A Spectral Generalization of Von Neumann Minimax Theorem

2019/05/23 by Bahman Kalantari, Kalantari, Bahman
Mathematics · Computer Science · Engineering · #Advanced Optimization Algorithms Research #Matrix Theory and Algorithms #Sparse and Compressive Sensing Techniques

paper · pdf · doi:10.48550/arxiv.1905.09762

Abstract

Given n × n real symmetric matrices A₁, …, A_m, the following \it spectral minimax property holds: min_{X ∈ \mathbfΔn} max_{y ∈ Sm} ∑_i=1^m y_iA_i \bullet X=max_{y ∈ Sm} min_{X ∈ \mathbfΔn} ∑_i=1^m y_iA_i \bullet X, where S_m is the simplex and \mathbfΔ_n the spectraplex. For diagonal A_i's this reduces to the classic minimax.

Related

• Continuous minimax theorems
• Counting Spectral Radii of Matrices with Positive Entries
• Combinatorial methods for the spectral p-norm of hypermatrices
• Spectrum maximizing products are not generically unique
• Minimal compact operators, subdifferential of the maximum eigenvalue and semi-definite programming