Beale

Problem definition

Objective function
f = @(x)  (1.5 - x(:,1) + x(:,1).*x(:,2)).^2 + ...
       (2.25 - x(:,1) + x(:,1).*x(:,2).^2).^2 + ...
       (2.625 - x(:,1) + x(:,1).*x(:,2).^3).^2
Optimization settings
o = struct	
% initializing struct

o.lb = -4.5	
% lower bounds

o.ub = 4.5	
% upper bounds
Graphic representation 
[x,y] = meshgrid(o.lb:0.4:o.ub)
surf(x,y,f([x(:),y(:)]))

Problem properties

convexity smoothness minimum
1 ∞ f(3,0.5) = 0

Optimization example with fminsearch

Optimization 
rng(0)	
% for tractability 

x0 = [1,1]	
% initial guess 

[xmin,fmin,info] = fminsearch(f,x0,o)	
% running minimization
Animation 
rng(0)
x0 = [1,1]
[~,~,info] = fminsearch(f,x0,o)
info.sol = [3 0.5 0]	
% solution 

info.animate = true	
% plot animation 

info.animfreq = 5	
% frame frequency 

optimview('fminsearch',info)

References

[1] E.M.L. Beale, "On an iterative method of finding a local minimum of a function of more than one variable", Princeton University Stat. Tech. Res. Group, Tech. Report 25, 1958

Related functions

fminsearch | meshgrid | surf