Conclusion

In this article, the subject of optimization using Mathematica tools is discussed, with an emphasis on models that could have multiple local optima of various quality: in such cases, we wish to find the best possible (global) solution. After introducing a general optimization modeling framework, several built-in Mathematica optimization functions and the MathOptimizer package are reviewed. To illustrate the usage of MathOptimizer, several point and multibody configuration analysis and design models are formulated and solved. In an illustrative test related to "best" circle packings, we also used NMinimize and MathOptimizer Professional.

As our results demonstrate, the usage of proper GO tools is both necessary and possible in order to provide globally established numerical solutions to difficult models of theoretical interest and practical relevance. To illustrate the practical use of our optimizers, let us mention here that MathOptimizer has been recently used to design high-quality acoustic equipment: consult, for example, [67]. The LGO solver system embedded in MathOptimizer Professional has already been used in a variety of advanced applications, including, for example, cancer therapy planning, laser equipment design, robotics design, industrial (shape) design, chemical product (material composition) design, water quality modeling, wastewater systems design and operations, computational chemistry, econometrics, financial modeling, and others. Most of these results are reported elsewhere: consult, for example, [29, 35-39]. We expect that a significant range of advanced optimization models developed using Mathematica will be successfully analyzed and solved using our packages.