R help - Dec 2005 - Understanding nonlinear optimization and Rosenbrock's banana valley function?

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

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
>
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