Convex programming involves a convex set F and a convex function c:F->R. The goal of convex programming is to find a point in F which minimizes c. In this paper, we introduce online convex programming. In online convex programming, the convex set is known in advance, but in each step of some repeated optimization problem, one must select a point in F before seeing the cost function for that step. This can be used to model factory production, farm production, and many other industrial optimization problems where one is unaware of the value of the items produced until they have already been constructed. We introduce an algorithm for this domain, apply it to repeated games, and show that it is really a generalization of infinitesimal gradient ascent, and the results here imply that generalized infinitesimal gradient ascent (GIGA) is universally consistent.

28 pages

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