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On Valid Inequalities for Quadratic Programming with Continuous Variables and Binary Indicators
"... Abstract. In this paper we study valid inequalities for a set that involves a continuous vector variable x ∈ [0, 1] n, its associated quadratic form xx T, and binary indicators on whether or not x> 0. This structure appears when deriving strong relaxations for mixed integer quadratic programs (MI ..."
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Abstract. In this paper we study valid inequalities for a set that involves a continuous vector variable x ∈ [0, 1] n, its associated quadratic form xx T, and binary indicators on whether or not x> 0. This structure appears when deriving strong relaxations for mixed integer quadratic programs (MIQPs). Valid inequalities for this set can be obtained by lifting inequalities for a related set without binary variables (QPB), that was studied by Burer and Letchford. After closing a theoretical gap about QPB, we characterize the strength of different classes of lifted QPB inequalities. We show that one class, liftedposdiagQPB inequalities, capture no new information from the binary indicators. However, we demonstrate the importance of the other class, called liftedconcaveQPB inequalities, in two ways. First, all lifted concaveQPB inequalities define the relevant convex hull for the case of convex quadratic programming with indicators. Second, we show that all perspective constraints are a special case of liftedconcaveQPB inequalities, and we further show that adding the perspective constraints to a semidefinite programming (SDP) relaxation of convex quadratic programs with binary indicators results in a problem whose bound is equivalent to the recent optimal diagonal splitting approach of Zheng et al.. Finally, we show the separation problem for liftedconcaveQPB inequalities is tractable if the number of binary variables involved in the inequality is small. Our study points out a direction to generalize perspective cuts to deal with nonseparable nonconvex quadratic functions with indicators in global optimization. Several interesting questions arise from our results, which we detail in our concluding section.
Computational Management Science manuscript No. (will be inserted by the editor) Optimization and Sustainable Development
, 2014
"... Abstract In this opinion paper, I argue that “optimization and sustainable development” indicates a set of specific engineering techniques rather than a unified discipline stemming from a unique scientific principle. On the other hand, I also propose a mathematical principle underlying at least some ..."
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Abstract In this opinion paper, I argue that “optimization and sustainable development” indicates a set of specific engineering techniques rather than a unified discipline stemming from a unique scientific principle. On the other hand, I also propose a mathematical principle underlying at least some of the concepts defining sustainability when optimizing a supply chain. The principle is based on the fact that since demand constraints are usually expressed as inequalities, those which are not active at the optimal solution imply the existence of some wasted activity, which may lead to an unsustainable solution. I propose using flowtype equation constraints instead, which help detect unsustainability through infeasibility.