A Spectral Generalization of Von Neumann Minimax Theorem

Abstract

Given $n\times n$ real symmetric matrices $A_1,\dots ,A_m$, the following {\it spectral minimax} property holds: $$\underset{X\in {\mathbf{\Delta }}_n}{\min }\underset{y\in S_m}{\max }\sum _{i=1}^m{y}_i{A}_i\bullet X=\underset{y\in S_m}{\max }\underset{X\in {\mathbf{\Delta }}_n}{\min }\sum _{i=1}^m{y}_i{A}_i\bullet X,$$ where $S_m$ is the simplex and ${\mathbf{\Delta }}_n$ the spectraplex. For diagonal $A_i$'s this reduces to the classic minimax.

4 pages