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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Bibliographic Details
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
Online Access:Get full text
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Summary: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 ]).
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-016-1036-0