From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming

We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite convex programs in which the performance of the approximation is quantified explicitly. To this end, we adopt the recent developments in two areas of randomized optimization and first order methods, leading to a priori as well as a posterior performance guarantees. We illustrate the generality and implications of our theoretical results in the special case of the long-run average cost and discounted cost optimal control problems for Markov decision processes on Borel spaces. The applicability of the theoretical results is demonstrated through a constrained linear quadratic optimal control problem and a fisheries management problem.Comment: 30 pages, 5 figure

Lp-norms, Log-barriers and Cramer transform in Optimization

We show that the Laplace approximation of a supremum by Lp-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem P and its dual appear naturally when using this simple approximation technique for the value function g of P or its Legendre-Fenchel conjugate. In addition, minimizing the LBF of the dual is just evaluating the Cramer transform of the Laplace approximation of g. Finally, this technique permits to sometimes define an explicit dual problem in cases when the Legendre-Fenchel conjugate of g cannot be derived explicitly from its definition