Grothendieck constant narrowed down using AI to two decimal digits
Primal/Dual Method for the Grothendieck Constant
Computational Complexity
Summary
The Grothendieck constant is a special number important in mathematics and computer science, but its exact value was not known. This paper finds that the constant is between about 1.773 and 1.7799, pinning down its hundredths place for the first time. The authors used a mix of math theories and artificial intelligence tools to guess and then rigorously verify their result. This improves understanding of this constant which appears in optimization and quantum information.
What this means in practice
- •For optimization engineers: Better estimates of the Grothendieck constant help refine bounds in optimization problems, improving algorithmic guarantees.
- •For quantum computing developers: Improved knowledge of the Grothendieck constant informs quantum information protocols related to operator inequalities.
A theory result. No direct application yet.
Authors
Steven Heilman, Chris Jones, Giulio Malavolta
Abstract
We determine the hundredths digit of the real Grothendieck constant by proving $1.773 \leq K_G \leq 1.7799$. Furthermore, numerical heuristics suggest the estimate $K_G \approx 1.779$. We start by simplifying the Gaussian duality theory of $K_G$, based on Hermite projection games and Krivine rounding schemes. The bounds are obtained by a heuristic primal/dual search for these objects, followed by rigorous {\em certification} of their values. The search and certification are performed using AI tools, and in particular the rigorous certificates are very large (analytical reductions to thousands of numerical inequalities).