Projected Perspective Reformulations for MIQP problems

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 restrictive, but satisfied in several practical applications—the PR of Mixed-Integer Quadratic Programs can also be reformulated as a piecewise linear-quadratic problem of roughly the same size of the standard continuous relaxation. Furthermore, if the original problem has some exploitable structure, this is typically preserved in the reformulation, allowing to construct 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.