Non-Convex Optimization via Real Algebraic Geometry

The high level purpose of this paper is to describe some recent advances in the field of Mathematics called Real Algebraic Geometry, and discuss some of its applications to complexity theory, and non-convex optimization. In particular, one of the questions underlying the entire development, is the crucial question: What makes an optimization problem difficult or easy? Along the way, we try to point out what we see as the promising research directions, and what we deem the difficulty and potential promise of these various directions to be.