nloptr (version 1.0.0)

nloptr-package: R interface to NLopt

Description

nloptr is an R interface to NLopt, a free/open-source library for nonlinear optimization started by Steven G. Johnson, providing a common interface for a number of different free optimization routines available online as well as original implementations of various other algorithms. The NLopt library is available under the GNU Lesser General Public License (LGPL), and the copyrights are owned by a variety of authors. Most of the information here has been taken from http://ab-initio.mit.edu/nlopt{the NLopt website}, where more details are available.

NLopt addresses general nonlinear optimization problems of the form:

min f(x) x in R^n

s.t. g(x) <= 0="" h(x)="0" lb="" <="x" where="" f="" is="" the="" objective="" function="" to="" be="" minimized="" and="" x="" represents="" n="" optimization="" parameters.="" this="" problem="" may="" optionally="" subject="" bound="" constraints="" (also="" called="" box="" constraints),="" ub.="" for="" partially="" or="" totally="" unconstrained="" problems="" bounds="" can="" take="" -inf="" inf.="" one="" also="" have="" m="" nonlinear="" inequality="" (sometimes="" a="" programming="" problem),="" which="" specified="" in="" g(x),="" equality="" that="" h(x).="" note="" not="" all="" of="" algorithms="" nlopt="" handle="" constraints.<="" p="">

An optimization problem can be solved with the general nloptr interface, or using one of the wrapper functions for the separate algorithms; auglag, bobyqa, cobyla, crs2lm, direct, lbfgs, mlsl, mma, neldermead, newuoa, sbplx, slsqp, stogo, tnewton, varmetric.

Details

ll{ Package: nloptr Type: Package Version: 0.9.9 Date: 2013-11-22 License: L-GPL LazyLoad: yes }

References

Steven G. Johnson, The NLopt nonlinear-optimization package, http://ab-initio.mit.edu/nlopt

See Also

optim nlm nlminb Rsolnp ssolnp nloptr auglag bobyqa cobyla crs2lm direct isres lbfgs mlsl mma neldermead newuoa sbplx slsqp stogo tnewton varmetric

Examples

# Example problem, number 71 from the Hock-Schittkowsky test suite.
#
# \min_{x} x1*x4*(x1 + x2 + x3) + x3
# s.t.
#    x1*x2*x3*x4 >= 25
#    x1^2 + x2^2 + x3^2 + x4^2 = 40
#    1 <= x1,x2,x3,x4 <= 5
# 
# we re-write the inequality as
#   25 - x1*x2*x3*x4 <= 0
#
# and the equality as
#   x1^2 + x2^2 + x3^2 + x4^2 - 40 = 0
#
# x0 = (1,5,5,1)
#
# optimal solution = (1.00000000, 4.74299963, 3.82114998, 1.37940829)


library('nloptr')

#
# f(x) = x1*x4*(x1 + x2 + x3) + x3
#
eval_f <- function( x ) { 
    return( list( "objective" = x[1]*x[4]*(x[1] + x[2] + x[3]) + x[3],
                  "gradient" = c( x[1] * x[4] + x[4] * (x[1] + x[2] + x[3]),
                                  x[1] * x[4],
                                  x[1] * x[4] + 1.0,
                                  x[1] * (x[1] + x[2] + x[3]) ) ) ) 
}

# constraint functions
# inequalities
eval_g_ineq <- function( x ) {
    constr <- c( 25 - x[1] * x[2] * x[3] * x[4] )
                 
    grad   <- c( -x[2]*x[3]*x[4],                       
                 -x[1]*x[3]*x[4],
                 -x[1]*x[2]*x[4],
                 -x[1]*x[2]*x[3] )
    return( list( "constraints"=constr, "jacobian"=grad ) )
}

# equalities
eval_g_eq <- function( x ) {
    constr <- c( x[1]^2 + x[2]^2 + x[3]^2 + x[4]^2 - 40 )
                 
    grad   <- c(  2.0*x[1],
                  2.0*x[2],
                  2.0*x[3],
                  2.0*x[4] )
    return( list( "constraints"=constr, "jacobian"=grad ) )
}

# initial values
x0 <- c( 1, 5, 5, 1 )

# lower and upper bounds of control
lb <- c( 1, 1, 1, 1 )
ub <- c( 5, 5, 5, 5 )


local_opts <- list( "algorithm" = "NLOPT_LD_MMA",
                    "xtol_rel"  = 1.0e-7 )
opts <- list( "algorithm" = "NLOPT_LD_AUGLAG",
              "xtol_rel"  = 1.0e-7,
              "maxeval"   = 1000,
              "local_opts" = local_opts )

res <- nloptr( x0=x0, 
               eval_f=eval_f, 
               lb=lb, 
               ub=ub, 
               eval_g_ineq=eval_g_ineq, 
               eval_g_eq=eval_g_eq, 
               opts=opts)               
print( res )