Exploiting algebraic structure in global optimization and the Belgian chocolate problem

Abstract

The Belgian chocolate problem involves maximizing a parameter \(\delta \) over a non-convex region of polynomials. In this paper we detail a global optimization method for this problem that outperforms previous such methods by exploiting underlying algebraic structure. Previous work has focused on iterative methods that, due to the complicated non-convex feasible region, may require many iterations or result in non-optimal \(\delta \). By contrast, our method locates the largest known value of \(\delta \) in a non-iterative manner. We do this by using the algebraic structure to go directly to large limiting values, reducing the problem to a simpler combinatorial optimization problem. While these limiting values are not necessarily feasible, we give an explicit algorithm for arbitrarily approximating them by feasible \(\delta \). Using this approach, we find the largest known value of \(\delta \) to date, \(\delta = 0.9808348\). We also demonstrate that in low degree settings, our method recovers previously known upper bounds on \(\delta \) and that prior methods converge towards the \(\delta \) we find.

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Acknowledgements

The authors would like to thank Bob Barmish for his valuable feedback, discussions, and advice. Zachary Charles was partially supported by the National Science Foundation Grant DMS-1502553. Nigel Boston was partially supported by the Simons Foundation Grant MSN179747.

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Correspondence to Zachary Charles.

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Charles, Z., Boston, N. Exploiting algebraic structure in global optimization and the Belgian chocolate problem. J Glob Optim 72, 241–254 (2018). https://doi.org/10.1007/s10898-018-0659-5

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Keywords

  • Global optimization
  • Control theory
  • Belgian chocolate problem
  • Stability
  • Simultaneous stabilization