In this work we design graph neural network architectures that capture optimal approximation algorithms for a large class of combinatorial optimization problems using powerful algorithmic tools from semidefinite programming (SDP). Concretely, we prove that polynomial-sized message passing algorithms can represent the most powerful polynomial time algorithms for Max Constraint Satisfaction Problems assuming the Unique Games Conjecture. We leverage this result to construct efficient graph neural network architectures, OptGNN, that obtain high-quality approximate solutions on landmark combinatorial optimization problems such as Max Cut and Minimum Vertex Cover. Finally, we take advantage of OptGNN's ability to capture convex relaxations to design an algorithm for producing dual certificates of optimality (bounds on the optimal solution) from the learned embeddings of OptGNN.