Examples:

     require(graphics)
     
     fr <- function(x) {   ## Rosenbrock Banana function
         x1 <- x[1]
         x2 <- x[2]
         100 * (x2 - x1 * x1)^2 + (1 - x1)^2
     }
     grr <- function(x) { ## Gradient of 'fr'
         x1 <- x[1]
         x2 <- x[2]
         c(-400 * x1 * (x2 - x1 * x1) - 2 * (1 - x1),
            200 *      (x2 - x1 * x1))
     }
     optim(c(-1.2,1), fr)
     optim(c(-1.2,1), fr, grr, method = "BFGS")
     optim(c(-1.2,1), fr, NULL, method = "BFGS", hessian = TRUE)
     ## These do not converge in the default number of steps
     optim(c(-1.2,1), fr, grr, method = "CG")
     optim(c(-1.2,1), fr, grr, method = "CG", control=list(type=2))
     optim(c(-1.2,1), fr, grr, method = "L-BFGS-B")
     
     flb <- function(x)
         { p <- length(x); sum(c(1, rep(4, p-1)) * (x - c(1, x[-p])^2)^2) }
     ## 25-dimensional box constrained
     optim(rep(3, 25), flb, NULL, method = "L-BFGS-B",
           lower=rep(2, 25), upper=rep(4, 25)) # par[24] is *not* at boundary
     
     ## "wild" function , global minimum at about -15.81515
     fw <- function (x)
         10*sin(0.3*x)*sin(1.3*x^2) + 0.00001*x^4 + 0.2*x+80
     plot(fw, -50, 50, n=1000, main = "optim() minimising 'wild function'")
     
     res <- optim(50, fw, method="SANN",
                  control=list(maxit=20000, temp=20, parscale=20))
     res
     ## Now improve locally {typically only by a small bit}:
     (r2 <- optim(res$par, fw, method="BFGS"))
     points(r2$par, r2$value, pch = 8, col = "red", cex = 2)
     
     ## Combinatorial optimization: Traveling salesman problem
     library(stats) # normally loaded
     
     eurodistmat <- as.matrix(eurodist)
     
     distance <- function(sq) {  # Target function
         sq2 <- embed(sq, 2)
         sum(eurodistmat[cbind(sq2[,2],sq2[,1])])
     }
     
     genseq <- function(sq) {  # Generate new candidate sequence
         idx <- seq(2, NROW(eurodistmat)-1)
         changepoints <- sample(idx, size=2, replace=FALSE)
         tmp <- sq[changepoints[1]]
         sq[changepoints[1]] <- sq[changepoints[2]]
         sq[changepoints[2]] <- tmp
         sq
     }
     
     sq <- c(1:nrow(eurodistmat), 1)  # Initial sequence: alphabetic
     distance(sq)
     # rotate for conventional orientation
     loc <- -cmdscale(eurodist, add=TRUE)$points
     x <- loc[,1]; y <- loc[,2]
     s <- seq_len(nrow(eurodistmat))
     tspinit <- loc[sq,]
     
     plot(x, y, type="n", asp=1, xlab="", ylab="",
          main="initial solution of traveling salesman problem", axes = FALSE)
     arrows(tspinit[s,1], tspinit[s,2], tspinit[s+1,1], tspinit[s+1,2],
            angle=10, col="green")
     text(x, y, labels(eurodist), cex=0.8)
     
     set.seed(123) # chosen to get a good soln relatively quickly
     res <- optim(sq, distance, genseq, method = "SANN",
                  control = list(maxit = 30000, temp = 2000, trace = TRUE,
                                 REPORT = 500))
     res  # Near optimum distance around 12842
     
     tspres <- loc[res$par,]
     plot(x, y, type="n", asp=1, xlab="", ylab="",
          main="optim() 'solving' traveling salesman problem", axes = FALSE)
     arrows(tspres[s,1], tspres[s,2], tspres[s+1,1], tspres[s+1,2],
            angle=10, col="red")
     text(x, y, labels(eurodist), cex=0.8)