# Generate gradient and Hessian for a function at given parameters. `gHgenb` is used to generate the gradient and Hessian of an objective function used for optimization. If a user-provided gradient function `gr` is available it is used to compute the gradient, otherwise package `numDeriv` is used. If a user-provided Hessian function `hess` is available, it is used to compute a Hessian. Otherwise, if `gr` is available, we use the function `jacobian()` from package `numDeriv` to compute the Hessian. In both these cases we check for symmetry of the Hessian. Computational Hessians are commonly NOT symmetric. If only the objective function `fn` is provided, then the Hessian is approximated with the function `hessian` from package `numDeriv` which guarantees a symmetric matrix. UTF-8 ```r gHgenb(par, fn, gr=NULL, hess=NULL, bdmsk=NULL, lower=NULL, upper=NULL, control=list(ktrace=0), ...) ``` ## Arguments - `par`: Set of parameters, assumed to be at a minimum of the function `fn`. - `fn`: Name of the objective function. - `gr`: (Optional) function to compute the gradient of the objective function. If present, we use the Jacobian of the gradient as the Hessian and avoid one layer of numerical approximation to the Hessian. - `hess`: (Optional) function to compute the Hessian of the objective function. This is rarely available, but is included for completeness. - `bdmsk`: An integer vector of the same length as `par`. When an element of this vector is 0, the corresponding parameter value is fixed (masked) during an optimization. Non-zero values indicate a parameter is free (1), at a lower bound (-3) or at an upper bound (-1), but this routine only uses 0 values. - `lower`: Lower bounds for parameters in `par`. - `upper`: Upper bounds for parameters in `par`. - `control`: A list of controls to the function. Currently asymptol (default of 1.0e-7 which tests for asymmetry of Hessian approximation (see code for details of the test); ktrace, a logical flag which, if TRUE, monitors the progress of gHgenb (default FALSE), and stoponerror, defaulting to FALSE to NOT stop when there is an error or asymmetry of Hessian. Set TRUE to stop. - ``...``: Extra data needed to compute the function, gradient and Hessian. ## Details None ## Returns `ansout` a list of four items, - **`gn`**: The approximation to the gradient vector. - **`Hn`**: The approximation to the Hessian matrix. - **`gradOK`**: TRUE if the gradient has been computed acceptably. FALSE otherwise. - **`hessOK`**: TRUE if the gradient has been computed acceptably and passes the symmetry test. FALSE otherwise. - **`nbm`**: The number of active bounds and masks. ## Examples ```r require(numDeriv) # genrose function code 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 return(gg) } genrose.h <- function(x, gs=NULL) { ## compute Hessian if(is.null(gs)) { gs=100.0 } n <- length(x) hh<-matrix(rep(0, n*n),n,n) for (i in 2:n) { z1<-x[i]-x[i-1]*x[i-1] z2<-1.0-x[i] hh[i,i]<-hh[i,i]+2.0*(gs+1.0) hh[i-1,i-1]<-hh[i-1,i-1]-4.0*gs*z1-4.0*gs*x[i-1]*(-2.0*x[i-1]) hh[i,i-1]<-hh[i,i-1]-4.0*gs*x[i-1] hh[i-1,i]<-hh[i-1,i]-4.0*gs*x[i-1] } return(hh) } maxfn<-function(x, top=10) { n<-length(x) ss<-seq(1,n) f<-top-(crossprod(x-ss))^2 f<-as.numeric(f) return(f) } negmaxfn<-function(x) { f<-(-1)*maxfn(x) return(f) } parx<-rep(1,4) lower<-rep(-10,4) upper<-rep(10,4) bdmsk<-c(1,1,0,1) # masked parameter 3 fval<-genrose.f(parx) gval<-genrose.g(parx) Ahess<-genrose.h(parx) gennog<-gHgenb(parx,genrose.f) cat("results of gHgenb for genrose without gradient code at ") print(parx) print(gennog) cat("compare to g =") print(gval) cat("and Hess\n") print(Ahess) cat("\n\n") geng<-gHgenb(parx,genrose.f,genrose.g) cat("results of gHgenb for genrose at ") print(parx) print(gennog) cat("compare to g =") print(gval) cat("and Hess\n") print(Ahess) cat("*****************************************\n") parx<-rep(0.9,4) fval<-genrose.f(parx) gval<-genrose.g(parx) Ahess<-genrose.h(parx) gennog<-gHgenb(parx,genrose.f,control=list(ktrace=TRUE), gs=9.4) cat("results of gHgenb with gs=",9.4," for genrose without gradient code at ") print(parx) print(gennog) cat("compare to g =") print(gval) cat("and Hess\n") print(Ahess) cat("\n\n") geng<-gHgenb(parx,genrose.f,genrose.g, control=list(ktrace=TRUE)) cat("results of gHgenb for genrose at ") print(parx) print(gennog) cat("compare to g =") print(gval) cat("and Hess\n") print(Ahess) gst<-5 cat("\n\nTest with full calling sequence and gs=",gst,"\n") gengall<-gHgenb(parx,genrose.f,genrose.g,genrose.h, control=list(ktrace=TRUE),gs=gst) print(gengall) top<-25 x0<-rep(2,4) cat("\n\nTest for maximization and top=",top,"\n") cat("Gradient and Hessian will have sign inverted") maxt<-gHgen(x0, maxfn, control=list(ktrace=TRUE), top=top) print(maxt) cat("test against negmaxfn\n") gneg <- grad(negmaxfn, x0) Hneg<-hessian(negmaxfn, x0) # gdiff<-max(abs(gneg-maxt$gn))/max(abs(maxt$gn)) # Hdiff<-max(abs(Hneg-maxt$Hn))/max(abs(maxt$Hn)) # explicitly change sign gdiff<-max(abs(gneg-(-1)*maxt$gn))/max(abs(maxt$gn)) Hdiff<-max(abs(Hneg-(-1)*maxt$Hn))/max(abs(maxt$Hn)) cat("gdiff = ",gdiff," Hdiff=",Hdiff,"\n") ```

gHgenb function