# The Dixon-Szego function in 2D optimisation

Jari Fowkes and Nick Trefethen, November 2010

(Chebfun example opt/DixonSzego.m)

The Chebfun example opt/Rosenbrock.m shows how Chebfun can be used to minimize a function of two variables over a rectangle. The present example is adapted from that one, and simply considers another function investigated by Dixon and Szego in 1975:

f = @(x,y) (4-2.1*x.^2+ x.^4/3).*x.^2 + x.*y + 4*(y.^2-1).*y.^2;

Over the rectangle [-2,2] x [-1.25,1.25], the function looks like this:

LW = 'linewidth'; FS = 'fontsize'; MS = 'markersize'; x = linspace(-2,2); y = linspace(-1.25,1.25); [xx,yy] = meshgrid(x,y); ff = f(xx,yy); figure, contour(x,y,ff,30,LW,1.2), colorbar axis([-2 2 -1.25 1.25]), hold on

Here is Chebfun code taken from opt/Rosenbrock.m to find the minimum and plot the point where it is attained. (The minimum is actually achieved at two points, because of symmetry, but Chebfun does not detect this.)

tic fminx0 = @(x0) min(chebfun(@(y) f(x0,y),[-1.25 1.25])); fminx = chebfun(fminx0,[-2 2],'vectorize','splitting','on'); [minf,minx] = min(fminx) [minf,miny] = min(chebfun(@(y) f(minx,y), [-1 3])) toc plot(minx,miny,'.k',MS,20)

minf = -1.0316 minx = 0.0898 minf = -1.0316 miny = -0.7127 Elapsed time is 3.303239 seconds.

Reference:

[1] L. C. W. Dixon and G. P. Szego, The global optimization problem: an introduction, in L. C. W. Dixon and G. P. Szego (eds.), Towards Global Optimisation 2, North-Holland, Amsterdam 1978, pp. 1-15.