R interface to NOMAD

snomadr is an R interface to NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search, Abramson, Audet, Couture and Le Digabel (2011)), an open source software C++ implementation of the Mesh Adaptive Direct Search (MADS, Le Digabel (2011)) algorithm designed for constrained optimization of blackbox functions.

NOMAD is designed to find (local) solutions of mathematical optimization problems of the form


   min     f(x)
x in R^n

s.t.       g(x) <= 0
           x_L <=  x   <= x_U

where \(f(x)\colon R^n \to R^k\) is the objective function, and \(g(x)\colon R^n \to R^m\) are the constraint functions. The vectors \(x_L\) and \(x_U\) are the bounds on the variables \(x\). The functions \(f(x)\) and \(g(x)\) can be nonlinear and nonconvex. The variables can be integer, continuous real number, binary, and categorical.

Kindly see https://www.gerad.ca/en/software/nomad and the references below for details.

Usage

snomadr(eval.f,
        n,
        bbin = NULL,
        bbout = NULL,
        x0 = NULL,
        lb = NULL,
        ub = NULL,
        nmulti = 0,
        random.seed = 0,
        opts = list(),
        display.nomad.progress = TRUE,
        information = list(),
        snomadr.environment = new.env(),
        ... )

Arguments

Objective, Dimension, And Evaluation Environment

These arguments identify the objective, dimension, black-box types, and evaluation environment.

bbin: types of variables. Variable types are 0 (CONTINUOUS), 1 (INTEGER), 2 (CATEGORICAL), 3 (BINARY)
bbout: types of output of eval.f. See the NOMAD User Guide https://nomad-4-user-guide.readthedocs.io/en/latest/#
eval.f: function that returns the value of the objective function
n: the number of variables
snomadr.environment: environment that is used to evaluate the functions. Use this to pass additional data or parameters to a function

Multi-Start Controls

These arguments control multistart execution and reproducibility settings.

nmulti: when it is missing, or it is equal to 0 and x0 is provided, snomadRSolve will be called to solve the problem. Otherwise, smultinomadRSolve will be called
random.seed: when it is not missing and not equal to 0, the initial points will be generated using this seed when nmulti > 0

NOMAD Options And Information Queries

These arguments control NOMAD options and optional information queries.

information

is a list. snomadr will call snomadRInfo to return the information about NOMAD according to the values of "info", "version" and "help".

"info"="-i": display the usage and copyright of NOMAD

"version"="-v": display the version of NOMAD you are using

"help"="-h": display all options

You also can display a specific option, for example, "help"="-h x0", this will tell you how to set x0

opts

list of options for NOMAD, see the NOMAD user guide https://nomad-4-user-guide.readthedocs.io/en/latest/#. Options can also be set by nomad.opt which should be in the folder where R (snomadr) is executed.

This interface uses NOMAD 4 option names directly (for example MIN_FRAME_SIZE).

A complete option catalog generated from the embedded NOMAD 4.5.0 definitions is bundled at system.file("nomad", "NOMAD_4_5_0_OPTIONS_REFERENCE.md", package="crs").

To improve comparability with historical NOMAD 3.9.1 behavior, snomadr applies a small NOMAD4 compatibility profile when these options are not user-supplied (user opts always take precedence): QUAD_MODEL_SEARCH, SGTELIB_MODEL_SEARCH, NM_SEARCH, SPECULATIVE_SEARCH, EVAL_OPPORTUNISTIC, EVAL_QUEUE_SORT, DIRECTION_TYPE, QUAD_MODEL_SEARCH_BOX_FACTOR, and QUAD_MODEL_BOX_FACTOR.

A few high-impact options are:

"MAX_BB_EVAL"=10000

"INITIAL_MESH_SIZE"=1

"MIN_MESH_SIZE"="r1.0e-10"

"MIN_FRAME_SIZE"=1

Note that the "r..." denotes relative measurement (relative to lb and ub)

Note that decreasing the maximum number of black box evaluations ("MAX_BB_EVAL") will terminate search sooner and may result in a less accurate solution. For complicated problems you may want to increase this value. When experimenting, "DISPLAY_DEGREE"=1 or "DISPLAY_DEGREE"=2 can provide additional native NOMAD output. Higher native display degrees are currently rejected by the embedded snomadr() interface because NOMAD4's verbose block output queue can abort repeated R-session solves.

Starting Values, Bounds, And Variable Types

These arguments identify starting values, bounds, and variable types.

lb: vector with lower bounds of the controls (use -1.0e19 for controls without lower bound)
ub: vector with upper bounds of the controls (use 1.0e19 for controls without upper bound)
x0: vector with starting values for the optimization. If it is provided and nmulti is bigger than 1, x0 will be the first initial point for multiple initial points

Additional Arguments

These arguments control displayed progress and pass-through values.

...: arguments that will be passed to the user-defined objective and constraints functions. See the examples below
display.nomad.progress: when FALSE, no output from snomadr is displayed on the screen. If the NOMAD option "DISPLAY_DEGREE"=0, is set, there will also be no output from NOMAD. Supported native "DISPLAY_DEGREE" values are 0, 1, and 2 in this embedded interface.

Details

snomadr is used in the crs package to numerically minimize an objective function with respect to the spline degree, number of knots, and optionally the kernel bandwidths when using crs with the option cv="nomad" (default). This is a constrained mixed integer combinatoric problem and is known to be computationally ‘hard’. See frscvNOMAD and krscvNOMAD for the functions called when cv="nomad" while using crs.

However, the user should note that for simple problems involving one predictor exhaustive search may be faster and potentially more accurate, so please bear in mind that cv="exhaustive" can be useful when using crs.

Naturally, exhaustive search is also useful for verifying solutions returned by snomadr. See frscv and krscv for the functions called when cv="exhaustive" while using crs.

Value

The return value contains a list with the inputs, and additional elements

call: the call that was made to solve
status: integer value with the status of the optimization
message: more informative message with the status of the optimization
bbe: number of blackbox evaluations reported by the interface
cache.hits: number of NOMAD native cache hits during the run
cache.size: number of points in NOMAD's native cache at the end of the run
callback.evaluations: number of evaluations sent to the R callback
total.evaluations: total evaluations represented by callback evaluations plus cache hits
iterations: number of iterations that were executed, if multiple initial points are set, this number will be the sum for each initial point.
objective: value if the objective function in the solution
solution: optimal value of the controls

References

Abramson, M.A. and C. Audet and G. Couture and J.E. Dennis Jr. and S. Le Digabel (2011), “The NOMAD project”. Software available at https://www.gerad.ca/en/software/nomad/

Le Digabel, S. (2011), “Algorithm 909: NOMAD: Nonlinear Optimization With The MADS Algorithm”. ACM Transactions on Mathematical Software, 37(4):44:1-44:15.

Author

Zhenghua Nie <niez@mcmaster.ca>

Examples

if (FALSE) { # \dontrun{
## List all options
snomadr(information=list("help"="-h"))

## Print given option,  for example,  MESH_SIZE
snomadr(information=list("help"="-h MESH_SIZE"))

## Print the version of NOMAD
snomadr(information=list("version"="-v"))

## Print usage and copyright
snomadr(information=list("info"="-i"))

## This is the example found in
## NOMAD/examples/basic/library/single_obj/basic_lib.cpp

eval.f <- function ( x ) {

    f <- c(Inf, Inf, Inf);
    n <- length (x);

    if ( n == 5 && ( is.double(x) || is.integer(x) ) ) {
        f[1] <- x[5];
        f[2] <- sum ( (x-1)^2 ) - 25;
        f[3] <- 25 - sum ( (x+1)^2 );
    }

    return ( as.double(f) );
}

## Initial values
x0 <- rep(0.0, 5 )

bbin <-c(1, 1, 1, 1, 1)
## Bounds
lb <- rep(-6.0,5 )
ub <- c(5.0, 6.0, 7.0, 1000000, 100000)

bbout <-c(0, 2, 1)
## Options
opts <-list("MAX_BB_EVAL"=500,
            "MIN_MESH_SIZE"=0.001,
            "INITIAL_MESH_SIZE"=0.1,
            "MIN_FRAME_SIZE"=1)

snomadr(eval.f=eval.f,n=5,  x0=x0, bbin=bbin, bbout=bbout, lb=lb, ub=ub, opts=opts)


## How to transfer other parameters into eval.f
##
## First example: supply additional arguments in user-defined functions
##

## objective function and gradient in terms of parameters
eval.f.ex1 <- function(x, params) {
    return( params[1]*x^2 + params[2]*x + params[3] )
}

## Define parameters that we want to use
params <- c(1,2,3)

## Define initial value of the optimization problem
x0 <- 0

## solve using snomadr
snomadr( n          =1,
        x0          = x0,
        eval.f      = eval.f.ex1,
        params      = params )


##
## Second example: define an environment that contains extra parameters
##

## Objective function and gradient in terms of parameters
## without supplying params as an argument
eval.f.ex2 <- function(x) {
    return( params[1]*x^2 + params[2]*x + params[3] )
}

## Define initial value of the optimization problem
x0 <- 0

## Define a new environment that contains params
auxdata        <- new.env()
auxdata$params <- c(1,2,3)

## pass The environment that should be used to evaluate functions to snomadr
snomadr(n                  =1,
        x0                 = x0,
        eval.f             = eval.f.ex2,
        snomadr.environment = auxdata )

## Solve using algebra
cat( paste( "Minimizing f(x) = ax^2 + bx + c\n" ) )
cat( paste( "Optimal value of control is -b/(2a) = ", -params[2]/(2*params[1]), "\n" ) )
cat( paste( "With value of the objective function f(-b/(2a)) = ",
           eval.f.ex1( -params[2]/(2*params[1]), params ), "\n" ) )

## The following example is NOMAD/examples/advanced/multi_start/multi.cpp
## This will call smultinomadRSolve to resolve the problem.
eval.f.ex1 <- function(x, params) {
    M<-as.numeric(params$M)
    NBC<-as.numeric(params$NBC)

    f<-rep(0, M+1)
    x<-as.numeric(x)

    x1 <- rep(0.0, NBC)
    y1 <- rep(0.0, NBC)

    x1[1]<-x[1]
    x1[2]<-x[2]
    y1[3]<-x[3]
    x1[4]<-x[4]
    y1[4]<-x[5]

    epi <- 6
    for(i in 5:NBC){
        x1[i]<-x[epi]
        epi <- epi+1
        y1[i]<-x[epi]
        epi<-epi+1
    }
    constraint <- 0.0
    ic <- 1
    f[ic]<-constraint
    ic <- ic+1

    constraint <- as.numeric(1.0)
    distmax <- as.numeric(0.0)
    avg_dist <- as.numeric(0.0)
    dist1<-as.numeric(0.0)

    for(i in 1:(NBC-1)){
        for (j in (i+1):NBC){
            dist1 <- as.numeric((x1[i]-x1[j])*(x1[i]-x1[j])+(y1[i]-y1[j])*(y1[i]-y1[j]))

            if((dist1 > distmax)) {distmax <- as.numeric(dist1)}
            if((dist1[1]) < 1) {constraint <- constraint*sqrt(dist1)}
            else if((dist1) > 14) {avg_dist <- avg_dist+sqrt(dist1)}
        }
    }

    if(constraint < 0.9999) constraint <- 1001.0-constraint
    else constraint = sqrt(distmax)+avg_dist/(10.0*NBC)

    f[2] <- 0.0
    f[M+1] <- constraint


    return(as.numeric(f) )
}

## Define parameters that we want to use
params<-list()
NBC <- 5
M <- 2
n<-2*NBC-3

params$NBC<-NBC
params$M<-M
x0<-rep(0.1, n)
lb<-rep(0, n)
ub<-rep(4.5, n)

eval.f.ex1(x0, params)

bbout<-c(2, 2, 0)
nmulti=5
bbin<-rep(0, n)
## Define initial value of the optimization problem

## Solve using snomadRSolve
snomadr(n            = as.integer(n),
        x0           = x0,
        eval.f       = eval.f.ex1,
        bbin         = bbin,
        bbout        = bbout,
        lb           = lb,
        ub           = ub,
        params       = params )

## Solve using smultinomadRSolve, if x0 is provided,  x0 will
## be the first initial point,  otherwise,  the program will
## check best_x.txt,  if it exists,  it will be read in as
## the first initial point. Other initial points will be
## generated by uniform distribution.
## nmulti represents the number of mads will run.
##
snomadr(n            = as.integer(n),
        eval.f       = eval.f.ex1,
        bbin         = bbin,
        bbout        = bbout,
        lb           = lb,
        ub           = ub,
        nmulti = as.integer(nmulti),
        display.nomad.progress = TRUE,
        params       = params )
} # }