Abstract
Generating polyhedral outer approximations and solving mixed-integer linear relaxations remains one of the main approaches for solving convex mixed-integer nonlinear programming (MINLP) problems. There are several algorithms based on this concept, and the efficiency is greatly affected by the tightness of the outer approximation. In this paper, we present a new framework for strengthening cutting planes of nonlinear convex constraints, to obtain tighter outer approximations. The strengthened cuts can give a tighter continuous relaxation and an overall tighter representation of the nonlinear constraints. The cuts are strengthened by analyzing disjunctive structures in the MINLP problem, and we present two types of strengthened cuts. The first type of cut is obtained by reducing the right-hand side value of the original cut, such that it forms the tightest generally valid inequality for a chosen disjunction. The second type of cut effectively uses individual right-hand side values for each term of the disjunction. We prove that both types of cuts are valid and that the second type of cut can dominate both the first type and the original cut. We use the cut strengthening in conjunction with the extended supporting hyperplane algorithm, and numerical results show that the strengthening can significantly reduce both the number of iterations and the time needed to solve convex MINLP problems.
Highlights
Mixed-integer nonlinear optimization (MINLP) arises in many applications across engineering, manufacturing, and the natural sciences (Boukouvala et al 2016)
We have presented a new framework for strengthening cuts to obtain tighter outer approximations for convex MINLP
The cut strengthening is based on analyzing disjunctive structures in the MINLP problem, and either strengthen the cut for the entire disjunction or separately for each term of the disjunction
Summary
Mixed-integer nonlinear optimization (MINLP) arises in many applications across engineering, manufacturing, and the natural sciences (Boukouvala et al 2016). The concept of using an outer approximation of the nonlinear constraints for MINLP problems, developed by (Duran and Grossmann 1986b; Geoffrion 1972), forms the core of several other convex MINLP algorithms, e.g., extended cutting plane (ECP) (Westerlund and Petterson 1995; Westerlund and Pörn 2002), feasibility pump (Bonami and Gonçalves 2012), extended supporting hyperplane (ESH) (Kronqvist et al 2016), and the center-cut algorithm (Kronqvist et al 2018a). This paper focuses on deriving strong cutting planes for convex MINLP problems, resulting in tight outer approximations, by exploiting disjunctive structures in the problem. 4. Section 5 presents an algorithm for solving convex MINLP problems that combines the ESH algorithm with the cut strengthening techniques.
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