# Truncated Newton minimization of an unconstrained function. An R implementation of the Truncated Newton method of Stephen Nash for driver to call the unconstrained function minimization. The algorithm is based on Nash (1979) This set of codes is entirely in R to allow users to explore and understand the method. UTF-8 ```r tn(x, fgfun, trace, ...) ``` ## Arguments - `x`: A numeric vector of starting estimates. - `fgfun`: A function that returns the value of the objective at the supplied set of parameters `par` using auxiliary data in `...`. The gradient is returned as attribute "gradient". The first argument of `fgfun` must be `par`. - `trace`: > 0 if progress output is to be presented. - ``...``: Further arguments to be passed to `fn`. ## Details Function `fgfun` must return a numeric value in list item `f` and a numeric vector in list item `g`. ## Returns A list with components: - **xstar**: The best set of parameters found. - **f**: The value of the objective at the best set of parameters found. - **g**: The gradient of the objective at the best set of parameters found. - **ierror**: An integer indicating the situation on termination. `0` indicates that the method believes it has succeeded; `2` that more than `maxfun` (default 150*n, where there are n parameters); `3` if the line search appears to have failed (which may not be serious); and `-1` if there appears to be an error in the input parameters. - **nfngr**: A number giving a measure of how many conjugate gradient solutions were used during the minimization process. ## References Stephen G. Nash (1984) "Newton-type minimization via the Lanczos method", SIAM J Numerical Analysis, vol. 21, no. 4, pages 770-788. For Matlab code, see http://www.netlib.org/opt/tn ## See Also `optim` ## Examples ```r ##################### ## All examples are in this .Rd file ## ## Rosenbrock Banana function fr <- function(x) { x1 <- x[1] x2 <- x[2] 100 * (x2 - x1 * x1)^2 + (1 - x1)^2 } gr <- function(x) { x1 <- x[1] x2 <- x[2] g1 <- -400 * (x2 - x1*x1) * x1 - 2*(1-x1) g2 <- 200*(x2 - x1*x1) gg<-c(g1, g2) } rosefg<-function(x){ f<-fr(x) g<-gr(x) attr(f, "gradient") <- g f } x<-c(-1.2, 1) ansrosenbrock <- tn(x, rosefg) print(ansrosenbrock) # use print to allow copy to separate file that cat("Compare to optim\n") ansoptrose <- optim(x, fr, gr) print(ansoptrose) genrose.f<- function(x, gs=NULL){ # objective function ## One generalization of the Rosenbrock banana valley function (n parameters) n <- length(x) if(is.null(gs)) { gs=100.0 } fval<-1.0 + sum (gs*(x[1:(n-1)]^2 - x[2:n])^2 + (x[2:n] - 1)^2) return(fval) } genrose.g <- function(x, gs=NULL){ # vectorized gradient for genrose.f # Ravi Varadhan 2009-04-03 n <- length(x) if(is.null(gs)) { gs=100.0 } gg <- as.vector(rep(0, n)) tn <- 2:n tn1 <- tn - 1 z1 <- x[tn] - x[tn1]^2 z2 <- 1 - x[tn] gg[tn] <- 2 * (gs * z1 - z2) gg[tn1] <- gg[tn1] - 4 * gs * x[tn1] * z1 gg } grosefg<-function(x, gs=100.0) { f<-genrose.f(x, gs) g<-genrose.g(x, gs) attr(f, "gradient") <- g f } n <- 100 x <- (1:100)/20 groseu<-tn(x, grosefg, gs=10) print(groseu) groseuo <- optim(x, fn=genrose.f, gr=genrose.g, method="BFGS", control=list(maxit=1000), gs=10) cat("compare optim BFGS\n") print(groseuo) lower<-1+(1:n)/100 upper<-5-(1:n)/100 xmid<-0.5*(lower+upper) grosec<-tnbc(xmid, grosefg, lower, upper) print(grosec) cat("compare L-BFGS-B\n") grosecl <- optim(par=xmid, fn=genrose.f, gr=genrose.g, lower=lower, upper=upper, method="L-BFGS-B") print(grosecl) ```

tn function