#
#
#  Use optim() to find min or max
#             
#               
########################################


#--- one parameter without derivative ---
  f <- function(x) -exp(-( (x-2)^2 ))  optim(1, f)



#--- one parameter with derivative ---
  f <- function(x) -exp(-( (x-2)^2 ))
  df <- function(x) -2*(x-2)*f(x)
  optim(1, f, df, method="CG")



#--- Two parameter case ---

  #-- 1. Rosenbrock function --
  f <- function(x1,y1) (1-x1)^2 + 100*(y1 - x1^2)^2
  x <- seq(-2,2,by=.15)
  y <- seq(-1,3,by=.15)
  z <- outer(x,y,f)
  persp(x,y,z,phi=45,theta=-45,col="yellow",shade=.00000001,ticktype="detailed")


  optim( c(0,0), f )


  #-- 2. Himmelblau's function --
  f <- function(x1,y1) (x1^2 + y1 - 11)^2 + (x1 + y1^2 - 7)^2
  x <- seq(-4.5,4.5,by=.2)
  y <- seq(-4.5,4.5,by=.2)
  z <- outer(x,y,f)
  persp(x,y,z,phi=-45,theta=45,col="yellow",shade=.65 ,ticktype="detailed")


  #-- rewrite f --
  f <- function(x) (x[1]^2 + x[2] - 11)^2 + (x[1] + x[2]^2 - 7)^2

  optim(c(-4,-4),f) #local
  optim(c(2,-2), f) #local
  optim(c(2,2),  f) #local
  optim(c(-4,4), f) #global