A minimax theorem with applications to machine learning, signal processing, and finance

This paper concerns a fractional function of the form x^Talpha/radicx^TBx, where B is positive definite. We consider the game of choosing x from a convex set, to maximize the function, and choosing (alpha, B) from a convex set, to minimize it. We prove the existence of a saddle point and describe an efficient method, based on convex optimization, for computing it. We describe applications in machine learning (robust Fisher linear discriminant analysis), signal processing (robust beam- forming, robust matched filtering), and finance (robust portfolio selection). In these applications, x corresponds to some design variables to be chosen, and the pair (alpha, B) corresponds to the statistical model, which is uncertain.