We develop a new class of approximate linear programs (ALPs) that project the high-dimensional value function of dynamic programs onto a class of basis functions, each defined as a network that represents aggregrations over the state space. The resulting ALP is a minimum-cost flow problem over an extended variable space that synchronizes flows across multiple … Read more
In the analysis of complex stochastic dynamic programs, we often seek strong theoretical guarantees on the suboptimality of heuristic policies. One technique for obtaining performance bounds is perfect information analysis: this approach provides bounds on the performance of an optimal policy by considering a decision maker who has access to the outcomes of all future … Read more
We describe a nonlinear generalization of dual dynamic programming theory and its application to value function estimation for deterministic control problems over continuous state and action (or input) spaces, in a discrete-time infinite horizon setting. We prove that the result of a one-stage policy evaluation can be used to produce nonlinear lower bounds on the … Read more
We consider the Resource Constrained Shortest Path problem arising as a subproblem in state-of-the-art Branch-Cut-and-Price algorithms for vehicle routing problems. We propose a variant of the bi-directional label correcting algorithm in which the labels are stored and extended according to so-called bucket graph. Such organization of labels helps to decrease significantly the number of dominance … Read more
Consider the problem of approximating the optimal policy of a Markov decision process (MDP) by sampling state transitions. In contrast to existing reinforcement learning methods that are based on successive approximations to the nonlinear Bellman equation, we propose a Primal-Dual π Learning method in light of the linear duality between the value and policy. The … Read more
Multi-stage stochastic optimization is a well-known quantitative tool for decision-making under uncertainty, which applications include financial and investment planning, inventory control, energy production and trading, electricity generation planning, supply chain management and similar fields. Theoretical solution of multi-stage stochastic programs can be found explicitly only in very exceptional cases due to the complexity of the … Read more
We consider dynamic selection problems, where a decision maker repeatedly selects a set of items from a larger collection of available items. A classic example is the dynamic assortment problem with demand learning, where a retailer chooses items to offer for sale subject to a display space constraint. The retailer may adjust the assortment over … Read more
Online bipartite matching (OBM) is a fundamental model underpinning many important applications, including search engine advertisement, website banner and pop-up ads, and ride-hailing. We study the i.i.d. OBM problem, where one side of the bipartition is fixed and known in advance, while nodes from the other side appear sequentially as i.i.d. realizations of an underlying … Read more
We study the problem of a retailer that maximizes profit through joint replenishment, pricing and removal decisions. This problem is motivated by the observation that retailers usually retain rights to remove inventory from their network either by returning it to the suppliers or through liquidation in the face of random demand and capacity constraints. We … Read more
We propose a Fully Polynomial-Time Approximation Scheme (FPTAS) for stochastic dynamic programs with multidimensional action, scalar state, convex costs and linear state transition function. The action spaces are polyhedral and described by parametric linear programs. This type of problems finds applications in the area of optimal planning under uncertainty, and can be thought of as … Read more