Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

Abstract

We consider in this paper a class of semi-continuous quadratic programming problems which arises in many real-world applications such as production planning, portfolio selection and subset selection in regression. We propose a lift-and-convexification approach to derive an equivalent reformulation of the original problem. This liftand-convexification approach lifts the quadratic term involving x only in the original objective function f (x, y) to a quadratic function of both x and y and convexifies this equivalent objective function. While the continuous relaxation of our new reformulation attains the same tight bound as achieved by the continuous relaxation of the well known perspective reformulation, the new reformulation also retains the linearly constrained quadratic programming structure of the original mix-integer problem. This prominent feature improves the performance of branch-and-bound algorithms by providing the same tightness at the root node as the state-of-the-art perspective reformulation and offering much faster processing time at children nodes. We further combine the liftand-convexification approach and the quadratic convex reformulation approach in the literature to form an even tighter reformulation. Promising results from our computational tests in both portfolio selection and subset selection problems numerically verify the benefits from these theoretical features of our new reformulations.

Tighter quadratically constrained convex reformulations for semi-continuous quadratic programming

“…It is then extended to unconstrained 0-1 quadratic programming (QP) by [2], to equality constrained 0-1 QP by [4], and to the general mixed-integer QP by [3]. Recently, the QCR method is recently applied in [19] to obtain an improved reformulation whose continuous relaxation bound is at most as tight as that of the perspective reformulation.…”

Review on Reformulation of the Mean-Variance Model with Real-life Trading Restrictions

“…Another efficient solution method for mixed-integer quadratic programming with semi-continuous variables and cardinality constraint is the lift-and-convexification reformulation (LCR) proposed by Wu et al (2015), where the original problem is: the original problem. The reformulated problem (P(u, v)) can be expressed as following:…”

“…Besides the perspective reformulation, another type of reformulation method called lift-and-convexification reformulation (LCR) has been put forward in (Wu, Sun, Li, & Zheng, 2015). The substance of this approach is to add a quadratic equivalent term multiplied by a parameter to the objective function and to convexify the objective function so that the resulting formulation equivalent to the original problem.…”

Ideal formulations for constrained convex optimization problems with indicator variables

“…In all cases, most of the approaches for (1) have focused on the q-sparsity constraint (or its Lagrangian relaxation). For example, a standard technique to improve the relaxations of (1) revolves around the use of the perspective reformulation [1,19,26,27,[30][31][32]34,36,42,54,61,64], an ideal formulation of a separable quadratic function with indicators (but no additional constraints). Recent work on obtaining ideal formulations for non-separable quadratic functions [4][5][6]27,35,44] also ignores additional constraints in Q.…”