A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities

TL;DR

This paper introduces a new successive linear relaxation method for solving complex mixed-integer nonlinear programming problems with multivariate Lipschitz continuous nonlinearities, improving computational efficiency and broadening application scope.

Contribution

The paper develops a novel iterative method that combines master and subproblems using Lipschitz constants, enhancing the handling of multivariate nonlinear constraints in MINLPs.

Findings

Proves the correctness and worst-case iteration bound of the method.

Demonstrates applicability to bilevel and gas transport optimization problems.

Shows improved computational performance over existing approaches.

Abstract

We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt et al. (2018, 2021) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems.