This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some o...

Balas introduced intersection cuts for mixed integer linear sets. Intersection cuts are given by closed form formulas and form an important class of cuts for solving mixed integer linear programs. In this paper we introduce an extension of intersection cuts to mixed integer conic quadratic sets. We identify the formula for the conic quadratic intersection cut by formulating a system of polynomi...

The Perspective Relaxation (PR) is a general approach for constructing tight approximations to Mixed-Integer NonLinear Problems with semicontinuous variables. The PR of a MINLP can be formulated either as a Mixed-Integer Second-Order Cone Program, provided that the original objective function is SOCP-representable, or as a Semi-Infinite MINLP. We show that under some further assumptions—rather ...

This paper develops a linear programming based branch-and-bound algorithm for mixed integer conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by Ben-Tal and Nemirovski. The algorithm is different from other linear programming based branch-and-bound algorithms for mixed integer nonlinear programs in ...