Engineering Optimization

Appendix EOptimization Challenge Problems (2‐D and Single OF)

E.1 Introduction

There is a need for convenient test functions for creators to test optimizers and for learners to explore issues. Many that have been offered are simple one‐line functions. Although these can embody certain aspects or features related to optimization, they misrepresent the complexity of real applications and are often contrived idealizations. This collection of test functions seeks to embody first‐principles models of optimization applications to reveal issues, within a step‐better reveal of reality, but remaining relatively simple.

This set of objective functions (OF) was created to provide sample challenges for testing optimizers. The examples are all two‐dimensional, having two decision variables (DVs) so as to provide visual understanding of the issues that they embody. Most are relatively simple to program and compute rather simply, for user convenience. Most represent physically meaningful situations, for the person who wants to see utility and relevance. Most should be understood by those with a STEM degree. All are presented with minimization as the objective.

In all equations that follow, x1 and x2 are the DVs, and f_of_x is the OF value. The DVs are programmed for the range [0, 10]. However, not all functions use DV values in that range. So, the DVs are scaled for the appropriate range and labeled x11 and x22. The OF value f_of_x is similarly scaled for a [0, 10] range. The 0–10 scaling ...