Slope and Geometry in Variational Mathematics

Abstract:

Various notions of the “slope” of a real-valued function pervade optimization and variational mathematics more broadly. In the semi-algebraic setting—an appealing model for concrete variational problems — the slope is particularly well-behaved. This talk sketches a variety of surprising applications, illustrating the unifying power of slope. High­lights include error bounds for level sets, the existence and regularity of steepest descent curves in complete metric spaces (following Ambrosio et al.), transversality and the con­ver­gence of von Neumann’s alternating projection algorithm, and the geometry underlying nonlinear programming active-set algorithms. This talk will be self-contained, requiring no familiarity with variational analysis, optimization theory, or semi-algebraic geometry. This is a joint work with A. Daniilidis, A.D. Ioffe, M. Larsson, and A.S. Lewis.

Posted on Nov 11, 2013 in Optimization and Equilibrium, Seminars