# Variable metric nonlinear function minimization, driver. A driver to call the unconstrained and bounds constrained versions of an R implementation of a variable metric method for minimization of nonlinear functions, possibly subject to bounds (box) constraints and masks (fixed parameters). The algorithm is based on Nash (1979) Algorithm 21 for main structure, which is itself drawn from Fletcher's (1970) variable metric code. This is also the basis of optim() method 'BFGS' which, however, does not deal with bounds or masks. In the present method, an approximation to the inverse Hessian (B) is used to generate a search direction t = - B %*% g, a simple backtracking line search is used until an acceptable point is found, and the matrix B is updated using a BFGS formula. If no acceptable point can be found, we reset B to the identity i.e., the search direction becomes the negative gradient. If the search along the negative gradient is unsuccessful, the method terminates. This set of codes 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. UTF-8 ```r Rvmmin(par, fn, gr, lower, upper, bdmsk, control = list(), ...) nvm(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. Note that a gradient function must generally be provided. However, to ensure compatibility with other optimizers, if `gr` is NULL, the forward gradient approximation from routine `grfwd` will be used. The use of numerical gradients for Rvmmin 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. - `bds`: A list of information resulting from function `bmchk` giving information on the status of the parameters and bounds and masks. - `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. - `control`: An optional list of control settings. - ``...``: Further arguments to be passed to `fn`. ## Details Note that `nvm` 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 source codes `Rvmmin` and `nvm` for R are still a work in progress, so users should watch the console output. The routine `nvm` attempts to use minimal checking and works only with a bounds constrained version of the algorithm, which may work as fast as a specific routine for unconstrained problems. This is an open question, and the author welcomes feedback. Function `fn` must return a numeric value. The `control` argument is a list. The `control` argument is a list. - **maxit**: A limit on the number of iterations (default 500 + 2*n where n is the number of parameters). This is the maximum number of gradient evaluations allowed. - **maxfevals**: A limit on the number of function evaluations allowed (default 3000 + 10*n). - **trace**: Set 0 (default) for no output, > 0 for diagnostic output (larger values imply more output). - **dowarn**: = TRUE if we want warnings generated by optimx. Default is TRUE. - **checkgrad**: = TRUE if we wish analytic gradient code checked against the approximations computed by `numDeriv`. Default is FALSE. - **checkbounds**: = TRUE if we wish parameters and bounds to be checked for an admissible and feasible start. Default is TRUE. - **keepinputpar**: = TRUE if we want bounds check to stop program when parameters are out of bounds. Else when FALSE, moves parameter values to nearest bound. Default is FALSE. - **maximize**: To maximize user_function, supply a function that computes (-1)*user_function. An alternative is to call Rvmmin via the package optimx. - **eps**: a tolerance used for judging small gradient norm (default = 1e-07). a gradient norm smaller than (1 + abs(fmin))*eps*eps is considered small enough that a local optimum has been found, where fmin is the current estimate of the minimal function value. - **acctol**: To adjust the acceptable point tolerance (default 0.0001) in the test ( f <= fmin + gradproj * steplength * acctol ). This test is used to ensure progress is made at each iteration. - **stepredn**: Step reduction factor for backtrack line search (default 0.2) - **reltest**: Additive shift for equality test (default 100.0) - **stopbadupdate**: A logical flag that if set TRUE will halt the optimization if the Hessian inverse cannot be updated after a steepest descent search. This indicates an ill-conditioned Hessian. A settign of FALSE causes Rvmmin methods to be aggressive in trying to optimize the function, but may waste effort. Default 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`. ## 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 vector of two integers giving the number of function and gradient evaluations. - **convergence**: An integer indicating the situation on termination of the function. `0` indicates that the method believes it has succeeded. Other values: - **`0`**: indicates successful termination to an acceptable solution - **`1`**: indicates that the iteration limit `maxit` had been reached. - **`2`**: indicates that a point with a small gradient norm has been found, which is likely a solution. - **`20`**: indicates that the initial set of parameters is inadmissible, that is, that the function cannot be computed or returns an infinite, NULL, or NA value. - **`21`**: indicates that an intermediate set of parameters is inadmissible. - **message**: A description of the situation on termination of the function. - **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 Fletcher, R (1970) A New Approach to Variable Metric Algorithms, Computer Journal, 13(3), pp. 317-322. Nash, J C (1979, 1990) Compact Numerical Methods for Computers: Linear Algebra and Function Minimisation, Bristol: Adam Hilger. Second Edition, Bristol: Institute of Physics Publications. ## See Also `optim` ## Examples ```r ##################### ## All examples for the Rvmmin package 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 } ansrosenbrock <- Rvmmin(fn=fr,gr="grfwd", par=c(1,2)) print(ansrosenbrock) cat("\n") cat("No gr specified as a test\n") ansrosenbrock0 <- Rvmmin(fn=fr, 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 # # The function is a sum of squares, but we impose the # constraints so that there are lower and upper bounds # away from zero, and parameter 6 is fixed at the initial # value 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) cat("Initial parameters:") print(xx) cat("Lower bounds:") print(lower) cat("upper bounds:") print(upper) cat("Masked (fixed) parameters:") print(which(bdmsk == 0)) ansbt<-Rvmmin(xx, bt.f, bt.g, lower, upper, bdmsk, control=list(trace=1)) print(ansbt) ##################### # A version of a generalized Rosenbrock problem 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<-Rvmmin(xx,genrose.f, genrose.g, gs=10) genrosenf<-Rvmmin(xx,genrose.f, gr="grfwd", gs=10) # use local numerical gradient genrosenullgr<-Rvmmin(xx,genrose.f, gs=10) # no gradient specified cat("genrosea uses analytic gradient\n") print(genrosea) cat("genrosenf uses grfwd standard numerical gradient\n") print(genrosenf) cat("genrosenullgr has no gradient specified\n") print(genrosenullgr) cat("Other numerical gradients can be used.\n") cat("timings B vs U\n") lo<-rep(-100,10) up<-rep(100,10) bdmsk<-rep(1,10) tb<-system.time(ab<-Rvmminb(xx,genrose.f, genrose.g, lower=lo, upper=up, bdmsk=bdmsk))[1] tu<-system.time(au<-Rvmminu(xx,genrose.f, genrose.g))[1] cat("times U=",tu," B=",tb,"\n") cat("solution Rvmminu\n") print(au) cat("solution Rvmminb\n") print(ab) cat("diff fu-fb=",au$value-ab$value,"\n") cat("max abs parameter diff = ", max(abs(au$par-ab$par)),"\n") # Test that Rvmmin will maximize as well as minimize maxfn<-function(x) { n<-length(x) ss<-seq(1,n) f<-10-(crossprod(x-ss))^2 f<-as.numeric(f) return(f) } negmaxfn<-function(x) { f<-(-1)*maxfn(x) return(f) } cat("test that maximize=TRUE works correctly\n") n<-6 xx<-rep(1,n) ansmax<-Rvmmin(xx,maxfn, gr="grfwd", control=list(maximize=TRUE,trace=1)) print(ansmax) cat("using the negmax function should give same parameters\n") ansnegmax<-Rvmmin(xx,negmaxfn, gr="grfwd", control=list(trace=1)) print(ansnegmax) ##################### cat("test bounds and masks\n") nn<-4 startx<-rep(pi,nn) lo<-rep(2,nn) up<-rep(10,nn) grbds1<-Rvmmin(startx,genrose.f, 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<-Rvmmin(startx,genrose.f, 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<-Rvmmin(startx,genrose.f, 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<-Rvmmin(startx,genrose.f, genrose.g, upper=up) print(grbds4) cat("test upper bound single value only\n") nn<-4 startx<-rep(pi,nn) grbds5<-Rvmmin(startx,genrose.f, 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<-Rvmmin(startx,genrose.f, 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<-Rvmmin(startx,genrose.f, 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<-Rvmmin(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)) } nn<-1 startx<-rep(0,nn) onepar<-Rvmmin(startx,sqtst, gr="grfwd", control=list(trace=1)) print(onepar) cat("Suppress warnings\n") oneparnw<-Rvmmin(startx,sqtst, gr="grfwd", control=list(dowarn=FALSE,trace=1)) print(oneparnw) ```

Rvmmin function