Rcplex_solve_QCP function

Solve quadratically constrained optimization problem with CPLEX

Interface to CPLEX solvers for quadratically constrained linear, quadratic, and mixed-integer programs. The general statement of the problem is [REMOVE_ME]$\min \frac{1}{2}x'Qx + c'xmin 0.5x'Qx + c'x [REMOVE_ME_2]$

[REMOVE_ME]$\mathrm{s.t} Ax \leq bs.t Ax <= b [REMOVE_ME_2]$

[REMOVE_ME]$\mathrm{and} a_i'x + x'Q_ix \leq r_i for i=1,\ldots,qand a_i'x + x'Q_i x <= r_i for i=1,...,q [REMOVE_ME_2]$

[REMOVE_ME]$lb \leq x \leq ublb <= x <= ub [REMOVE_ME_2]$

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.

Description

Interface to CPLEX solvers for quadratically constrained linear, quadratic, and mixed-integer programs. The general statement of the problem is

$$\min \frac{1}{2}x'Qx + c'xmin 0.5x'Qx + c'x$$ $$\mathrm{s.t} Ax \leq bs.t Ax <= b$$ $$\mathrm{and} a_i'x + x'Q_ix \leq r_i for i=1,\ldots,qand a_i'x + x'Q_i x <= r_i for i=1,...,q$$ $$lb \leq x \leq ublb <= x <= ub$$

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.

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

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

• QC: a list with three elements: QC, dir, and b. The element QC is a list with the quadratic part Q, a matrix, and the linear part of the constraint L, a numeric (currently nonzero values are not supported). dir has the same meaning as argument sense and b as bvec.

• 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.

Details

See function link[Rcplex]{Rcplex}() for more information about sparse matrix representation and control arguments.

Returns

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

References

IBM ILOG CPLEX Optimization Studio documentation

Author(s)

Hector Corrada Bravo and Stefan Theussl

See Also

Rcplex.close, optim

Examples

## objective function
c <- c(1, 2, 3)
Q <- matrix(c(-33, 6, 0, 6, -22, 11.5, 0, 11.5, -11), nrow = 3)

## constraints

## linear part
A <- matrix(c(-1, 1, 1, -3, 1, 1), nrow = 2)
dir <- c("L", "L")
b <- c(20, 30)

## quadratic part
QC <- list(QC = list(Q = list(diag(1, nrow = 3)), L = NULL), dir = "L", b = 1)

## bounds
ub <- c(40, Inf, Inf)

## solve
res <- Rcplex_solve_QCP(c,A, b, Q, ub = ub, QC = QC, sense = dir, objsense = "max")
print(res)

## solve MIQCP
res <- Rcplex_solve_QCP(c, A, b, Q, ub = ub, QC = QC,
                        sense = dir, objsense = "max", vtype = c("C", "I", "C"))

## quadratic and linear part
QC <- list(QC = list(Q = list(diag(1, nrow = 3)), L = list(c(3,4,-3))), dir = "L", b = 1)

## solve
res <- Rcplex_solve_QCP(c,A, b, Q, ub = ub, QC = QC, sense = dir, objsense = "max")
print(res)

Rcplex.close()