Mathematics of Operations Research current issue

An Asymptotically Optimal Policy for a Quantity-Based Network Revenue Management Problem

Reiman, M. I., Wang, Q. — 2008-06-18

We consider a canonical revenue management problem in a network setting where the goal is to find a customer admission policy to maximize the total expected revenue over a fixed finite horizon. There is a set of resources, each of which has a fixed capacity. There are several customer classes, each with an associated arrival process, price, and resource consumption vector. If a customer is accepted, it effectively removes the resources that it consumes from the system. The exact solution cannot be obtained for reasonable-sized problems due to the curse of dimensionality. Several (approximate) solution techniques have been proposed in the literature. One way to analytically compare policies is via an asymptotic analysis where both resource sizes and arrival rates grow large. Many of the proposed policies are asymptotically optimal on the fluid scale. However, as we demonstrate in this paper, these policies may fail to be optimal on the more sensitive diffusion scale even for quite simple problem instances. We develop a new policy that achieves diffusion-scale optimality. The policy starts with a probabilistic admission rule derived from the optimization of the fluid model, embeds a trigger function that tracks the difference between the actual and expected customer acceptance, and sets threshold values for the trigger function, the violation of which invokes the reoptimization of the admission rule. We show that re-solving the fluid model, which needs to be performed at most once, is required for extending the asymptotic optimality from the fluid scale to the diffusion scale. We demonstrate the implementation of the policy by numerical examples.

Total Dual Integrality of Rothblum's Description of the Stable-Marriage Polyhedron

Kiraly, T., Pap, J. — 2008-06-18

Rothblum showed that the convex hull of the stable matchings of a bipartite preference system can be described by an elegant system of linear inequalities. In this paper we prove that the description given by Rothblum is totally dual integral. We give a constructive proof based on the results of Gusfield and Irving on rotations, which gives rise to a strongly polynomial algorithm for finding an integer optimal dual solution.

Local Indices for Degenerate Variational Inequalities

Simsek, A., Ozdaglar, A., Acemoglu, D. — 2008-06-18

We provide an index formula for solutions of variational inequality problems defined by a continuously differentiable function F over a convex set M represented by a finite number of inequality constraints. Our index formula can be applied when the solutions are nonsingular and possibly degenerate, as long as they also satisfy the injective normal map (INM) property, which is implied by strong stability. We show that when the INM property holds, the degeneracy in a solution can be removed by perturbing the function F slightly, i.e., the index of a degenerate solution is equal to the index of a nondegenerate solution of a slightly perturbed variational inequality problem. We further show that our definition of the index is equivalent to the topological index of the normal map at the zero corresponding to the solution. As an application of our index formula, we provide a global index theorem for variational inequalities which holds even when the solutions are degenerate.

Lattice Theory and the Consumer's Problem

Mirman, L. J., Ruble, R. — 2008-06-18

This paper explores and explains the application of the lattice theoretic approach to classic comparative statics in consumer theory. Through examples of preferences that are not quasiconcave, or not differentiable, or not continuous, the approach is shown to characterize income effects more powerfully than the standard approach. The underlying partial order is key to applying the method. Therefore, several adapted partial orders are introduced and discussed.

A Geometrical Characterization of Multidimensional Hausdorff Polytopes with Applications to Exit Time Problems

Helmes, K., Rohl, S. — 2008-06-18

We present a formula for the corner points of the multidimensional Hausdorff polytopes and show how this result can be used to improve linear programming models for computing, e.g., moments of exit time distributions of diffusion processes. Specifically, we compute the mean exit time of two-dimensional Brownian motion from the unit square, as well as higher moments of the exit time of time-space Brownian motion, i.e., the two-dimensional process composed of a one-dimensional Wiener process and the time component, from a rectangle. The corner point formula is complemented by a convergence result, which provides the analytical underpinning of the numerical method that we use.

Truncation Strategies in Matching Markets

Ehlers, L. — 2008-06-18

Roth and Rothblum [Roth, A. E., U. G. Rothblum. 1999. Truncation strategies in matching markets—In search of advice for participants. Econometrica 67 21–43] showed that for matching markets using the deferred acceptance algorithm a physician with symmetric (incomplete) information possibly gains only by truncating her true ranking. We show that in symmetric information environments this result is identical for all priority mechanisms and all linear programming mechanisms introduced in British entry-level medical markets and in public school choice in some American cities.

An Analysis of Monotone Follower Problems for Diffusion Processes

Bayraktar, E., Egami, M. — 2008-06-18

We consider a singular stochastic control problem, which is called the monotone follower stochastic control problem, and give sufficient conditions for the existence and uniqueness of a local-time type optimal control. To establish this result, we use a methodology that has not been employed to solve singular control problems. We first confine ourselves to local-time strategies. We then apply a transformation to the total reward accrued by reflecting the diffusion at a given boundary and show that it is linear in its continuation region. Now, the problem of finding the optimal boundary becomes a nonlinear optimization problem: The slope of the linear function and an obstacle function need to be simultaneously maximized. The necessary conditions of optimality come from first-order derivative conditions. We show that under some weak assumptions these conditions become sufficient. We also show that the local-time strategies are optimal in the class of all monotone increasing controls.

As a by-product of our analysis, we give sufficient conditions for the value function to be C² on all of its domain. We solve two dividend payment problems to show that our sufficient conditions are satisfied by the examples considered in the mainstream literature. We show that our assumptions are satisfied not only when capital of a company is modeled by a Brownian motion with drift, but also when we change the modeling assumptions and use a square root process to model the capital.

A 2-Approximation Algorithm for Stochastic Inventory Control Models with Lost Sales

Levi, R., Janakiraman, G., Nagarajan, M. — 2008-06-18

In this paper, we describe the first computationally efficient policies for stochastic inventory models with lost sales and replenishment lead times that admit worst-case performance guarantees.

In particular, we introduce dual-balancing policies for lost-sales models that are conceptually similar to dual-balancing policies recently introduced for a broad class of inventory models in which demand is backlogged rather than lost. That is, in each period, we balance two opposing costs: the expected marginal holding costs against the expected marginal lost-sales cost. Specifically, we show that the dual-balancing policies for the lost-sales models provide a worst-case performance guarantee of two under relatively general demand structures. In particular, the guarantee holds for independent (not necessarily identically distributed) demands and for models with correlated demands such as the AR(1) model and the multiplicative autoregressive demand model. The policies and the worst-case guarantee extend to models with capacity constraints on the size of the order and stochastic lead times. Our analysis has several novel elements beyond the balancing ideas for backorder models.

Fluid Limits for Processor-Sharing Queues with Impatience

Gromoll, H. C., Robert, P., Zwart, B. — 2008-06-18

We investigate a processor-sharing queue with renewal arrivals and generally distributed service times. Impatient jobs may abandon the queue or renege before completing service. The random time representing a job's patience has a general distribution and may be dependent on its initial service time requirement. A scaling procedure that gives rise to a fluid model with nontrivial yet tractable steady state behavior is presented. This fluid model captures many essential features of the underlying stochastic model, and it is used to analyze the impact of impatience in processor-sharing queues.

Stochastic Games on a Product State Space

Flesch, J., Schoenmakers, G., Vrieze, K. — 2008-06-18

We examine so-called product-games with an aperiodic transition structure, with respect to the average reward, for which we present an approach based on communicating states. For the general n-player case, we establish the existence of 0-equilibria. In addition, for the special case of two-player zero-sum games of this type, we show that both players have stationary 0-optimal strategies.

Lowner's Operator and Spectral Functions in Euclidean Jordan Algebras

Sun, D., Sun, J. — 2008-06-18

We study analyticity, differentiability, and semismoothness of Löwner's operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any symmetric cone defined in a Euclidean Jordan algebra is shown to be strongly semismooth. The research also raises several open questions, whose answers would be of strong interest for optimization research.

Optimal Multiple Stopping of Linear Diffusions

Carmona, R., Dayanik, S. — 2008-06-18

Motivated by the analysis of financial instruments with multiple exercise rights of American type and mean reverting underlyers, we formulate and solve the optimal multiple-stopping problem for a general linear regular diffusion process and a general reward function. Instead of relying on specific properties of geometric Brownian motion and call and put option payoffs as in most of the existing literature, we use general theory of optimal stopping for diffusions, and we illustrate the resulting optimal exercise policies by concrete examples and constructive recipes.

Approximation Algorithms for the Capacitated Multi-Item Lot-Sizing Problem via Flow-Cover Inequalities

Levi, R., Lodi, A., Sviridenko, M. — 2008-06-18

We study the classical capacitated multi-item lot-sizing problem with hard capacities. There are N items, each of which has a specified sequence of demands over a finite planning horizon of T discrete periods; the demands are known in advance but can vary from period to period. All demands must be satisfied on time. Each order incurs a time-dependent fixed ordering cost regardless of the combination of items or the number of units ordered, but the total number of units ordered cannot exceed a given capacity C. On the other hand, carrying inventory from period to period incurs holding costs. The goal is to find a feasible solution with minimum overall ordering and holding costs.

We show that the problem is strongly NP-hard, and then propose a novel facility location type LP relaxation that is based on an exponentially large subset of the well-known flow-cover inequalities; the proposed LP can be solved to optimality in polynomial time via an efficient separation procedure for this subset of inequalities. Moreover, the optimal solution of the LP can be rounded to a feasible integer solution with cost that is at most twice the optimal cost; this provides a 2-approximation algorithm which is the first constant approximation algorithm for the problem. We also describe an interesting on-the-fly variant of the algorithm that does not require solving the LP a priori with all the flow-cover inequalities. As a by-product we obtain the first theoretical proof regarding the strength of flow-cover inequalities in capacitated inventory models.

Bayesian Sequential Change Diagnosis

Dayanik, S., Goulding, C., Poor, H. V. — 2008-06-18

Sequential change diagnosis is the joint problem of detection and identification of a sudden and unobservable change in the distribution of a random sequence. In this problem, the common probability law of a sequence of i.i.d. random variables suddenly changes at some disorder time to one of finitely many alternatives. This disorder time marks the start of a new regime, whose fingerprint is the new law of observations. Both the disorder time and the identity of the new regime are unknown and unobservable. The objective is to detect the regime-change as soon as possible, and, at the same time, to determine its identity as accurately as possible. Prompt and correct diagnosis is crucial for quick execution of the most appropriate measures in response to the new regime, as in fault detection and isolation in industrial processes, and target detection and identification in national defense. The problem is formulated in a Bayesian framework. An optimal sequential decision strategy is found, and an accurate numerical scheme is described for its implementation. Geometrical properties of the optimal strategy are illustrated via numerical examples. The traditional problems of Bayesian change-detection and Bayesian sequential multi-hypothesis testing are solved as special cases. In addition, a solution is obtained for the problem of detection and identification of component failure(s) in a system with suspended animation.

A Characterization of Box-Mengerian Matroid Ports

Chen, X., Ding, G., Zang, W. — 2008-06-18

Let M be a matroid on E{l}, where l E is a distinguished element of M. The l-port of M is the set P= {P: P E with P{l} a circuit of M}. Let A be the P-E incidence matrix. Let U_{2, 4} be the uniform matroid on four elements of rank two, let F₇ be the Fano matroid, let F₇* be the dual of F₇, and let F₇⁺ be the unique series extension of F₇. In this paper, we prove that the system Ax≥1, x≥0 is box-totally dual integral (box-TDI) if and only if M has no U_{2, 4}-minor using l, no F₇*-minor using l, and no F₇⁺-minor using l as a series element. Our characterization yields a number of interesting results in combinatorial optimization.