Projected Perspective Reformulations with Applications in Design Problems

The Perspective Relaxation (PR) is a general approach for constructing tight approximations to MixedInteger 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 SOCPrepresentable), 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 Mixed-Integer Quadratic Program can also be reformulated as a piecewise linear-quadratic problem, 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 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.