From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming
For researchers in optimization and control, this provides a principled approximation framework with quantified performance guarantees, though it is an incremental extension of existing randomized optimization and first-order methods.
The paper develops a method to approximate infinite-dimensional linear programs with tractable finite convex programs, providing explicit error bounds. It demonstrates applicability to optimal control problems in Markov decision processes and two case studies.
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.