Comparing perspective reformulations for piecewise-convex optimization

TL;DR

This paper compares different reformulation techniques for piecewise-convex optimization problems within the context of solving MINLPs, highlighting their theoretical and computational differences when using perspective reformulations.

Contribution

It provides a comparative analysis of three standard formulations for piecewise-convex MINLP relaxations, emphasizing the impact of perspective reformulations on their equivalence.

Findings

Perspective reformulations improve relaxation bounds in convex segments.

The three formulations are not equivalent when perspective reformulation is applied.

Computational experiments demonstrate differences in solution quality and efficiency.

Abstract

Our study is motivated by the solution of Mixed-Integer Non-Linear Programming (MINLP) problems with separable non-convex functions via the Sequential Convex MINLP technique, an iterative method whose main characteristic is that of solving, for bounding purposes, piecewise-convex MINLP relaxations obtained by identifying the intervals in which each univariate function is convex or concave and then relaxing the concave parts with piecewise-linear relaxations of increasing precision. This process requires the introduction of new binary variables for the activation of the intervals where the functions are defined. In this paper we compare the three different standard formulations for the lower bounding subproblems and we show, both theoretically and computationally, that -- unlike in the piecewise-linear case -- they are not equivalent when the perspective reformulation is applied to reinforce the formulation in the segments where the original functions are convex.