# An R implementation of a nonlinear conjugate gradient algorithm with the Dai / Yuan update and restart. Based on Nash (1979) Algorithm 22 for its main structure. The purpose of `Rcgmin` is to minimize an unconstrained or bounds (box) and mask constrained function of many parameters by a nonlinear conjugate gradients method. This code is entirely in R to allow users to explore and understand the method. It also allows bounds (or box) constraints and masks (equality constraints) to be imposed on parameters. `Rcgmin` is a wrapper that calls `Rcgminu` for unconstrained problems, else `Rcgminb`. UTF-8 ```r Rcgmin(par, fn, gr, lower, upper, bdmsk, control = list(), ...) ncg(par, fn, gr, bds, control = list()) ``` ## Arguments - `par`: A numeric vector of starting estimates. - `fn`: A function that returns the value of the objective at the supplied set of parameters `par` using auxiliary data in `...`. The first argument of `fn` must be `par`. - `gr`: A function that returns the gradient of the objective at the supplied set of parameters `par` using auxiliary data in `...`. The first argument of `fn` must be `par`. This function returns the gradient as a numeric vector. If `gr` is not provided or is NULL, then the simple forward gradient code `grfwd` is used. However, we recommend carefully coded and checked analytic derivatives for Rcgmin. The use of numerical gradients for Rcgmin is discouraged. First, the termination test uses a size measure on the gradient, and numerical gradient approximations can sometimes give results that are too large. Second, if there are bounds constraints, the step(s) taken to calculate the approximation to the derivative are NOT checked to see if they are out of bounds, and the function may be undefined at the evaluation point. There is also the option of using the routines `grfwd`, `grback`, `grcentral` or `grnd`. The last of these calls the `grad()` function from package numDeriv. These are called by putting the name of the (numerical) gradient function in quotation marks, e.g., gr="grfwd" to use the standard forward difference numerical approximation. Note that all but the `grnd` routine use a stepsize parameter that can be redefined in a special scratchpad storage variable `deps`. The default is `deps = 1e-07`. However, redefining this is discouraged unless you understand what you are doing. - `lower`: A vector of lower bounds on the parameters. - `upper`: A vector of upper bounds on the parameters. - `bdmsk`: An indicator vector, having 1 for each parameter that is "free" or unconstrained, and 0 for any parameter that is fixed or MASKED for the duration of the optimization. - `bds`: A list of information resulting from function `bmchk` giving information on the status of the parameters and bounds and masks. - `control`: An optional list of control settings. - ``...``: Further arguments to be passed to `fn`. ## Details Functions `fn` must return a numeric value. Note that `ncg` is to be called from `optimr` and does NOT allow dot arguments. It is intended to use the internal functions `efn` and `egr` generated inside `optimr()` along with bounds information from `bmchk()` available there. The `control` argument is a list. - **maxit**: A limit on the number of iterations (default 500). Note that this is used to compute a quantity `maxfeval`<-round(sqrt(n+1)*maxit) where n is the number of parameters to be minimized. - **trace**: Set 0 (default) for no output, >0 for trace output (larger values imply more output). - **eps**: Tolerance used to calculate numerical gradients. Default is 1.0E-7. See source code for `Rcgmin` for details of application. - **`dowarn`**: = TRUE if we want warnings generated by optimx. Default is TRUE. - **`tol`**: Tolerance used in testing the size of the square of the gradient. Default is 0 on input, which uses a value of tolgr = npar*npar*.Machine$double.eps in testing if crossprod(g) <= tolgr * (abs(fmin) + reltest). If the user supplies a value for `tol` that is non-zero, then that value is used for tolgr. reltest=100 is only alterable by changing the code. fmin is the current best value found for the function minimum value. Note that the scale of the gradient means that tests for a small gradient can easily be mismatched to a given problem. The defaults in Rcgmin are a "best guess". - **`checkgrad`**: = TRUE if we want gradient function checked against numerical approximations. Default is FALSE. - **`checkbounds`**: = TRUE if we want bounds verified. Default is TRUE. As of 2011-11-21 the following controls have been REMOVED - **usenumDeriv**: There is now a choice of numerical gradient routines. See argument `gr`. - **maximize**: To maximize user_function, supply a function that computes (-1)*user_function. An alternative is to call Rcgmin via the package optimx, where the MAXIMIZE field of the OPCON structure in package `optfntools` is used. ## Returns A list with components: - **par**: The best set of parameters found. - **value**: The value of the objective at the best set of parameters found. - **counts**: A two-element integer vector giving the number of calls to 'fn' and 'gr' respectively. This excludes those calls needed to compute the Hessian, if requested, and any calls to 'fn' to compute a finite-difference approximation to the gradient. - **convergence**: An integer code. '0' indicates successful convergence. '1' indicates that the function evaluation count 'maxfeval' was reached. '2' indicates initial point is infeasible. - **message**: A character string giving any additional information returned by the optimizer, or 'NULL'. - **bdmsk**: Returned index describing the status of bounds and masks at the proposed solution. Parameters for which bdmsk are 1 are unconstrained or "free", those with bdmsk 0 are masked i.e., fixed. For historical reasons, we indicate a parameter is at a lower bound using -3 or upper bound using -1. ## References Dai, Y. H. and Y. Yuan (2001). An efficient hybrid conjugate gradient method for unconstrained optimization. Annals of Operations Research 103 (1-4), 33–47. Nash JC (1979). Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation. Adam Hilger, Bristol. Second Edition, 1990, Bristol: Institute of Physics Publications. Nash, J. C. and M. Walker-Smith (1987). Nonlinear Parameter Estimation: An Integrated System in BASIC. New York: Marcel Dekker. See https://www.nashinfo.com/nlpe.htm for a downloadable version of this plus some extras. ## See Also `optim` ## Examples ```r ##################### require(numDeriv) ## Rosenbrock Banana function fr <- function(x) { x1 <- x[1] x2 <- x[2] 100 * (x2 - x1 * x1)^2 + (1 - x1)^2 } grr <- function(x) { ## Gradient of 'fr' x1 <- x[1] x2 <- x[2] c(-400 * x1 * (x2 - x1 * x1) - 2 * (1 - x1), 200 * (x2 - x1 * x1)) } grn<-function(x){ gg<-grad(fr, x) } ansrosenbrock0 <- Rcgmin(fn=fr,gr=grn, par=c(1,2)) print(ansrosenbrock0) # use print to allow copy to separate file that # can be called using source() ##################### # Simple bounds and masks test bt.f<-function(x){ sum(x*x) } bt.g<-function(x){ gg<-2.0*x } n<-10 xx<-rep(0,n) lower<-rep(0,n) upper<-lower # to get arrays set bdmsk<-rep(1,n) bdmsk[(trunc(n/2)+1)]<-0 for (i in 1:n) { lower[i]<-1.0*(i-1)*(n-1)/n upper[i]<-1.0*i*(n+1)/n } xx<-0.5*(lower+upper) ansbt<-Rcgmin(xx, bt.f, bt.g, lower, upper, bdmsk, control=list(trace=1)) print(ansbt) ##################### 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 } # analytic gradient test xx<-rep(pi,10) lower<-NULL upper<-NULL bdmsk<-NULL genrosea<-Rcgmin(xx,genrose.f, genrose.g, gs=10) genrosen<-optimr(xx, genrose.f, "grfwd", method="Rcgmin", gs=10) genrosenn<-try(Rcgmin(xx,genrose.f, gs=10)) # use local numerical gradient cat("genrosea uses analytic gradient\n") print(genrosea) cat("genrosen uses default gradient approximation\n") print(genrosen) cat("timings B vs U\n") lo<-rep(-100,10) up<-rep(100,10) bdmsk<-rep(1,10) tb<-system.time(ab<-Rcgminb(xx,genrose.f, genrose.g, lower=lo, upper=up, bdmsk=bdmsk))[1] tu<-system.time(au<-Rcgminu(xx,genrose.f, genrose.g))[1] cat("times U=",tu," B=",tb,"\n") cat("solution Rcgminu\n") print(au) cat("solution Rcgminb\n") print(ab) cat("diff fu-fb=",au$value-ab$value,"\n") cat("max abs parameter diff = ", max(abs(au$par-ab$par)),"\n") maxfn<-function(x) { n<-length(x) ss<-seq(1,n) f<-10-(crossprod(x-ss))^2 f<-as.numeric(f) return(f) } gmaxfn<-function(x) { gg<-grad(maxfn, x) } negmaxfn<-function(x) { f<-(-1)*maxfn(x) return(f) } cat("test that maximize=TRUE works correctly\n") n<-6 xx<-rep(1,n) ansmax<-Rcgmin(xx,maxfn, gmaxfn, control=list(maximize=TRUE,trace=1)) print(ansmax) cat("using the negmax function should give same parameters\n") ansnegmaxn<-optimr(xx,negmaxfn, "grfwd", method="Rcgmin", control=list(trace=1)) print(ansnegmaxn) ##################### From Rvmmin.Rd cat("test bounds and masks\n") nn<-4 startx<-rep(pi,nn) lo<-rep(2,nn) up<-rep(10,nn) grbds1<-Rcgmin(startx,genrose.f, gr=genrose.g,lower=lo,upper=up) print(grbds1) cat("test lower bound only\n") nn<-4 startx<-rep(pi,nn) lo<-rep(2,nn) grbds2<-Rcgmin(startx,genrose.f, gr=genrose.g,lower=lo) print(grbds2) cat("test lower bound single value only\n") nn<-4 startx<-rep(pi,nn) lo<-2 up<-rep(10,nn) grbds3<-Rcgmin(startx,genrose.f, gr=genrose.g,lower=lo) print(grbds3) cat("test upper bound only\n") nn<-4 startx<-rep(pi,nn) lo<-rep(2,nn) up<-rep(10,nn) grbds4<-Rcgmin(startx,genrose.f, gr=genrose.g,upper=up) print(grbds4) cat("test upper bound single value only\n") nn<-4 startx<-rep(pi,nn) grbds5<-Rcgmin(startx,genrose.f, gr=genrose.g,upper=10) print(grbds5) cat("test masks only\n") nn<-6 bd<-c(1,1,0,0,1,1) startx<-rep(pi,nn) grbds6<-Rcgmin(startx,genrose.f, gr=genrose.g,bdmsk=bd) print(grbds6) cat("test upper bound on first two elements only\n") nn<-4 startx<-rep(pi,nn) upper<-c(10,8, Inf, Inf) grbds7<-Rcgmin(startx,genrose.f, gr=genrose.g,upper=upper) print(grbds7) cat("test lower bound on first two elements only\n") nn<-4 startx<-rep(0,nn) lower<-c(0,1.1, -Inf, -Inf) grbds8<-Rcgmin(startx,genrose.f,genrose.g,lower=lower, control=list(maxit=2000)) print(grbds8) cat("test n=1 problem using simple squares of parameter\n") sqtst<-function(xx) { res<-sum((xx-2)*(xx-2)) } gsqtst<-function(xx) { gg<-2*(xx-2) } ######### One dimension test nn<-1 startx<-rep(0,nn) onepar<-Rcgmin(startx,sqtst, gr=gsqtst,control=list(trace=1)) print(onepar) cat("Suppress warnings\n") oneparnw<-Rcgmin(startx,sqtst, gr=gsqtst,control=list(dowarn=FALSE,trace=1)) print(oneparnw) ```

Rcgmin function