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.