This CRAN task view contains a list of packages which offer
facilities for solving optimization problems.
Although every regression model in statistics solves an optimization
problem they are not part of this view. If you are looking for regression
methods, the following views will contain useful starting points:
Multivariate,
SocialSciences,
Robust
among others.
The focus of this task view is on
Optimization Infrastructure Packages
,
General Purpose Continuous
Solvers
,
Mathematical
Programming Solvers
and
Specific
Applications in Optimization
. Packages are categorized in these
four sections.
Many packages provide functionality for more than one
of the subjects listed at the end of this task view. E.g., mixed
integer linear programming solvers typically offer standard linear programming
routines like the simplex algorithm. Therefore following each package
description a list of abbreviations describes the typical features
of the optimizer (i.e., the problems which can be solved). The
full names of the abbreviations given in square brackets can be
found at the end of this task view under
Classification According to
Subject
.
If you think that some package is missing from the list, please
let us know.

Trying to unify optimization algorithms via a single wrapper
function,
optimx
helps to proper specify the
(nonlinear) optimization problem including objective function,
gradient function, and scaling. This package supports the
(local) optimization of smooth, nonlinear functions with at most
box constraints (bounds).
optimx
depends not only on
packages and/or functions mentioned in this section of this task
view but also on two packages implemented by the author(s),
namely
Rcgmin
and
Rvmmin. Both are "pure
R" implementations of conjugate gradient minimization and
variable metric nonlinear function minimization algorithms,
respectively.

The R Optimization Infrastructure (ROI) package
provides a framework for handling optimization problems in R. It
uses an object oriented approach to define and solve various
optimization tasks in R which can be from different problem
classes (e.g., linear, quadratic, nonlinear programming
problems). This makes optimization transparent for the R user as
the corresponding workflow is completely abstracted from the
underlying solver. Furthermore, this approach allows for easy
switching between solvers, given that corresponding solver plugins
are available, and thus enhances comparability.

Package stats offers several general purpose optimization routines.
First, function
optim()
provides an implementation of
the BroydenFletcherGoldfarbShanno (BFGS) method,
bounded BFGS, conjugate gradient, NelderMead, and simulated
annealing (SANN) optimization
methods. It utilizes gradients, if provided, for faster
convergence. Typically it is used for unconstrained optimization
but includes an option for boxconstrained
optimization.
Additionally, for minimizing a function subject to
linear inequality constraints stats contains the routine
constrOptim().
For onedimensional unconstrained function optimization there is
optimize()
which searches an interval
for a minimum or maximum.
Then there is
nlm
which is used for solving nonlinear
unconstrained minimization problems. Eventually,
nlminb()
offers unconstrained and constrained
optimization using PORT routines. [RGA, QN]

Package
nloptr
provides access to NLopt, an LGPL licensed library of
various nonlinear optimization algorithms. It includes local derivativefree
(COBYLA, NelderMead, Subplex) and gradientbased (e.g., BFGS) methods, and also
the augmented Lagrangian approach for nonlinear constraints. [DF, GO, QN]

Implementations of the augmented lagrange multiplier method for general nonlinear
optimization can be found in packages
alabama
and
Rsolnp.

Package
blowtorch
offers routines for building and solving
constrained optimization problems via stochastic gradient descent.

clue
contains the function
sumt()
for
solving constrained optimization problems via the sequential
unconstrained minimization technique (SUMT).

Package
dfoptim, derivativefree optimization
procedures, contains quite efficient R implementations of the
NelderMead and HookeJeeves algorithms (unconstrained and
boundsconstrained). [DF]

GrassmannOptim
is a package for Grassmann manifold
optimization. The implementation uses gradientbased algorithms
and embeds a stochastic gradient method for global search.

Package
gsl
provides BFGS, conjugate gradient,
steepest descent, and NelderMead algorithms. It uses a "line
search" approach via the function
multimin(). It is
based on the GNU Scientific Library (GSL). [RGA, QN]

Two optimization algorithms are implemented in package
lbfgs: Limitedmemory BroydenFletcherGoldfarbShann
(LBFGS) and OrthantWise Limitedmemory QuasiNewton
(OWLQN). The latter allows for objectives with a
nondifferentiable penalty. [QN]

Package
maxLik
provides a general purpose
NewtonRaphson optimizer
maxNR()
as well as wrapper
to methods, implemented in
optim(). It supports fitting a
submodel by specifying constant parameters.

An R port of the Scilab neldermead module is packaged in
neldermead
offering several direct search algorithms based on the simplex approach.

Several derivativefree optimization algorithms are provided with package
minqa; e.g., the functions
bobyqa(),
newuoa(),
and
uobyqa()
allow to minimize a function of many variables by a trust
region method that forms quadratic models by interpolation.
bobyqa()
additionally permits box constraints (bounds) on the parameters. [DF]

Package
minpack.lm
provides a function
nls.lm()
for solving nonlinear leastsquares problems by a modification of the
LevenbergMarquardt algorithm, as found in MINPACK.

Package
powell
optimizes functions using Powell's
UObyQA algorithm (Unconstrained Optimization by Quadratic
Approximation).

RCEIM
implements a stochastic heuristic method for performing
multidimensional function optimization.

subplex
provides unconstrained function
optimization based on a subspace searching simplex method.

Package
ucminf
implements an algorithm of quasiNewtonian type
for nonlinear unconstrained optimization. [QN]

In package
trust, a routine with the same name offers
local optimization based on the "trust region" approach.

trustOptim
implements a "trust region" algorithm for
unconstrained nonlinear optimization. The algorithm is optimized for
objective functions with sparse Hessians. This makes the algorithm highly
scalable and efficient, in terms of both time and memory footprint.

Package
nleqslv
provides a function
nleqslv()
implementing Newton and Broyden methods with line search and trust region global
strategies for solving medium sized system of nonlinear equations.

Package
genalg
contains
rbga(), an implementation
of a genetic algorithm for multidimensional function optimization.

Machine coded genetic algorithm (MCGA) provided by package
mcga
is a tool which solves optimization problems based
on byte representation of variables.

Package
rgenoud
offers
genoud(), a routine which is
capable of solving complex function minimization/maximization problems by combining
evolutionary algorithms with a derivativebased (quasiNewtonian) approach.

An R implementation of the SelfOrganising Migrating Algorithm
(SOMA) is available in package
soma. The approach of
this stochastic optimization algorithm is similar to that of
genetic algorithms and can be applied to any costminimization
problem with a bounded parameter space, and is robust to local minima.
Semidefinite and Convex Solvers

CSDP is a library of routines that implements a primaldual barrier method
for solving semidefinite programming problems; it is interfaced in the
Rcsdp
package. [SDP]

The
CLSOCP
package provides an implementation of a onestep
smoothing Newton method for the solution of second order cone programming
(SOCP) problems.

cccp
contains routines for solving cone constrained
convex problems by means of interiorpoint methods (partially ported
from Python's CVXOPT).

The DSDP library implements an interiorpoint method for semidefinite
programming with primal and dual solutions; it is interfaced in package
Rdsdp. [SDP]
Global Optimization

Package
DEoptim
provides a global optimizer based on the
Differential Evolution algorithm.
RcppDE
provides a C++ implementation
(using Rcpp) of the same
DEoptim()
function.

DEoptimR
provides an implementation of the jDE variant of the
differential evolution stochastic algorithm for nonlinear programming problems
(It allows to handle constraints in a flexible manner.)

GenSA
is a package providing a function for generalized Simulated
Annealing which can be used to search for the global minimum of a quite complex
nonlinear objective function with a large number of optima.

GA
provides functions for optimization using Genetic Algorithms
in both, the continuous and discrete case, whether constrained or not. This package
allows to run corresponding optimization tasks in parallel.

A particle swarm optimizer (PSO) is implemented in package
pso.
Another (parallelized) implementation of the PSO algorithm can be found in package
ppso available from
rforge.net/ppso
.

Package
hydroPSO
implements the latest Standard Particle Swarm
Optimization algorithm (SPSO2011); it is parallelcapable, and includes
several finetuning options and postprocessing functions.

Package
cmaes
implements an effective global optimization procedure
using a covariance matrix adapting evolutionary strategy (CMAES).

Package
Rmalschains
implements an algorithm family for
continuous optimization called memetic algorithms with local search chains
(MALSChains).

nloptr
especially supports several global optimization routines,
such as DIRECT, controlled random search (CRS), multilevel singlelinkage (MLSL),
improved stochastic ranking (ISRES), or stochastic global optimization (StoGO).

The
NMOF
package provides implementations of differential evolution,
particle swarm optimization, local search and threshold accepting (a variant of
simulated annealing). The latter two methods also work for discrete optimization
problems, as does the implementation of a genetic algorithm that is included in
the package.
This section provides an overview of open source as well as commercial
optimizers. Which type of mathematical programming problem can be solved
by a certain package or function can be seen from the abbreviations in square
brackets. For a
classification by subject
see the list at the end of this task view.

Package
linprog
solves linear programming problems using
the function
solveLP()
(the solver is based on
lpSolve) and can read model files in MPS
format. [LP]

In package
quadprog
solve.QP()
solves
quadratic programming problems
with linear equality and inequality constraints. [QP]

BB
contains the function
spg()
providing a
spectral projected gradient method for large scale optimization with
simple constraints. It takes a nonlinear objective function as an
argument as well as basic constraints. Furthermore,
BB
contains two functions (
dfsane()
and
sane()) for using the spectral gradient method
for solving a nonlinear system of equations.

In the
boot
package there is a routine called
simplex()
which realizes the twophase tableau simplex
method for (relatively small) linear programming problems. [LP]

kernlab
contains the function
ipop
for
solving quadratic programming problems using interior point
methods. [IPM, QP]

limSolve
offers to solve linear or quadratic
optimization functions. [LP, QP]

LowRankQP
primal/dual interior point method solving
quadratic programming problems [IPM, QP]

Package
rcdd
offers the function
lpcdd()
for solving linear programs with exact arithmetic using the
GNU Multiple Precision (GMP)
library. [LP]

In package Rdonlp2 (see the
rmetrics
project)
function
donlp2(), a wrapper for the DONLP2
solver, offers the minimization of smooth nonlinear functions and
constraints. DONLP2 can be used freely for any kind of research
purposes, otherwise it requires licensing. [GO, NLP]

The
NEOS Server
for Optimization
provides online access to stateoftheart
optimization problem solvers. Package
rneos
enables the
user to pass optimization problems to NEOS and retrieve results
within R.
Interfaces to Open Source Optimizers

Package
clpAPI
provides high level access from R to
lowlevel API routines of the
COIN OR Clp
solver
library. [LP]

Package
lpSolve
contains the routine
lp()
to solve LPs and MILPs by calling the freely
available solver
lp_solve
. This solver is
based on the revised simplex method and a branchandbound (B&B)
approach. It supports semicontinuous variables and Special Ordered
Sets (SOS). Furthermore
lp.assign()
and
lp.transport()
are aimed at solving assignment problems
and transportation problems, respectively. Additionally, there is the
package
lpSolveAPI
which provides an R interface to the
low level API routines of lp_solve (see also project
lpsolve
on RForge).
lpSolveAPI
supports reading linear programs from files in lp
and MPS format. [BP, IP, LP, MILP, SPLP]

Packages
glpkAPI
as well as
package
Rglpk
provide an interface to the
GNU Linear Programming
Kit
(GLPK). Whereas the former provides high level access
to low level routines the latter offers a high level routine
Rglpk_solve_LP()
to solve MILPs using GLPK. Both
packages offer the possibility to use models formulated in the MPS
format. [BP, IP, IPM, LP, MILP]

Rsymphony
provides the routine
Rsymphony_solve_LP()
which
interfaces the SYMPHONY solver for mixed integer linear
programs. SYMPHONY applies LP
relaxation with a branchandcut approach. It is part of the
Computational
Infrastructure for Operations Research
(COINOR) project which is an initiative to spur the
development of open source software for the operations research
community. [LP, IP, MILP]

The NOMAD solver is implemented in the
crs
package
for solving mixed integer programming problems. This algorithm is
accessible via the
snomadr()
function and is
primarily designed for constrained optimization of blackbox
functions.
Interfaces to Commercial Optimizers
This section surveys interfaces to commercial solvers. Typically, the
corresponding libraries have to be installed separately.

Packages
cplexAPI
and
Rcplex
provide an
interface to the CPLEX solver package from
IBM
. CPLEX
provides dual/primal simplex optimizers as well as a barrier
optimizer for solving large scale linear and quadratic
programs. Furthermore, it offers a mixed integer optimizer to
solve difficult mixed integer programs. Note that CPLEX is
not
free
and you have to get a license from
IBM
. Academics will receive a free
licence upon request. [LP, IP, BP, QP, MILP, MIQP, IPM]

Package
Rmosek
offers an interface to the
commercial optimizer from
MOSEK
. It provides dual/primal
simplex optimizers as well as a barrier optimizer. In addition to
solving LP and QP problems this solver can handle SOCP and
quadratically constrained programming (QPQC) tasks. Furthermore,
it offers a mixed integer optimizer to solve difficult mixed
integer programs (MILP, MISOCP, etc.). Note that you have to get a
license from
MOSEK
. Academic
licenses are free of charge. [LP, IP, BP, QP, MILP, MIQP,
IPM]

Gurobi Optimization ships an R binding with their 5.0 release
that allows to solve LP, MIP, QP, MIQP, SOCP, and MISOCP models
from within R. See the
R
reference manual
from the Gurobi documentation for details.

The
localsolver
package provides an interface to the hybrid
mathematical programming software LocalSolver from Innovation 24.
LocalSolver is a commercial product, academic licenses are available on
request. [LP, MIP, QP, NLP, HEUR]

Package
adagio
provides functions for single and
multiple knapsack problems and solves subset sum and assignment
tasks.

In package
clue
solve_LSAP()
enables
the user to solve the linear sum assignment problem (LSAP) using an
efficient C implementation of the Hungarian algorithm. [SPLP]

The
CEoptim
package provides an optimization solver,
based on the CrossEntropy method, for continuous and combinatorial
optimization problems. It can be used to solve multiextremal problems,
accepts linear constraints,and handles discrete and continuous variables.
[COP]

The data cloning algorithm is a global optimization approach
and a variant of simulated annealing which has been implemented
in package
dclone. The package provides low level
functions for implementing maximum likelihood estimating
procedures for complex models using data cloning and Bayesian
Markov chain Monte Carlo methods.

Objective functions for benchmarking the performance of
global optimization algorithms can be found in package
globalOptTests.

cec2013
and
cec2005benchmark
contain many test
functions for global optimization from the 2005 and 2013 special sessions
on realparameter optimization at the IEEE CEC congresses on evolutionary
computation.

Package
goalprog
provides some functions for
lexicographic linear goal
programming and optimization. Goal programming is a branch of
multiobjective, multicriteria decision analysis. [MOP]

igraph, a package for graph and network analysis,
uses the very fast igraph C library. It can be used to calculate
shortest paths, maximal network flows, minimum
spanning trees, etc. [GRAPH]

irace
contains an optimization algorithm for optimizing the
parameters of other optimization algorithms. This problem is called "(offline)
algorithm configuration". [GO]

matchingR
and
matchingMarkets
implement the GaleShapley
algorithm for the stable marriage and the college admissions problem, the stable
roommates and the house allocation problem. [COP, MM]

Package
kofnGA
uses a genetic algorithm to choose a
subset of a fixed size k from the integers 1:n, such that a user
supplied objective function is minimized at that subset.

Besides functionality for solving general isotone regression problems,
package
isotone
provides a framework of active set methods for
isotone optimization problems with arbitrary order restrictions.

maxLik
adds a likelihoodspecific layer on top of a number of
maximization routines like BrendtHallHallHausman (BHHH) and NewtonRaphson
among others. It includes summary and print methods which extract the standard
errors based on the Hessian matrix and allows easy swapping of maximization
algorithms. It also provides a function to check whether an analytic derivative
is computed directly.

Multicriteria optimization problems can be solved using
package
mco
which implements genetic
algorithms. [MOP]

Package
optmatch
provides routines for solving
matching problems by translating them into minimumcost flow problems,
which are in turn solved optimally by the RELAXIV codes of
Bertsekas and Tseng (free for research). [SPLP]

Package
quantreg
contains variations of simplex and
of interior point routines (
nlrq(),
crq()). It provides an interface to L1 regression
in the R code of function
rq(). [SPLP, LP, IPM]

The
desirability
package contains S3 classes for
multivariate optimization using the desirability function approach
of Harrington (1965) using functional forms described by
Derringer and Suich (1980).

Package
sna
contains the function
lab.optim()
which is the frontend to a series of heuristic routines for optimizing
some bivariate graph statistic. [GRAPH]

Package
TSP
provides basic infrastructure for
handling and solving the
traveling salesperson problem (TSP). The main routine
solve_TSP()
solves the TSP
through several heuristics. In addition, it provides an interface
to the
Concorde TSP
Solver
, which has to be
downloaded separately. [SPLP]
What follows is an attempt to provide a bysubject overview of
packages. The full name of the subject as well as the corresponding
MSC
2010
code (if available) are given in brackets.

LP (Linear programming, 90C05):
boot,
clpAPI,
cplexAPI,
glpkAPI,
limSolve,
linprog,
lpSolve,
lpSolveAPI,
quantreg,
rcdd,
Rcplex,
Rglpk,
Rmosek,
Rsymphony

GO (Global Optimization):
DEoptim,
DEoptimR,
GenSA,
GA,
pso,
hydroPSO,
cmaes,
nloptr,
NMOF

SPLP (Special problems of linear programming like transportation,
multiindex, etc., 90C08):
clue,
lpSolve,
lpSolveAPI,
optmatch,
quantreg,
TSP

BP (Boolean programming, 90C09):
cplexAPI,
glpkAPI,
lpSolve,
lpSolveAPI,
Rcplex,
Rglpk

IP (Integer programming, 90C10):
cplexAPI,
glpkAPI,
lpSolve,
lpSolveAPI,
Rcplex,
Rglpk,
Rmosek,
Rsymphony

MIP (Mixed integer programming and its variants MILP for LP
and MIQP for QP, 90C11):
cplexAPI,
glpkAPI,
lpSolve,
lpSolveAPI,
Rcplex,
Rglpk,
Rmosek,
Rsymphony

QP (Quadratic programming, 90C20):
cplexAPI,
kernlab,
limSolve,
LowRankQP,
quadprog,
Rcplex,
Rmosek

SDP (Semidefinite programming, 90C22):
Rcsdp,
Rdsdp

CP (Convex programming, 90C25):
cccp,
CLSOCP

COP (Combinatorial optimization, 90C27):
adagio,
CEoptim,
TSP,
matchingR

MOP (Multiobjective and goal programming, 90C29):
goalprog,
mco

NLP (Nonlinear programming, 90C30):
nloptr,
alabama,
Rsolnp, Rdonlp2

GRAPH (Programming involving graphs or networks, 90C35):
igraph,
sna

IPM (Interiorpoint methods, 90C51):
cplexAPI,
kernlab,
glpkAPI,
LowRankQP,
quantreg,
Rcplex

RGA (Methods of reduced gradient type, 90C52): stats
(
optim()),
gsl

QN (Methods of quasiNewton type, 90C53): stats
(
optim()),
gsl,
lbfgs,
nloptr,
ucminf

DF (Derivativefree methods, 90C56):
dfoptim,
minqa,
nloptr

HEUR (Approximation methods and heuristics, 90C59):
irace