# Profile extrema with BFGS optimization Evaluate profile extrema for a set of matrices allPsi with full optimization. ```r getProfileExtrema(f, fprime = NULL, d, allPsi, opts = NULL) ``` ## Arguments - `f`: the function to be evaluated - `fprime`: derivative of the function - `d`: dimension of the input domain - `allPsi`: a list containing the matrices Psi (dim $pxd$) for which to compute the profile extrema - `opts`: a list containing the options for this function and the subfunctions getProfileSup_optim , getProfileInf_optim . The options only for getProfileExtrema are * `limits:`an optional list containing `lower` and `upper`, two vectors with the limits of the input space. If NULL then `limits=list(upper=rep(1,d),lower=rep(0,d))` * `discretization:`an optional integer representing the discretization size for the profile computation for each dimension of eta. Pay attention that this leads to a grid of size `discretization^p`. * `heavyReturn:`If TRUE returns also all minimizers, default is FALSE. * `plts:`If TRUE and p==1 for all Psi in allPsi, plots the profile functions at each Psi, default is FALSE. * `verb:`If TRUE, outputs intermediate results, default is FALSE. ## Returns a list of two data frames (min, max) of the evaluations of $P^sup_Psi f(eta) = sup_{Psi x = \eta} f(x) $ and $P^inf_Psi f(eta) = inf_{Psi x = \eta} f(x) $ discretized over 50 equally spaced points for each dimension for each Psi in `allPsi`. This number can be changed by defining it in options$discretization. ## Examples ```r # Compute the oblique profile extrema with full optimization on 2d example # Define the function testF <- function(x,params,v1=c(1,0),v2=c(0,1)){ return(sin(crossprod(v1,x)*params[1]+params[2])+cos(crossprod(v2,x)*params[3]+params[4])-1.5) } testFprime <- function(x,params,v1=c(1,0),v2=c(0,1)){ return(matrix(c(params[1]*v1[1]*cos(crossprod(v1,x)*params[1]+params[2])- params[3]*v2[1]*sin(crossprod(v2,x)*params[3]+params[4]), params[1]*v1[2]*cos(crossprod(v1,x)*params[1]+params[2])- params[3]*v2[2]*sin(crossprod(v2,x)*params[3]+params[4])),ncol=1)) } # Define the main directions of the function theta=pi/6 pparams<-c(1,0,10,0) vv1<-c(cos(theta),sin(theta)) vv2<-c(cos(theta+pi/2),sin(theta+pi/2)) # Define optimizer friendly function f <-function(x){ return(testF(x,pparams,vv1,vv2)) } fprime <- function(x){ return(testFprime(x,pparams,vv1,vv2)) } # Define list of directions where to evaluate the profile extrema all_Psi <- list(Psi1=vv1,Psi2=vv2) # Evaluate profile extrema along directions of all_Psi allOblique<-getProfileExtrema(f=f,fprime = fprime,d = 2,allPsi = all_Psi, opts = list(plts=FALSE,discretization=100,multistart=8)) # Consider threshold=0 threshold <- 0 # Plot oblique profile extrema functions plotMaxMin(allOblique,allOblique$Design,threshold = threshold) ## Since the example is two dimensional we can visualize the regions excluded by the profile extrema # evaluate the function at a grid for plots inDes<-seq(0,1,,100) inputs<-expand.grid(inDes,inDes) outs<-apply(X = inputs,MARGIN = 1,function(x){return(testF(x,pparams,v1=vv1,v2=vv2))}) # obtain the points where the profiles take the threshold value cccObl<-getChangePoints(threshold = threshold,allRes = allOblique,Design = allOblique$Design) # visualize the functions and the regions excluded image(inDes,inDes,matrix(outs,ncol=100),col=grey.colors(20),main="Example and oblique profiles") contour(inDes,inDes,matrix(outs,ncol=100),add=TRUE,nlevels = 20) contour(inDes,inDes,matrix(outs,ncol=100),add=TRUE,levels = c(threshold),col=4,lwd=1.5) plotOblique(cccObl$alwaysEx$`0`[[1]],all_Psi[[1]],col=3) plotOblique(cccObl$alwaysEx$`0`[[2]],all_Psi[[2]],col=3) plotOblique(cccObl$neverEx$`0`[[1]],all_Psi[[1]],col=2) plotOblique(cccObl$neverEx$`0`[[2]],all_Psi[[2]],col=2) ``` ## Author(s) Dario Azzimonti

getProfileExtrema function