Combining QCR and CHR for convex quadratic pure 0–1 programming problems with linear constraints

Abstract

The convex hull relaxation (CHR) method (Albornoz in Doctoral Dissertation, 1998, Ahlatçıoglu in Summer paper, 2007, Ahlatçıoglu and Guignard in OPIM Dept. Report, 2010 provides lower bounds and feasible solutions on convex 0-1 nonlinear programming problems with linear constraints. In the quadratic case, these bounds may often be improved by a preprocessing step that adds to the quadratic objective function terms that are equal to 0 for all 0-1 feasible solutions yet increase its continuous minimum. Prior to computing CHR bounds, one may use Plateau's quadratic convex reformulation (QCR) method ( 2006), or one of its weaker predecessors designed for unconstrained problems, the eigenvalue method of Hammer and Rubin (RAIRO 3:67-79, 1970) or the method of Billionnet and Elloumi (Math. Program, Ser. A 109:55-68, 2007). In this paper, we first describe the CHR method, and then present the QCR reformulation methods. We present computational results for convex GQAP problems.M. Guignard was partially supported under NSF Grant DMI-0400155.

Quadratic Convex Reformulations for Semicontinuous Quadratic Programming

“…Ahlatçıoglu et al [1] proposed to combine QCR and the convex hull relaxation to solve problem (BQP). The geometric investigation in Li et al [36] for binary quadratic programs provides some theoretical support for QCR from another angle.…”

“…Problem (P) is in general NP-hard (see [7]). Its difficulty arises from the discrete structure induced by the constraint in (1). This constraint is used to model the situation where x i must rest inside an interval if it is not zero, that is, x i ∈ {0} ∪ [a i , b i ].…”

A mixed-integer programming formulation and Lagrangean relaxation for the cross-dock door assignment problem

“…The problem is formulated as a bilinear integer program and transformed into a special case of the generalised quadratic three-dimensional assignment problem to use the branch-and-bound algorithm presented in Hahn, Smith, and Zhu (2010) to solve it. Guignard et al (2012) present two different heuristic algorithms to solve the same problem. The first approach is a local search method in which generalised assignment problems are iteratively solved to determine the best assignment of origins or destinations to dock doors.…”

“…The first approach is a local search method in which generalised assignment problems are iteratively solved to determine the best assignment of origins or destinations to dock doors. The second, however, is an adaptation of the convex hull heuristic introduced in Ahlafçioglu et al (2012). We refer to Downloaded by [University of Saskatchewan Library] at 19:13 16 March 2015 Luo and Noble (2012) and Choy et al (2012) for other CDAPs that incorporate additional features of real applications such as a limited capacity on storage and staging areas and random arrivals of inbound trucks.…”

“…However, very little work has been done for the development of mathematical programming formulations that can be used with general purpose optimisation solvers and with decomposition techniques to efficiently solve them and to provide performance guaranties on the quality of the obtained solutions. In this paper, we study the CDAP considered in Zhu et al (2009) and Guignard et al (2012). From now on, we refer to this problem as the CDAP.…”