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Exploiting Algebraic Structure in Global Optimization and the Belgian Chocolate Problem
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.Comment: 15 page