Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations

A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher-dimensional space by introducing variables Y ij to represent each of the products x i x j of variables appearing in a quadratic form. One advantage of such ex...

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Vydáno v:	Mathematical programming Ročník 130; číslo 2; s. 359 - 413
Hlavní autoři:	Saxena, Anureet, Bonami, Pierre, Lee, Jon
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer-Verlag 01.12.2011 Springer Springer Nature B.V
Témata:	Algorithms Applied sciences Calculus of Variations and Optimal Control; Optimization Combinatorics Computation Constraints Construction Exact sciences and technology Full Length Paper Heuristic Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Numerical Analysis Operational research and scientific management Operational research. Management science Optimization Quadratic forms Studies Theoretical Variables global optimization 90C26 Nonconvex programming Non convex programming Subgradient optimization Constraint satisfaction Global optimum Linear programming Mixed integer programming Semi definite programming Quadratic programming Cut generation Integer programming Time average Disjunctive programming Heuristic method Relaxation method Non convex analysis
ISSN:	0025-5610, 1436-4646
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Abstract	A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher-dimensional space by introducing variables Y ij to represent each of the products x i x j of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint and disjunctive programming. On the other hand, the main drawback of such an extended formulation is its huge size, even for problems for which the number of x i variables is moderate. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen-reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on box-QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 s, even for large instances with 100 variables; the SDP+RLT relaxations for the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver.
AbstractList	A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher-dimensional space by introducing variables Y sub( )ijto represent each of the products x sub( )ix sub( )jof variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y - x x T [sccue] 0 and disjunctive programming. On the other hand, the main drawback of such an extended formulation is its huge size, even for problems for which the number of x sub( )ivariables is moderate. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen-reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on box-QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 s, even for large instances with 100 variables; the SDP+RLT relaxations for the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver. A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher-dimensional space by introducing variables Y ij to represent each of the products x i x j of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint and disjunctive programming. On the other hand, the main drawback of such an extended formulation is its huge size, even for problems for which the number of x i variables is moderate. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen-reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on box-QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 s, even for large instances with 100 variables; the SDP+RLT relaxations for the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver. A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher-dimensional space by introducing variables Y ^sub ij^ to represent each of the products x ^sub i^ x ^sub j^ of variables appearing in a quadratic form. One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y − x x T 0 and disjunctive programming. On the other hand, the main drawback of such an extended formulation is its huge size, even for problems for which the number of x ^sub i^ variables is moderate. In this paper, we study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations. To do so, we use projection techniques pioneered in the context of the lift-and-project methodology. We show how the extended formulation can be algorithmically projected to the original space by solving linear programs. Furthermore, we extend the technique to project the SDP relaxation by solving SDPs. In the case of an MIQCP with a single quadratic constraint, we propose a subgradient-based heuristic to efficiently solve these SDPs. We also propose a new eigen-reformulation for MIQCP, and a cut generation technique to strengthen this reformulation using polarity. We present extensive computational results to illustrate the efficiency of the proposed techniques. Our computational results have two highlights. First, on the GLOBALLib instances, we are able to generate relaxations that are almost as strong as those proposed in our companion paper even though our computing times are about 100 times smaller, on average. Second, on box-QP instances, the strengthened relaxations generated by our code are almost as strong as the well-studied SDP+RLT relaxations and can be solved in less than 2 s, even for large instances with 100 variables; the SDP+RLT relaxations for the same set of instances can take up to a couple of hours to solve using a state-of-the-art SDP solver.[PUBLICATION ABSTRACT]
Author	Lee, Jon Bonami, Pierre Saxena, Anureet
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Keywords	global optimization 90C26 Nonconvex programming Non convex programming Subgradient optimization Constraint satisfaction Global optimum Linear programming Mixed integer programming Semi definite programming Quadratic programming Cut generation Integer programming Time average Disjunctive programming Heuristic method Relaxation method Non convex analysis
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SubjectTerms	Algorithms Applied sciences Calculus of Variations and Optimal Control; Optimization Combinatorics Computation Constraints Construction Exact sciences and technology Full Length Paper Heuristic Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Mixed integer Numerical Analysis Operational research and scientific management Operational research. Management science Optimization Quadratic forms Studies Theoretical Variables
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Title	Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations
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