Mixed-integer quadratic programming is in NP

Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it...

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Published in:	Mathematical programming Vol. 162; no. 1-2; pp. 225 - 240
Main Authors:	Pia, Alberto Del, Dey, Santanu S., Molinaro, Marco
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017 Springer Nature B.V
Subjects:	Calculus of Variations and Optimal Control; Optimization Combinatorics Complexity theory Economics Feasibility Full Length Paper Inequality Integer programming Lectures Linear programming Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Polynomials Quadratic programming Studies Theoretical Integer programming 90C20 Quadratic programming 90C60 90C11 Complexity
ISSN:	0025-5610, 1436-4646
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Abstract	Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP (Vavasis in Inf Process Lett 36(2):73–77 [ 17 ]) and integer linear programming is in NP (Borosh and Treybig in Proc Am Math Soc 55:299–304 [ 1 ], von zur Gathen and Sieveking in Proc Am Math Soc 72:155–158 [ 18 ], Kannan and Monma in Lecture Notes in Economics and Mathematical Systems, vol. 157, pp. 161–172. Springer [ 9 ], Papadimitriou in J Assoc Comput Mach 28:765–768 [ 15 ]).
AbstractList	Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP (Vavasis in Inf Process Lett 36(2):73-77 [17]) and integer linear programming is in NP (Borosh and Treybig in Proc Am Math Soc 55:299-304 [1], von zur Gathen and Sieveking in Proc Am Math Soc 72:155-158 [18], Kannan and Monma in Lecture Notes in Economics and Mathematical Systems, vol. 157, pp. 161-172. Springer [9], Papadimitriou in J Assoc Comput Mach 28:765-768 [15]). Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP (Vavasis in Inf Process Lett 36(2):73–77 [ 17 ]) and integer linear programming is in NP (Borosh and Treybig in Proc Am Math Soc 55:299–304 [ 1 ], von zur Gathen and Sieveking in Proc Am Math Soc 72:155–158 [ 18 ], Kannan and Monma in Lecture Notes in Economics and Mathematical Systems, vol. 157, pp. 161–172. Springer [ 9 ], Papadimitriou in J Assoc Comput Mach 28:765–768 [ 15 ]).
Author	Dey, Santanu S. Molinaro, Marco Pia, Alberto Del
Author_xml	– sequence: 1 givenname: Alberto Del surname: Pia fullname: Pia, Alberto Del organization: Department of Industrial and Systems Engineering & Wisconsin Institute for Discovery, University of Wisconsin-Madison – sequence: 2 givenname: Santanu S. surname: Dey fullname: Dey, Santanu S. organization: Computer Science Department, PUC-Rio – sequence: 3 givenname: Marco surname: Molinaro fullname: Molinaro, Marco email: mmolinaro@inf.puc-rio.br organization: Computer Science Department, PUC-Rio
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SubjectTerms	Calculus of Variations and Optimal Control; Optimization Combinatorics Complexity theory Economics Feasibility Full Length Paper Inequality Integer programming Lectures Linear programming Mathematical and Computational Physics Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Polynomials Quadratic programming Studies Theoretical
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