Computational Optimization and Applications
The split closure of a strictly convex body
Operations Research Letters
On linear programs with linear complementarity constraints
Journal of Global Optimization
Semidefinite relaxations for mixed 0-1 second-order cone program
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
GloMIQO: Global mixed-integer quadratic optimizer
Journal of Global Optimization
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This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from the equation Y = x x T . We use the non-convex constraint $${ Y - x x^T \preccurlyeq 0}$$ to derive disjunctions of two types. The first ones are directly derived from the eigenvectors of the matrix Y − x x T with positive eigenvalues, the second type of disjunctions are obtained by combining several eigenvectors in order to minimize the width of the disjunction. We also use the convex SDP constraint $${ Y - x x^T \succcurlyeq 0}$$to derive convex quadratic cuts, and we combine both approaches in a cutting plane algorithm. We present computational results to illustrate our findings.