,y),x0,options) %min 

The y referred to there is being "captured" from the workspace variable y which is the lhsnorm that you defined earlier.

With y being non-scalar, your multinomial Z function is going to return multiple values for each input x value, but for fminsearch you need to return a vector.

Perhaps what you need is

[xmin,fval]=fminsearch(@(xy)Z(xy(1),xy(2)),x0,options) %min 

[sogol bandekian]

thanks alot for your answer

But if I define only y how about x?will x generate randome variable for input Z?since I want to show the effect of uncertainty on bivariate function

I change my code also ,please look at them again

[Torsten]

In your code above, Z is a matrix of values. To use fminsearch it must be a function handle. Thus your code will not work.

Maybe you could explain what you are trying to do.

[sogol bandekian]

I tried to generate same uncertain points by x and y then implement them to Z ,for showing the effect of uncertain parameters on a bi-variate function(f(x,y)=z)and also detemine the maxima and minima with using color map

[Walter Roberson]

x = 0:0.1:1;
y = 0:0.1:1;
[X,Y] = meshgrid(x,y);
Z = 2.*X.^3+Y.^3-3.*X.^2-12.*X-3.*Y;
Z = Z + (-0.1+0.2*rand(size(Z)));
fun = @(x)interp2(X, Y, Z, x(1), x(2), 'spline');
options = optimset('Display','iter','PlotFcns',@optimplotfval);
x0=[x(1),y(1)]
lb = [min(x), min(y)];
ub = [max(x), max(y)];
[xmin,fval] = fmincon(fun, x0, [], [], [], [], lb, ub, [], options); %min 

You will get a location near (1,1) and a fval about -14.9 .

The actual minimum for the non-noisy function is -15 at (1,1) exactly, and with the noise might be slightly less than that. But it does not find the minima that closely, as the minima is exactly at the bounds.

Consider your interpolation for a moment. Is there ever a case where the interpolated value can be less than the minimum Z (noisy) Z value?

If you were using linear interpolation or bilinear interpolation, the answer would be NO -- but on the other hand linear and bilinear interpolation do not satisfy the continuity requirements.

Is it possible for cubic interpolation to interpolate a value that is less than the minimum Z value? The answer to that is Yes. If you consider an edge such as

     + +
         +
    +

then in order to fit a quadratic or cubic to that, the interpolated value needs to come up from the left, pass upwards though the second, pass downwards through the third, and then on through the fourth. That implies that between the second and third, the projection would be for a value above the maximum (second, third) point. Any time that the third point does not fall off as quickly as the rise between the first and second points, quadratic or cubic fit needs to project a maximum between the second and third points. This discussion is in terms of max, but flip it upside down and you can see the same issue applies to minima.

Thus, in order to have the possibility of a projected minima being less than the minima present in the array, you have to be counting on the details of how the interpolation algorithm overshoots beyond the actual points.

If you do not want to be exploring the details of interpolation overshoot between close points, then there would be no possibility of interpolation getting a value less than the minima in the array, and so there would be no point in doing a search -- the minima would have to be at one of the locations that is exactly equal to the minimum in the array.

[sogol bandekian]

thank you so muc for your completed explaination .for getting max what can I do?and how I can use colormap for showing max and minima??

[Walter Roberson]

create a new function which is the negative of your existing function, and minimize that, then take the negative of the resulting value.

I suggest that plot() or plot3() or text() to mark the points of interest.

[sogol bandekian]

x = 0:0.2:1;
y = 0:0.2:1;
[X,Y] = meshgrid(x,y);
Z = 2.*X.^3+Y.^3-3.*X.^2-12.*X-3.*Y;
Z2 = -(2.*X.^3+Y.^3-3.*X.^2-12.*X-3.*Y);
Z = Z +(-0.1+0.2*rand(size(Z)));
Z2 = Z2 +(-0.1+0.2*rand(size(Z2)));
fun = @(x)interp2(X,Y,Z,x(1),x(2));
fun2= @(x)interp2(X,Y,Z2,x(1),x(2));%Interpolation for 2-D gridded data in meshgrid format
options = optimset('Display','iter','PlotFcns',@optimplotfval);
x0=[x(1),y(1)]
lb = [min(x), min(y)];
ub = [max(x), max(y)];
[xmin,fval] = fmincon(fun, x0, [], [], [], [], lb, ub, [], options); %min
[xmax,fval]= fmincon(fun2, x0, [], [], [], [], lb, ub, [], options); %max





for getting max values I did this ,but the plotpoints are getting less and this is not showign max and minima

[Walter Roberson]

x = 0:0.2:1;
y = 0:0.2:1;
[X,Y] = meshgrid(x,y);
Z = 2.*X.^3+Y.^3-3.*X.^2-12.*X-3.*Y;
Z = Z + (-0.1+0.2*rand(size(Z)));
fun = @(x)interp2(X,Y,Z,x(1),x(2), 'spline');
fun2 = @(x)-fun(x);
options = optimset('Display','iter','PlotFcns',@optimplotfval);
x0=[x(1),y(1)]
lb = [min(x), min(y)];
ub = [max(x), max(y)];
[xmin,fvalmin] = fmincon(fun, x0, [], [], [], [], lb, ub, [], options); %min
[xmax,fvalmax]= fmincon(fun2, x0, [], [], [], [], lb, ub, [], options); %max
fvalmax = -fvalmax;

surf(X, Y, Z, 'edgecolor', 'none') 
hold on 
plot3(xmin(1), xmin(2), fvalmin, 'rv') 
plot3(xmax(1), xmax(2), fvalmax, 'r^') 
hold off

[sogol bandekian]

thank you so much