Mixed-integer quadratic programming is in NP - Pia, Alberto Del; Dey, Santanu S.; Molinaro, Marco (Mathematical programming, 2017)
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 ]).