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

The convex hull relaxation (CHR) method (Albornoz in Doctoral Dissertation, 1998, Ahlatcioglu in Summer paper, 2007, Ahlatcioglu 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.