#simple monte carlo optimization#
par(mfrow=c(1,2))
#The function to be optimized
mci <- function(x){(cos(50*x)+sin(20*x))^2}
plot(function(x)mci(x), xlim=c(0,1),ylim=c(0,4),lwd=2)
optimize(mci, c(0, 1), tol = 0.0001, maximum=TRUE)
#The monte carlo maximum
nsim<-5000;u<-runif(nsim);
max(mci(u))
plot(u,mci(u))
#The "exact" value is 3.8325#