# Optimizing a multi-modal function with a two step anneal method.

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I have been working on a reliable optimization method for this crazy function.

f.egg<-function(x,y){ (2+cos(x)+cos(y))/(100+x^2+y^2) }

I noticed that if I had a large variance in the random normal generator, the optimizer would jump all over the place but would not settle down in the best optimum. So I added in a second step that takes the best of the attempted values and uses them as a second set of start values. Then, a second smaller variance is used that does not allow for major jumps. The two step anneal function can be seen below.

twostep.anneal<-function(f,mu,n=1000,sig=c(1,.1),t=1,g=0.999){ xm = mu[1] ym = mu[2] fm = f(xm,ym) x = rep(NA,n*2) y = rep(NA,n*2) fx = rep(NA,n*2) for(k in 1:2){ s=sig[k] if(k==2){ xm=best[1] ym=best[2] } for (i in 1:n){ dxm = xm+rnorm(1,0,s) dym = ym+rnorm(1,0,s) fdm = f(dxm, dym) t = t*g if (runif(1) < (fdm/fm)^(1/t)){ xm = dxm ym = dym fm = fdm } x[(i+(k-1)*n)] = xm y[(i+(k-1)*n)] = ym fx[(i+(k-1)*n)] = fm } ii = which(fx==max(fx, na.rm=TRUE))[1] best=c(x[ii], y[ii]) } list(x = x, y = y, fx = fx, best = best , fbest = fx[ii], t=t) }

To run the function and see the path of the anneal function use this code.

x<-seq(-30,30,.5) y<-seq(-30,30,.5) z<-outer(x,y, f.egg) aa<-twostep.anneal(f.egg, n=1000, sig=c(2, 2), mu=c(5,5), t=2, g=.99) contour(x,y,z) lines(aa$x, aa$y, col=2)

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