Amélie Lambert

An efficient compact quadratic convex reformulation for general integer quadratic programs

We address the exact solution of general integer quadratic programs with linear constraints. These programs constitute a particular case of mixed-integer quadratic programs for which we introduce in Billionnet et al. (Math. Program., 2010) a general solution method based on quadratic convex reformulation, that we called MIQCR. This reformulation consists in designing an equivalent quadratic program with a convex objective function. The problem reformulated by MIQCR has a relatively important size that penalizes its solution time. In this paper, we propose a convex reformulation less general than MIQCR because it is limited to the general integer case, but that has a significantly smaller size. We call this approach Compact Quadratic Convex Reformulation (CQCR). We evaluate CQCR from the computational point of view. We perform our experiments on instances of general integer quadratic programs with one equality constraint. We show that CQCR is much faster than MIQCR and than the general non-linear solver BARON (Sahinidis and Tawarmalani, User’s manual, 2010) to solve these instances. Then, we consider the particular class of binary quadratic programs. We compare MIQCR and CQCR on instances of the Constrained Task Assignment Problem. These experiments show that CQCR can solve instances that MIQCR and other existing methods fail to solve.

Exact quadratic convex reformulations of mixed-integer quadratically constrained problems

We propose a solution approach for the general problem (QP) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of (QP). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex.

Linear Reformulations of Integer Quadratic Programs

Let (QP) be an integer quadratic program that consists in minimizing a quadratic function subject to linear constraints. In this paper, we present several linearizations of (QP). Many linearization methods for the quadratic 0-1 programs are known. A natural approach when considering (QP) is to reformulate it into a quadratic 0-1 program. However, this method, that we denote BBL (Binary Binary Linearization), leads to a quadratic program with a large number of variables and constraints.

A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation

Let

Global solution of non-convex quadratically constrained quadratic programs

(MQP)

Extending the QCR method to general mixed-integer programs

be a general mixed-integer quadratic program that consists of minimizing a quadratic function

Convex reformulations of Integer Quadratically Constrained Problems

f(x) = x^TQx +c^Tx