How difficult is nonlinear optimization? A practical solver tuning approach, with illustrative results

Nonlinear optimization (NLO) encompasses a vast range of problems, from very simple to theoretically intractable instances. For this reason, it is impossible to offer guaranteed—while practically meaningful—advice to users of NLO software. This issue becomes apparent, when facing exceptionally hard and/or previously unexplored NLO challenges. We propose a heuristic quadratic meta-model based approach, and suggest corresponding key option settings to use with the Lipschitz global optimizer (LGO) solver suite. These LGO option settings are directly related to estimating the sufficient computational effort to handle a broad range of NLO problems. The proposed option settings are evaluated experimentally, by solving (numerically) a representative set of NLO test problems which are based on real-world optimization applications and non-trivial academic challenges. Our tests include also a set of scalable optimization problems which are increasingly difficult to handle as the size of the model-instances increases. Based on our computational results, it is possible to offer generally valid, practical advice to LGO users. Arguably (and mutatis mutandis), comparable advice can be given to users of other NLO software products with a similarly broad mandate to LGO’s. An additional benefit of such aggregated tests is that their results can effectively assist the rapid evaluation and verification of NLO solver performance during software development phases.