Rcplex (version 0.3-3)

# Rcplex: Solve optimization problem with CPLEX

## Description

Interface to CPLEX solvers for linear quadratic and (linear or quadratic) mixed-integer programs. The general statement of the problem is $$\min \frac{1}{2}x'Qx + c'x$$ $$\mathrm{s.t} Ax \leq b$$ $$lb \leq x \leq ub$$

If Q==NULL then the problem is linear, if any value of the vtype argument is "B" or "I" then the problem is a mixed-integer program. The control argument is used to set CPLEX's many parameters. See details. The objsense determines if the problem is a maximization or minimization problem. The sense argument is used to set the constraint directions.

## Usage

Rcplex(cvec, Amat, bvec, Qmat = NULL, lb = 0, ub = Inf, control = list(), objsense = c("min", "max"), sense = "L", vtype = NULL, n = 1)

## Arguments

cvec
The linear coefficient of the objective function
Amat
The constraint matrix (requires ncol(Amat)==length(cvec))
bvec
The constraints right-hand side (requires length(bvec)==nrow(Amat))
Qmat
The quadratic coefficient of the objective function. If NULL the problem is linear. If not NULL, it must be a symmetric positive semidefinite matrix of size length(cvec) by length(cvec). Default NULL
lb
Lower bound on the problem variables. If length(lb)==1 then lb is the lower bound of all variables. Otherwise, length(lb)==length(cvec). Set lb=-Inf to have no lower bound. Default 0.
ub
Upper bound on the problem variables. See lb for further details. Default Inf.
control
A list of CPLEX parameters. See *Details*
objsense
Either "max" or "min", determines the optimization direction. Default "min"
sense
The direction of the inequality in each constraint. If length(sense)==1 then the same value is taken for each constraint. Can be one of "L" (less than or equal), "G" (reater than or equal) or "E" (equal). Requires length(sense)==length(bvec). Default "L".
vtype
Determines the type of each problem variable. Can be one of "C" (continuous), "I" (integer) or "B" (binary). If length(vtype)==1 the same value is taken for all variables. Otherwise, requires length(vtype)==length(ctype). Default "C".
n
Determines the maximal number of solutions the solver should return in case of an MIP with more than one solution at optimum. If CPLEX should search for "all" solutions then n has to be set to NA. In CPLEX this is also called populating the solution pool. The parameters solnpoolagap, solnpoolgap, and solnpoolintensity influence the search for multiple solutions (see also the control argument below for details). Available from CPLEX 11.0 on. Rcplex() raises a warning if an older version of CPLEX is used and n>1. Default 1.

## Value

Returns a list with the following components, or, if n > 1 a list of length equal to the number of optimal solutions containing the following components for each solution:
xopt
Values of problem variables at optimum.
obj
Value of objective function at optimum.
status
Solution status. See CPLEX documentation for meaning of status codes.
extra
List with extra information about solution with components
slack:
Values of slack variables for inequality constraints.
nodecnt:
(IF MIP PROBLEM) Number of nodes in the search tree evaluated
lambda:
(IF NOT MIP PROBLEM) Values of dual variables at optimum

## Details

Matrices A and C may be sparse matrices from a class in the hierarchy defined by the Matrix package. In that case, the internal casting functions are used to create the proper data structures to pass to CPLEX, which is similar to the column-major storage mode defined by the dgCMatrix-class defined by the Matrix package.

We also provide a simple S3-style class for sparse matrices simple_triplet_matrix, as used in the relations package. Matrices A and C can be objects of this class. See the examples for example usage. simple_triplet_matrix objects MUST be in column-major order. The control argument can be used to set CPLEX's many parameters, including the particular algorithm used for solving the given problem. See the ILOG CPLEX Parameters guide for further details. The following parameters are supported:

trace:
Turn CPLEX output on (1) or off(0). Default 1.

maxcalls:
Number of calls to the CPLEX optimizer before license is released. Set to 1 to get a new license on every call to Rcplex. Can be any positive number. Default 500.

method:
Algorithm to use (Default 0):
0:
Automatic: CPLEX chooses algorithm automatically

1:
Primal Simplex

2:
Dual Simplex

3:
Network Simplex

4:
Barrier

preind:
Turn presolver on (1) or off (0). Default 1.

aggind:
Limit on the number of applications of the aggregator. Possible Values: -1 (automatic), 0 (do not use), any positive integer

itlim:
Maximum number of simplex iterations. Can be any nonnegative number. Default 1e8.

epagap:
Absolute MIP optimality gap tolerance. Can be any nonnegative number. Default 1e-6.

epgap:
Relative MIP optimality gap tolerance. Can be any nonnegative number. Default 1e-4.

tilim:
Time limit in seconds of call to optimizer. Can be any nonnegative number. Default 1e75.

disjcuts:
Indicator for disjunctive cuts used in MIP solver. Must be in -1:3. Default 0 (automatic).

mipemphasis:
Indicator for MIP solver emphasis. Must be in 0:4. Default 0 (balance optimality and feasibility)

cliques:
Indicator for clique cuts in MIP solver. Must be in -1:2. Default 0 (automatic)

nodesel:
Node selection strategy in MIP solver. Must be in 0:3. Default 1 (best-bound search).

probe:
Probe level in MPI solver. Must be -1:3. Default 0 (automatic)

varsel:
Variable selection strategy in MIP solver. Must be in -1:4. Default 0 (choose best method automatically).

flowcovers:
Indicator for flowcover cuts in MIP solver. Must be in -1:2. Default 0 (automatic).

solnpoolagap:
Sets an absolute tolerance on the objective value for the solutions in the solution pool. Can be any nonnegative real number. Ignored in versions < 11.0 of CPLEX. Default 0

solnpoolgap:
Sets a relative tolerance on the objective value for the solutions in the solution pool. Can be any nonnegative real number. Ignored in versions < 11.0 of CPLEX. Default 0

solnpoolintensity:
Controls the trade-off between the number of solutions generated for the solution pool and the amount of time and memory consumed. Must be in 0:4. Ignored in versions < 11.0 of CPLEX. Default 0 (automatic).

round:
Flag indicating if integer solutions for MIPs should be rounded before returning. In some cases, CPLEX returns slightly infeasible integer solutions. Setting this option to 1 ensures that the returned solution is integral by rounding. Default 0 (no rounding).

## References

IBM ILOG CPLEX Optimization Studio documentation

Rcplex.close, optim

## Examples

## A linear program (this is lpex1.c in the CPLEX examples)
cvec <- c(1,2,3)
Amat <- matrix(c(-1,1,1,-1,3,-1),byrow=TRUE,nc=3)
bvec <- c(20,-30)
ub <- c(40,Inf,Inf)

res <- Rcplex(cvec,Amat,bvec,ub=ub,objsense="max",sense=c('L','G'))
print(res)

## A linear program with random data
## use the barrier method
n = 20; m = 25
nnz <- trunc(.2 * m * n)

## entries in simple_triplet_matrix clas
##  *must* be in column major order
nnz <- sort(sample(m*n,nnz,replace=FALSE)-1)
Amat <- simple_triplet_matrix(
i = (nnz %% m) + 1,
j = trunc(nnz/m) + 1,
v = rnorm(nnz),
nrow=m,ncol=n)

x0 <- runif(n)
b <- as.matrix(Amat) %*% x0
cvec <- rnorm(n)

res <- Rcplex(cvec,Amat,b,sense='E',control=list(method=4))
print(res)

## A quadratic problem (this is qpex1.c in the CPLEX examples)
cvec <- c(1,2,3)
Qmat <- matrix(c(-33,6,0,
6,-22,11.5,
0,11.5,-11),
byrow=TRUE,
nc=3)
Amat <- matrix(c(-1,1,1,
1,-3,1),
byrow=TRUE,nc=3)
bvec <- c(20,30)
ub <- c(40,Inf,Inf)

res <- Rcplex(cvec,Amat,bvec,Qmat,ub=ub,objsense="max")
print(res)

## A mixed integer linear program (mipex1.c in the CPLEX examples)
cvec <- c(1,2,3,1)
Amat <- matrix(c(-1,1,1,10,
1,-3,1,0,
0,1,0,-3.5),
byrow=TRUE, nc=4)
bvec <- c(20,30,0)
lb <- c(0,0,0,2)
ub <- c(40,Inf,Inf,3)
vtype <- c(rep("C",3),"I")

res <- Rcplex(cvec,Amat,bvec,lb=lb,ub=ub,sense=c("L","L","E"),
objsense="max",vtype=vtype)
print(res)

## A mixed integer quadratic program
cvec <- c(1,2,3,1)
Qmat <- matrix(c(-33,6,0,0,
6,-22,11.5,0,
0,11.5,-11,0,
0,0,0,0),
byrow=TRUE, nc=4)
Amat <- matrix(c(-1,1,1,10,
1,-3,1,0,
0,1,0,-3.5),
byrow=TRUE, nc=4)
bvec <- c(20,30,0)
ub <- c(40,Inf,Inf,3)
vtype <- c(rep("C",3),"I")

res <- Rcplex(cvec,Amat,bvec,Qmat=Qmat,ub=ub,sense=c("L","L","E"),
objsense="max",vtype=vtype)
print(res)
Rcplex.close()