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The Belgian chocolate problem involves maximizing a parameter {\delta} over anon-convex region of polynomials. In this paper we detail a global optimizationmethod for this problem that outperforms previous such methods by exploitingunderlying algebraic structure. Previous work has focused on iterative methodsthat, due to the complicated non-convex feasible region, may require manyiterations or result in non-optimal {\delta}. By contrast, our method locatesthe largest known value of {\delta} in a non-iterative manner. We do this byusing the algebraic structure to go directly to large limiting values, reducingthe problem to a simpler combinatorial optimization problem. While theselimiting values are not necessarily feasible, we give an explicit algorithm forarbitrarily approximating them by feasible {\delta}. Using this approach, wefind the largest known value of {\delta} to date, {\delta} = 0.9808348. We alsodemonstrate that in low degree settings, our method recovers previously knownupper bounds on {\delta} and that prior methods converge towards the {\delta}we find.

Exploiting Algebraic Structure in Global Optimization and the Belgian Chocolate Problem