A computational comparison of reformulations of the perspective relaxation: SOCP vs. cutting planes

Abstract

The Perspective Reformulation generates tight approximations to MINLP problems with semicontinuous variables. It can be implemented either as a Second-Order Cone Program, or as a Semi-Infinite Linear Program. We compare the two reformulations on two MIQPs in the context of exact or approximate Branch-and-Cut algorithms.

Perspective Reformulations of the CTA Problem with L2 Distances

“…• (SOCP) and (SOCP+) have been tested but were regularly worse than (P/C) and (P/C+), respectively, for single-threaded executions, confirming the results of [13]; therefore, the corresponding results have not been reported.…”

“…First of all, the corresponding model would be a SOCP with up to three SOCP constraints for each sensitive cell; the standard formulation (SOCP), which already has only two of them, is typically not competitive with (P/C) [13], a fact that we directly verified to be true for CTA also. Furthermore, the rationale of [14] is to exploit structural properties in the original problem, which are absent here for general tabular data since the matrix A lacks exploitable characteristics.…”

“…This is possible since (14) is a lower approximation to (13); furthermore, the two objective function obviously coincide on integer solutions. The model 10.…”

“…Either (3) or any finite approximation to (4)-typically, to be iteratively refined-can be used as models of (2), whose continuous relaxation is significantly stronger than that of (1) and that therefore is a more convenient starting point to develop exact and approximate solution algorithms [11,12,19,1,15]. Somewhat surprisingly, the potentially very large and approximated (4) appears to be most often preferable to the compact and exact (3) in the context of exact or approximate enumerative solution approaches [13], likely due to the better reoptimization capabilities of simplex methods for linear programs w.r.t. those of interior point methods for conic programs.…”

On interval-subgradient and no-good cuts

“…In [18], lifting techniques are discussed in the framework of NLP; [20] discusses an extension of the RLT to convex Mixed-Integer Programming (MIP). A certain attention has been devoted to conic MIP [8,2]; in part, this is due to the fact that Lift&Project techniques (see, e.g., [3]) to compute valid inequalities for the union of two convex sets can easily be extended to the nonlinear setting [9], and this may produce strong conical reformulations of MIPs [22,12] out of which effective cuts may be obtained [11].…”