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This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic construction also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with existing approaches, this method is based on convex optimization, and thus has polynomial-time complexity. As an application, we use this method to study powers of the construction given by Coppersmith and Winograd Journal of Symbolic Computation, 1990 and obtain the upper bound ω < 2.3728639 on the exponent of square matrix multiplication, which slightly improves the best known upper bound.
François Le Gall (Tue,) studied this question.