Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.Comment: 19 pages, 4 figures. This version of the paper will appear in the Journal of the ACM. The earlier conference version in STOC'12 had the title "Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds

Distributionally Robust Optimization: A Review

The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization