Projected Perspective Reformulations for NonLinear Network Design 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. While these reformulations significantly improve the lower bound and the running times of the corresponding enumerative approaches, they may spoil some valuable structure of the problem, such as the presence of network constraints. In this paper, we show that under some further assumptions the PR of a 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 and where the (network) structure of the original problem is safeguarded. We apply this approach to a quadratic-cost singlecommodity network design problem, comparing the newly developed algorithm with those based on both the standard continuous relaxation and the two usual variants of PR relaxation.