Reformulation and solution approach for non-separable integer quadratic programs

Abstract

We consider quadratic programs with pure general integer variables. The objective function is quadratic and convex and the constraints are linear. An exact solution approach is proposed. It is decomposed into two phases. In the first phase, the initial problem is reformulated into an equivalent problem with a separable objective function. This is done by use of a Gauss decomposition of the Hessian matrix of the initial problem and requires the addition of some continuous variables and constraints. In the second phase, the reformulated problem is linearized by an approximation of each squared term by a set of K linear functions that correspond to the tangents of a hyperbola in K points. We give a proof of the intuitive property that when K is large enough, the optimal value of the obtained linear program is very close to optimal value of the two previous problems, the initial problem and the reformulated separable problem. The reminder is dedicated to the implementation of a branch-and-bound algorithm for the solution of linearized problem, and its application to a set of instances. Several points are considered among which choice of the right value for parameter K and the implementation of a sophisticated heuristic solution algorithm. The numerical comparison is done with CPLEX 12.2 since, in this case, the initial problem as well as the problem reformulated by the first step can be solved by CPLEX. We show that with our approach, the total CPU time is divided by a factor ranging from 1.2 to 131.6 for instances with 40–60 variables.