Projected Perspective Reformulations with Applications in Design Problems

The Perspective Relaxation (PR) is a general approach for constructing tight approximations to Mixed Integer Non Linear Programs (MINLP) with semi-continuous variables. The PR of a MINLP can be formulated either as a Mixed Integer Second Order Cone Program (MI-SOCP), provided that the original objective function is SOCP-representable, or as a Semi-Infinite MINLP. In this paper, we show that under some further assumptions (rather restrictive, but satisfied in several practical applications), the PR of a Mixed Integer Quadratic Program (MIQP) can also be reformulated as a piecewise Quadratic Program (QP), ultimately yielding a QP relaxation of roughly the same size of the standard continuous relaxation. Furthermore, if the original problem has some exploitable structure, then this structure is typically preserved in the reformulation, thus allowing the construction of specialized approaches for solving the PR. We report on implementing these ideas on two MIQPs with appropriate structure: a sensor placement problem and a quadratic-cost (single-commodity) network design problem.