One of the most exciting areas of theoretical computer science is to understand the approximability of fundamental optimization problems. A prominent example is the traveling salesman problem, for which a long-standing conjecture states that a certain algorithm gives a better guarantee than current methods can prove. The resolution of this conjecture and of many other fundamental problems is intimately related to an increased understanding of strong convex relaxations.Although these problems have resisted numerous attempts, recent breakthrough results, in which the PI has played a central role, indicate new research directions with the potential to resolve some of our most exciting open questions. We propose three research directions to revolutionize our understanding of these problems and more generally of the use of convex relaxations in approximation algorithms:(I) develop new approaches to analyze and harness the power of existing convex relaxations;(II) understand the power of automatically generated relaxations; and(III) prove the optimality of algorithms based on convex relaxations.The proposed research lies in the frontier of approximation algorithms and optimization with connections to major problems in complexity theory, such as the unique games conjecture. Any progress will be a significant contribution to theoretical computer science and mathematical optimization.