Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization

TLDR

The study builds on the need to merge gradient‑based local NLP solvers with the global optimization capabilities of OptQuest, addressing large benchmark problems with over 100 variables and constraints. The authors develop OQNLP, a heuristic that seeks global optima for pure and mixed integer nonlinear problems with many constraints and variables. OQNLP generates starting points via OptQuest scatter search and then applies a gradient‑based local NLP solver to each point while holding discrete variables fixed to locate local solutions. On 155 benchmark NLP and MINLP problems, OQNLP achieved global solutions for almost all cases, typically within one or two local solver calls.

Abstract

The algorithm described here, called OptQuest/NLP or OQNLP, is a heuristic designed to find global optima for pure and mixed integer nonlinear problems with many constraints and variables, where all problem functions are differentiable with respect to the continuous variables. It uses OptQuest, a commercial implementation of scatter search developed by OptTek Systems, Inc., to provide starting points for any gradient-based local solver for nonlinear programming (NLP) problems. This solver seeks a local solution from a subset of these points, holding discrete variables fixed. The procedure is motivated by our desire to combine the superior accuracy and feasibility-seeking behavior of gradient-based local NLP solvers with the global optimization abilities of OptQuest. Computational results include 155 smooth NLP and mixed integer nonlinear program (MINLP) problems due to Floudas et al. (1999), most with both linear and nonlinear constraints, coded in the GAMS modeling language. Some are quite large for global optimization, with over 100 variables and 100 constraints. Global solutions to almost all problems are found in a small number of local solver calls, often one or two.