Beyond convexity: local search and equilibrium computation

It is well known that an equilibrium point of a zero-sum two-player game can be computed in polynomial time using linear programming while the computation of a Nash equilibrium in a general two-player game is PPAD complete. In parallel, an Arrow-Debreu equilibrium price of an exchange market of traders with linear utilities is polynomial-time computable using convex programming, while the computation of an equilibrium price of an exchange market with linearly separable piece-wise linear utilities is PPAD complete. This convexity based dichotomy is fascinating in my view. In this talk I would like to discuss some of our recent work about the mathematical and complexity structure of equilibria, both for fixed-point-based Nash and market equilibria and for potential-function-based network equilibria that can be found by any local search procedure. I would also like to touch on the questions such as "Is local search fundamentally easier than fixed point computation?"