Papers for
optimization algorithm designers
Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.
Finding specific Nash equilibria in games is computationally hard
Finding a Positive Index Nash Equilibrium is PPADS-Complete
Abstract: Every nondegenerate bimatrix game has a Nash equilibrium of Shapley index +1, since all equilibria are isolated, have index +1 or -1, and their indices sum to +1. We prove that the following promise search problem is PPADS-complete: given a rational bimatrix game promised to be nondegenerate, find an exact Nash equilibrium of index +1. To our knowledge, this is the first PPADS-complete equilibrium search problem whose instances are explicit rational normal form payoff matrices, rather than succinct circuits or Turing machines, and thereby addresses an open question posed by Daskalakis [Daskalakis, 2019].
Elementary proof shows balanced sign choices keep vectors small
An elementary proof of the Komlós conjecture
Abstract: We give an elementary proof of the Komlós conjecture by simplifying the recent proof of Guo, Fang, and Lu. We show that any vectors $v_1,\ldots,v_n\in\mathbb{R}^d$ with $\|v_i\|_2\le1$ admit signs $\varepsilon_i\in\{-1,1\}$ such that $\|\sum_{i=1}^n\varepsilon_i v_i\|_\infty\le36$. The proof uses only elementary combinatorial and probabilistic arguments and basic calculus.