The cost of not knowing enough: mixed-integer optimization with implicit Lipschitz nonlinearities

Martin Schmidt,Mathias Sirvent,Winnifried Wollner

doi:10.1007/s11590-021-01827-9

Abstract

It is folklore knowledge that nonconvex mixed-integer nonlinear optimization problems can be notoriously hard to solve in practice. In this paper we go one step further and drop analytical properties that are usually taken for granted in mixed-integer nonlinear optimization. First, we only assume Lipschitz continuity of the nonlinear functions and additionally consider multivariate implicit constraint functions that cannot be solved for any parameter analytically. For this class of mixed-integer problems we propose a novel algorithm based on an approximation of the feasible set in the domain of the nonlinear function—in contrast to an approximation of the graph of the function considered in prior work. This method is shown to compute approximate global optimal solutions in finite time and we also provide a worst-case iteration bound. In some first numerical experiments we show that the “cost of not knowing enough” is rather high by comparing our approach with the open-source global solver SCIP. This reveals that a lot of work is still to be done for this highly challenging class of problems and we thus finally propose some possible directions of future research.

Highlights

Mixed-integer nonlinear optimization problems (MINLPs) are one of the most important classes of models in mathematical optimization
The most important distinction is to be made between convex and nonconvex MINLPs
One typically needs to derive convex underestimators and concave overestimators that yield convex relaxations of the nonconvex nonlinearities [9,10,11]. These approaches and most other algorithms for computing global optima of such nonconvex MINLPs are based on spatial branching

Summary

Introduction

Mixed-integer nonlinear optimization problems (MINLPs) are one of the most important classes of models in mathematical optimization. Spatial branching based on convex underestimators and concave overestimators usually exploit known analytical properties of the nonconvex nonlinear functions, which is obviously not possible if these properties, like, e.g., differentiability, are not known or even knowledge about the explicit representation is missing. In this case, one typically tries to resort to Lipschitz assumptions about the nonlinearities, which leads to the field of global Lipschitz optimization; see, e.g., [12,13,14,15,16,17,18] to name only a few.

Problem definition

Algorithm

Stationary gas transport optimization

Results

Conclusion