Abstract
Different versions of polyhedral outer approximation are used by many algorithms for mixed-integer nonlinear programming (MINLP). While it has been demonstrated that such methods work well for convex MINLP, extending them to solve nonconvex problems has traditionally been challenging. The Supporting Hyperplane Optimization Toolkit (SHOT) is a solver based on polyhedral approximations of the nonlinear feasible set of MINLP problems. SHOT is an open source COIN-OR project, and is currently one of the most efficient global solvers for convex MINLP. In this paper, we discuss some extensions to SHOT that significantly extend its applicability to nonconvex problems. The functionality include utilizing convexity detection for selecting the nonlinearities to linearize, lifting reformulations for special classes of functions, feasibility relaxations for infeasible subproblems and adding objective cuts to force the search for better feasible solutions. This functionality is not unique to SHOT, but can be implemented in other similar methods as well. In addition to discussing the new nonconvex functionality of SHOT, an extensive benchmark of deterministic solvers for nonconvex MINLP is performed that provides a snapshot of the current state of nonconvex MINLP.
Highlights
Mixed-integer nonlinear programming (MINLP) is one of the most versatile optimization paradigms with many applications across engineering, manufacturing and the natural sci-Journal of Global Optimization ences [7,21,27,42,67]
We present some heuristic techniques that have recently been added to Supporting Hyperplane Optimization Toolkit (SHOT) to improve its performance on nonconvex MINLP problems; some of these improvements were briefly mentioned in the conference paper [48]
We have described some new features added to the SHOT solver
Summary
Mixed-integer nonlinear programming (MINLP) is one of the most versatile optimization paradigms with many applications across engineering, manufacturing and the natural sci-. Gurobi recently introduced functionality to globally optimize nonconvex mixed-integer quadratically constrained quadratic programming (MIQCQP) problems [31]. These global solvers mainly rely on spatial branch and bound, where convex understimators and concave overestimators are refined in nodes of a branching tree. Sometimes optimization software users are mainly interested in finding a good-enough feasible solution to the optimization problem within a reasonable computation time In such situations a local MINLP solver, or a heuristic MINLP technique [13,45], might be the best option. Solvers based on a POA technique solve a sequence of linear relaxations of the MINLP problem, where cutting or supporting hyperplanes form an outer approximation of the nonlinear feasible set.
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