MATLAB: Direct search in Matlab

direct searchmapleoptimization

Hello, I'm trying to get Matlab to give me the the same results that I am able get in Maple.
I have an equation with two unknowns, x and y. They are both angles and I am trying to find the solutions to that equation. If i use a package named DirectSearch in Maple i can get exactly what i want. I can set constraints on x and y. I can choose number of solutions and the distance between them.
In Matlab i have tried to use multistart, nonlcon and various functions from the optimization toolbox. But i can't get Matlab to find the solutions(x and y) which gives me zero. In other words i want the solutions which gives me a function value that equals zero. Here is an example of what i have tried so far
clear all, clc, format compact, format short
ms = MultiStart;
opts = optimoptions(@fmincon,'Algorithm','interior-point');
fun = @(x) (((0.2284963440e-1*sin(x(1))*cos(x(2))*sin(x(2))+0.2284963440e-1*cos(x(1))...
    *sin(x(2))^2+(-0.2944413003e-1*cos(x(1))-0.2777543403e-2*sin(x(1)))*sin(x(2)))...
    *sqrt(3.2186761*10^7-3.1957320*10^7*cos(x(2)-.1584758961))-56.24910000*sin(x(1))...
    *cos(x(2))*sin(x(2))-56.24910000*cos(x(1))*sin(x(2))^2+(72.48281461*cos(x(1))+6.837497436*sin(x(1)))...
    *sin(x(2)))*cos(-1.*acos((6561.744237*(cos(x(2)-.1584758961)-.9222315718))...
    /sqrt(3.2186761*10^7-3.1957320*10^7*cos(x(2)-.1584758961)))+x(2))+((-0.2284963440e-1*sin(x(1))...
    *cos(x(2))^2+(-0.2284963440e-1*cos(x(1))*sin(x(2))+0.2944413003e-1*cos(x(1))+0.2777543403e-2...
    *sin(x(1)))*cos(x(2)))*sqrt(3.2186761*10^7-3.1957320*10^7*cos(x(2)-.1584758961))+56.24910000*sin(x(1))...
    *cos(x(2))^2+(-72.48281461*cos(x(1))-6.837497436*sin(x(1))+56.24910000*cos(x(1))*sin(x(2)))*cos(x(2)))...
    *sin(-1.*acos((6561.744237*(cos(x(2)-.1584758961)-.9222315718))...
    /sqrt(3.2186761*10^7-3.1957320*10^7*cos(x(2)-.1584758961)))+x(2))-.1112132575...
    *sin(x(1))*cos(x(2))^2+(.1433098472*cos(x(1))-.1112132575*cos(x(1))...
    *sin(x(2))+.9563372292*sin(x(1)))*cos(x(2))+.9428184287*cos(x(1))*sin(x(2)))...
    /(sin(x(1))*cos(x(2))+cos(x(1))*sin(x(2)));
%problem = createOptimProblem('fmincon','x0',[0,0],'objective',fun,'lb',[0.85,0.5],'ub',[1.35,0.85], 'options',opts);
problem2 = createOptimProblem('fmincon','x0',[0,0],'objective',fun,'lb',[0.85,0.5],'ub',[1.35,0.85],'nonlcon',@mycon, 'options',opts);
%[xminm,fminm,flagm,outptm,manyminsm] = run(ms,problem,50)
[xminm,fminm,flagm,outptm,manyminsm] = run(ms,problem2,50)
function [c,ceq] = mycon(x)
c = [] 
ceq =@(x)(((0.2284963440e-1*sin(x(1))*cos(x(2))*sin(x(2))+0.2284963440e-1*cos(x(1))*...
    sin(x(2))^2+(-0.2944413003e-1*cos(x(1))-0.2777543403e-2*sin(x(1)))*sin(x(2)))*...
    sqrt(3.2186761*10^7-3.1957320*10^7*cos(x(2)-.1584758961))-56.24910000*sin(x(1))*cos(x(2))...
    *sin(x(2))-56.24910000*cos(x(1))*sin(x(2))^2+(72.48281461*cos(x(1))+6.837497436*sin(x(1)))...
    *sin(x(2)))*cos(-1.*acos((6561.744237*(cos(x(2)-.1584758961)-.9222315718))...
    /sqrt(3.2186761*10^7-3.1957320*10^7*cos(x(2)-.1584758961)))+x(2))+((-0.2284963440e-1...
    *sin(x(1))*cos(x(2))^2+(-0.2284963440e-1*cos(x(1))*sin(x(2))+0.2944413003e-1...
    *cos(x(1))+0.2777543403e-2*sin(x(1)))*cos(x(2)))*sqrt(3.2186761*10^7-3.1957320*10^7*cos(x(2)...
    -.1584758961))+66.2491*sin(x(1))*cos(x(2))^2+(-72.48281461*cos(x(1))-6.837497436*sin(x(1))...
    +56.24910000*cos(x(1))*sin(x(2)))*cos(x(2)))*sin(-1.*acos((6561.744237*(cos(x(2)-.1584758961)-.9222315718))...
    /sqrt(3.2186761*10^7-3.1957320*10^7*cos(x(2)-.1584758961)))+x(2))-.1112132575*sin(x(1))*...
    cos(x(2))^2+(.1433098472*cos(x(1))-.1112132575*cos(x(1))*sin(x(2))+.9563372292*sin(x(1)))*...
    cos(x(2))+.9428184287*cos(x(1))*sin(x(2)))/(sin(x(1))*cos(x(2))+cos(x(1))*sin(x(2)))
end

Best Answer

FMINCON is not a root finder. Minimizing fun will not necessarily give you zero. You need to use fsolve, or lsqnonlin depending on the x,y constraints you are trying to implement.
Also, it is overkill to use MultiStart for a 2-variable problem. Simply make a surf plot of your objective function and see where it gets close to zero. Or use min(abs(z)) to search the surface samples.
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