Duality for Borel measurable cost functions

Average-Cost Optimality Results for Borel-Space Markov Decision Processes with Universally Measurable Policies

We consider discrete-time Markov Decision Processes with Borel state and action spaces and universally measurable policies. For several long-run average cost criteria, we establish the following optimality results: the optimal average cost functions are lower semianalytic, there exist universally measurable semi-Markov or history-dependent $ε$-optimal policies, and similar results hold for the minimum average costs achievable by Markov or stationary policies. We then analyze the structure of the optimal average cost functions, proving sufficient conditions for them to be constant almost everyw…

Average Cost Optimality Inequality for Markov Decision Processes with Borel Spaces and Universally Measurable Policies

We consider average-cost Markov decision processes (MDPs) with Borel state and action spaces and universally measurable policies. For the nonnegative cost model and an unbounded cost model with a Lyapunov-type stability character, we introduce a set of new conditions under which we prove the average cost optimality inequality (ACOI) via the vanishing discount factor approach. Unlike most existing results on the ACOI, our result does not require any compactness and continuity conditions on the MDPs. Instead, the main idea is to use the almost-uniform-convergence property of a pointwise converge…

A Striktpositivstellensatz for measurable functions (corrected version)

A weighted sums of squares decomposition of positive Borel measurable functions on a bounded Borel subset of the Euclidean space is obtained via duality from the spectral theorem for tuples of commuting self-adjoint operators. The analogous result for polynomials or certain rational functions was amply exploited during the last decade in a variety of applications.

支配超平面的Borel选择

Borel selection of dominating hyperplanes

研究在标准均匀化定理不直接适用的情况下，是否存在Borel可测方式选择支配仿射泛函，并证明在半解析正则性假设下确实存在。

Duality in convex problems of Bolza over functions of bounded variation

This paper studies convex problems of Bolza in the conjugate duality framework of Rockafellar. We parameterize the problem by a general Borel measure which has direct economic interpretation in problems of financial economics. We derive a dual representation for the optimal value function in terms of continuous dual arcs and we give conditions for the existence of solutions. Combined with well-known results on problems of Bolza over absolutely continuous arcs, we obtain optimality conditions in terms of extended Hamiltonian conditions.

A General Duality Theorem for the Monge--Kantorovich Transport Problem

The duality theory of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $μ$ and $ν$. The transport cost function $c:X\times Y \to [0,\infty]$ is assumed to be Borel. Our main result states that in this setting there is no duality gap, provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses $1-\varepsilon$ from $(X,μ)$ to $(Y, ν)$, as $\varepsilon >0$ tends to zero…