[R] Understanding nonlinear optimization and Rosenbrock's banana valley function?

[Spencer Graves]

GENERAL REFERENCE ON NONLINEAR OPTIMIZATION?

	  What are your favorite references on nonlinear optimization?  I like 
Bates and Watts (1988) Nonlinear Regression Analysis and Its 
Applications (Wiley), especially for its key insights regarding 
parameter effects vs. intrinsic curvature.  Before I spent time and 
money on several of the refences cited on the help pages for "optim", 
"nlm", etc., I thought I'd ask you all for your thoughts.

ROSENBROCK'S BANANA VALLEY FUNCTION?

	  Beyond this, I wonder if someone help me understand the lessons one 
should take from Rosenbrock's banana valley function:

banana <- function(x){
   100*(x[2]-x[1]^2)^2+(1-x[1])^2
}

	  This a quartic x[1] and a parabola in x[2] with a unique minimum at 
x[2]=x[1]=1.  Over the range (-1, 2)x(-1,1), it looks like a long, 
curved, deep, narrow banana-shaped valley.  It is a known hard problem 
in nonlinear regression, but these difficulties don't affect "nlm" or 
"nlminb" until the hessian is provided analytically (with R 2.2.0 under 
Windows XP):

nlm(banana, c(-1.2, 1)) # found the minimum in 23 iterations
nlminb(c(-1.2, 1), banana)# found the min in 35 iterations

Dbanana <- function(x){
   c(-400*x[1]*(x[2] - x[1]^2) - 2*(1-x[1]),
     200*(x[2] - x[1]^2))
}
banana1 <- function(x){
   b <- 100*(x[2]-x[1]^2)^2+(1-x[1])^2
   attr(b, "gradient") <- Dbanana(x)
   b
}

nlm(banana1, c(-1.2, 1)) # solved the problem in 24 iterations
nlminb(c(-1.2, 1), banana, Dbanana)# solution in 35 iterations

D2banana <- function(x){
         a11 <- (2 - 400*(x[2] - x[1]^2) + 800*x[2]*x[1]^2)
         a21 <- (-400*x[1])
         matrix(c(a11,a21,a21,200),2,2)
}
banana2 <- function(x){
   b <- 100*(x[2]-x[1]^2)^2+(1-x[1])^2
   attr(b, "gradient") <- Dbanana(x)
   attr(b, "hessian") <- D2banana(x)
   b
}

nlm(banana2, c(-1.2, 1))
# Found the valley but not the minimum
# in the default 100 iterations.
nlm(banana2, c(-1.2, 1), iterlim=10000)
# found the minimum to 3 significant digits in 5017 iterations.

nlminb(c(-1.2, 1), banana, Dbanana, D2banana)
# took 95 iterations to find the answer to double precision.

	  To understand this better, I wrote my own version of "nlm" (see 
below), and learned that the hessian is often indefinite, with one 
eigenvalue positive and the other negative.  If I understand correctly, 
a negative eigenvalue of the hessian tends to push the next step towards 
increasing rather than decreasing the function.  I tried a few things 
that accelerated the convergence slightly, but but my "nlm." still had 
not converged after 100 iterations.

	  What might be done to improve the performance of something like "nlm" 
without substantially increasing the overhead for other problems?

	  Thanks.
	  spencer graves
#############################
nlm. <- function(f=fgh, p=c(-1.2, 1),
   gradtol=1e-6, steptol=1e-6, iterlim=100){
# R code version of "nlm"
# requiring analytic gradient and hessian
#
# Initial evaluation
   f.i <- f(p)
   f0 <- f.i+1
# Iterate
   for(i in 1:iterlim){
     df <- attr(f.i, "gradient")
#   Gradient sufficiently small?
     if(sum(df^2)<(gradtol^2)){
       return(list(minimum=f.i, estimate=p+dp,
           gradient=df, hessian=d2f, code=1,
           iterations=i))
     }
#
     d2f <- attr(f.i, "hessian")
     dp <- (-solve(d2f, df))
#   Step sufficiently small?
     if(sum(dp^2)<(steptol^2)){
       return(list(minimum=f.i, estimate=p+dp,
           gradient=df, hessian=d2f, code=2,
           iterations=i))
     }
#   Next iter
     f0 <- f.i
     f.i <- f(p+dp)
#   Step size control
     if(f.i>=f0){
       for(j in 1:iterlim){
         {
           if(j==1){
             d2f.eig <- eigen(d2f, symmetric=T)
             cat("\nstep size control; i=", i,
                 "; p=", round(p, 3), "; dp=", signif(dp, 2),
                 "; eig(hessian)=",signif(d2f.eig$values, 4))
             v.max <- (1+max(abs(d2f.eig$values)))
             v.adj <- pmax(.001*v.max, abs(d2f.eig$values))
             evec.df <- (t(d2f.eig$vectors) %*% df)
             dp <- (-(d2f.eig$vectors %*%
                      (evec.df/(1+v.adj))))
           }
           else{
             cat(".")
             dp <- dp/2
           }
         }
         f.i <- f(p+dp)
         f2 <- f(p+dp/2)
         if(f2<f.i){
           dp <- dp/2
           f.i <- f2
         }
         if(f.i<f0)break # j
       }
       if(f.i>=f0){
         cat("\n")
         return(list(minimum=f0, estimate=p,
            gradient=attr(f0, "gradient"),
            hessian=attr(f0, "hessian"), code=3,
            iterations=i))
       }
     }
     p <- p+dp
     cat(i, p, f.i, "\n")
   }
   return(list(minimum=f.i, estimate=p,
       gradient=df, hessian=d2f, code=4,
       iterations=i))
}

-- 
Spencer Graves, PhD
Senior Development Engineer
PDF Solutions, Inc.
333 West San Carlos Street Suite 700
San Jose, CA 95110, USA

spencer.graves at pdf.com
www.pdf.com <http://www.pdf.com>
Tel:  408-938-4420
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