A combined linear and nonlinear presolve for nonlinear optimization

Abstract

Presolve techniques have been experimentally shown to significantly accelerate the performance of optimization solvers, achieving speedups of several orders of magnitude in widely used benchmarks. Building on the success of these techniques in linear and mixed-integer linear optimization problems, we introduce novel presolve methods specifically designed for nonlinear optimization. These methods aim to reduce model size and nonlinearity while preserving convexity and ensuring global optimality. We propose a combined linear and nonlinear presolve approach that integrates classical linear presolve strategies with novel methods for reformulating nonlinear expressions and simplifying models. For instance, monotonicity arguments are used to fix nonlinear variables, and linear constraints are exploited to tighten bilinear products. Computational experiments on diverse nonlinear benchmarks and continuous relaxations of discrete nonlinear problems demonstrate the efficacy of our approach. The results show that the proposed methods significantly enhance the performance of one global and four local nonlinear optimization solvers.