Unbounded Convex Sets for Non-Convex Mixed-Integer Quadratic Programming Samuel Burer * Adam N. Letchford † Published in Mathematical Programming January 2014 Abstract This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic program- ming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a face of a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some other well-studied con- vex sets. Several classes of valid and facet-inducing inequalities are also derived. Key Words: mixed-integer non-linear programming, global optimisa- tion, polyhedral combinatorics, convex analysis. 1 Introduction A Mixed-Integer Quadratic Program (MIQP) is an optimisation problem that can be written in the following form: min  c T x + x T Qx : Ax = b, x ∈ Z n1 + × R n2 +  , (1) where n = n 1 +n 2 , c ∈ Q n , Q ∈ Q n×n , A ∈ Q m×n , b ∈ Q m , and Q is symmetric without loss of generality. MIQPs are a generalisation of Mixed-Integer Linear Programs and therefore NP -hard to solve. On the other hand, they can be regarded as a special kind of Mixed-Integer Non-Linear Program (MINLP). If Q is positive semidefinite (psd), then the objective function is convex, and one can use any method for convex MINLPs (such as those described in [4, 15]). Otherwise, the objective function is non-convex, and even solving the continuous relaxation of the MIQP is an NP -hard global optimisation problem (see, e.g., [38, 42]). * Department of Management Sciences, Tippie College of Business, University of Iowa. E-mail: samuel-burer@uiowa.edu. Research supported in part by NSF Grant CCF-0545514. † Department of Management Science, Lancaster University, United Kingdom. A.N.Letchford@lancaster.ac.uk 1