Computing a bound for the rectangular parameter α via the new constraint program

Obtain a quantitative bound on the rectangular matrix multiplication parameter α—defined as the largest real number such that n×n^α×n multiplication can be performed in O(n^{2+ε}) time for all ε>0—by solving the constraint program induced by the authors’ asymmetric laser-method analysis.

Background

The authors’ method yields large and complex non-convex constraint programs that must be solved numerically to extract exponents for rectangular matrix multiplication. They report that their nonlinear solver struggles and that they were unable to solve the program for α, a classical parameter introduced by Coppersmith that governs the largest rectangle width allowing near-quadratic time.

This unresolved computational step prevents the paper from reporting an improved bound on α using their framework, making the numerical solution of the α-optimization a concrete open task tied to their methodology.