Mixed-integer quadratic programming is in NP

Abstract

Mixed-integer quadratic programming (MIQP) 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 [11] and integer linear programming is in NP [1,12,4,9].

Feedback scheduling of sensor network activity using a Hybrid Dynamical Systems approach

“…Moreover, the MPC approach requires solving MIQP problem, which is known to be NP-hard [11]. This usually implies a large computational cost.…”

Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

“…We assume that P I is unbounded, because otherwise Problem (1) can be solved using Theorem 4.9. If the degree of f is not greater than two, then Problem (1) can be solved by [8]. Thus we will assume that the degree of f is exactly three.…”

“…When f : R 2 → R is a quadratic polynomial, [8] proves Theorem 1.1 using the fact that P can be divided into regions where f is quasiconvex and quasiconcave. We use a similar approach for homogeneous polynomials and determine quasiconvexity and quasiconcavity by analyzing the bordered Hessian.…”

“…Recently, Del Pia et al [7] showed that the decision version of mixed-integer quadratic programming is in NP. Del Pia and Weismantel [8] showed that for polynomials in two variables of degree two, the problem is solvable in polynomial time.…”

The multiplicative weights update algorithm for mixed integer nonlinear programming: theory, applications, and limitations

“…MINLP is NP-hard because it includes Mixed Integer Linear Programming (MILP) [102,139] and Mixed Integer Quadratic Programming (MIQP) [69] as special cases, when constraints are affine and objective function is linear or quadratic, respectively. MINLP is, in general, undecidable [130], even for p 10, when the objective function is linear and the constraints are polynomial [67].…”