# Copyright (C) 1997-2009, 2017 The R Core Team ### Helical Valley Function ### Page 362 Dennis + Schnabel require(stats); require(graphics); require(utils) theta <- function(x1,x2) (atan(x2/x1) + (if(x1 <= 0) pi else 0))/ (2*pi) ## but this is easier : theta <- function(x1,x2) atan2(x2, x1)/(2*pi) f <- function(x) { f1 <- 10*(x[3] - 10*theta(x[1],x[2])) f2 <- 10*(sqrt(x[1]^2+x[2]^2)-1) f3 <- x[3] return(f1^2 + f2^2 + f3^2) } ## explore surface {at x3 = 0} x <- seq(-1, 2, length.out=50) y <- seq(-1, 1, length.out=50) z <- apply(as.matrix(expand.grid(x, y)), 1, function(x) f(c(x, 0))) contour(x, y, matrix(log10(z), 50, 50)) str(nlm.f <- nlm(f, c(-1,0,0), hessian = TRUE)) points(rbind(nlm.f\$estim[1:2]), col = "red", pch = 20) stopifnot(all.equal(nlm.f\$estimate, c(1, 0, 0))) ### the Rosenbrock banana valley function fR <- function(x) { x1 <- x[1]; x2 <- x[2] 100*(x2 - x1*x1)^2 + (1-x1)^2 } ## explore surface fx <- function(x) { ## `vectorized' version of fR() x1 <- x[,1]; x2 <- x[,2] 100*(x2 - x1*x1)^2 + (1-x1)^2 } x <- seq(-2, 2, length.out=100) y <- seq(-0.5, 1.5, length.out=100) z <- fx(expand.grid(x, y)) op <- par(mfrow = c(2,1), mar = 0.1 + c(3,3,0,0)) contour(x, y, matrix(log10(z), length(x))) str(nlm.f2 <- nlm(fR, c(-1.2, 1), hessian = TRUE)) points(rbind(nlm.f2\$estim[1:2]), col = "red", pch = 20) ## Zoom in : rect(0.9, 0.9, 1.1, 1.1, border = "orange", lwd = 2) x <- y <- seq(0.9, 1.1, length.out=100) z <- fx(expand.grid(x, y)) contour(x, y, matrix(log10(z), length(x))) mtext("zoomed in");box(col = "orange") points(rbind(nlm.f2\$estim[1:2]), col = "red", pch = 20) par(op) with(nlm.f2, stopifnot(all.equal(estimate, c(1,1), tol = 1e-5), minimum < 1e-11, abs(gradient) < 1e-6, code %in% 1:2)) fg <- function(x) { gr <- function(x1, x2) c(-400*x1*(x2 - x1*x1)-2*(1-x1), 200*(x2 - x1*x1)) x1 <- x[1]; x2 <- x[2] structure(100*(x2 - x1*x1)^2 + (1-x1)^2, gradient = gr(x1, x2)) } nfg <- nlm(fg, c(-1.2, 1), hessian = TRUE) str(nfg) with(nfg, stopifnot(minimum < 1e-17, all.equal(estimate, c(1,1)), abs(gradient) < 1e-7, code %in% 1:2)) ## or use deriv to find the derivatives fd <- deriv(~ 100*(x2 - x1*x1)^2 + (1-x1)^2, c("x1", "x2")) fdd <- function(x1, x2) {} body(fdd) <- fd nlfd <- nlm(function(x) fdd(x[1], x[2]), c(-1.2,1), hessian = TRUE) str(nlfd) with(nlfd, stopifnot(minimum < 1e-17, all.equal(estimate, c(1,1)), abs(gradient) < 1e-7, code %in% 1:2)) fgh <- function(x) { gr <- function(x1, x2) c(-400*x1*(x2 - x1*x1) - 2*(1-x1), 200*(x2 - x1*x1)) h <- function(x1, x2) { a11 <- 2 - 400*x2 + 1200*x1*x1 a21 <- -400*x1 matrix(c(a11, a21, a21, 200), 2, 2) } x1 <- x[1]; x2 <- x[2] structure(100*(x2 - x1*x1)^2 + (1-x1)^2, gradient = gr(x1, x2), hessian = h(x1, x2)) } nlfgh <- nlm(fgh, c(-1.2,1), hessian = TRUE) str(nlfgh) ## NB: This did _NOT_ converge for R version <= 3.4.0 with(nlfgh, stopifnot(minimum < 1e-15, # see 1.13e-17 .. slightly worse than above all.equal(estimate, c(1,1), tol=9e-9), # see 1.236e-9 abs(gradient) < 7e-7, code %in% 1:2)) # g[1] = 1.3e-7