Understanding nonlinear optimization and Rosenbrock's banana valley function?

Henry Wolkowicz (google his page for lots of optimization references)
mentioned to me that that function is a standard example to show
that first order methods (e.g. steepest descent) can fail by repeatedly
crossing back and forth over the valley whereas second order methods
(damped Newton, damped quasi-Newton, trust region) use curvature
info to get convergence in a few iterations.

On 12/4/05, Spencer Graves <spencer.graves at pdf.com> wrote:

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
Fax: 408-280-7915