Author
Zhou, ZhihongIssue Date
2012Keywords
Optimization SimulationStagewise Independence
Stochastic Decomposition
Stochastic Dual Dynamic Programming
Systems & Industrial Engineering
Multistage Stochastic Decomposition
Multistage Stochastic Program
Advisor
Sen, SuvrajeetBayraksan, Guzin
Metadata
Show full item recordPublisher
The University of Arizona.Rights
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.Abstract
In this dissertation, we focus on developing samplingbased algorithms for solving stochastic linear programs. The work covers both two stage and multistage versions of stochastic linear programs. In particular, we first study the two stage stochastic decomposition (SD) algorithm and present some extensions associated with SD. Specifically, we study two issues: a) are there conditions under which the regularized version of SD generates a unique solution? and b) in cases where a user is willing to sacrifice optimality, is there a way to modify the SD algorithm so that a user can tradeoff solution times with solution quality? Moreover, we present our preliminary approach to address these questions. Secondly, we investigate the multistage stochastic linear programs and propose a new approach to solving multistage stochastic decision models in the presence of constraints. The motivation for proposing the multistage stochastic decomposition algorithm is to handle large scale multistage stochastic linear programs. In our setting, the deterministic equivalent problems of the multistage stochastic linear program are too large to be solved exactly. Therefore, we seek an asymptotically optimum solution by simulating the SD algorithmic process, which was originally designed for twostage stochastic linear programs (SLPs). More importantly, when SD is implemented in a timestaged manner, the algorithm begins to take the flavor of a simulation leading to what we refer to as optimization simulation. As for multistage stochastic decomposition, there are a couple of advantages that deserve mention. One of the benefits is that it can work directly with sample paths, and this feature makes the new algorithm much easier to be integrated within a simulation. Moreover, compared with other samplingbased algorithms for multistage stochastic programming, we also overcome certain limitations, such as a stagewise independence assumption.Type
textElectronic Dissertation
Degree Name
Ph.D.Degree Level
doctoralDegree Program
Graduate CollegeSystems & Industrial Engineering
Degree Grantor
University of ArizonaCollections
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Selfoptimizing stochastic systems: Applications to stochastic shortest path problem, stochastic traveling salesman problem, and queueing systems.Yakowitz, Sidney; Jayawardena, Thusitha Senadirage.; Sen, Suvrajeet (The University of Arizona., 1990)We investigate stochastic systems which have a set of control parameters and a performance criterion. By operating the system at fixed control parameters, noisy performance values are observed. (The values are noisy due to the inherent stochastic nature of the system.) Certain relevant distributions of the system are assumed unavailable. The task is to develop algorithms that guide the system to optimal parameter settings based on its operating history. Consider the stochastic shortest path problem, where the time to traverse an edge is given by a random variable whose distribution is unavailable explicitly. The optimality criterion (to be maximized) is the probability of going from a given source node to a given terminal node within a specified critical time period. By choosing a particular path and traversing it, realizations of the distributions, i.e. time to go from the source node to the terminal node on that path are observed. Or consider the M/M/1 queue. Here, the control parameter is the average service time. The performance criterion is the sum of cost of the server and cost of system time of a customer in steadystate. By choosing a particular average service time and serving accordingly, a noisy observation of the total cost is obtained. We combine an asymptotically optimal random search method for finding the global optimum of a function with problemspecific local search techniques. Such a combination results in efficient solution procedures for the above problems. This conclusion is reached by applying the procedure to problems for which the optimum solutions are known. The main contribution of the study is in demonstrating that "reasonable" performance can be achieved for the proposed optimization problems in "reasonable" time by exploiting problemspecific structures to advantage. The generality of the method should allow others to use it in different optimization settings than ours. Also, the selfoptimizing aspect of these methods and the stochastic versions of the local search techniques are new.

An exploration of stochastic decomposition algorithms for stochastic linear programs with recourse.Lowe, Wing Wah.; Higle, Julia L.; Duckstein, Lucien; Sen, Suvrajeet (The University of Arizona., 1994)Stochastic linear programs are linear programs in which some of the problem data are random variables. The particular kind of programs that we study belong to the recourse model. Under this model, some decisions are postponed until better information becomes available (e.g., an outcome of a random variable is realized), while other decisions must be made 'here and now.' For example, in a telecommunication network planning problem, decisions regarding the addition of network capacity have to be made before knowing customer demand (i.e., 'here and now'). Once the demand is realized, efficient usage of the network can then be determined. This work explores algorithms for the solution of such programs: stochastic linear programs with recourse. The algorithms investigated can be described as decomposition based cutting plane methods in which the cuts are estimated from random samples. Moreover, the algorithms all use the incremental sampling plan inherent to the Stochastic Decomposition (SD) algorithm developed by Higle and Sen in 1991. Our study includes both two stage and multistage programs. For the solution of two stage programs, we present the Conditional Stochastic Decomposition (CSD) algorithm, a multicut version of the SD algorithm. CSD is most suitable for situations in which data are difficult to obtain and may be computationally intense. Because of this potential intensity, we explore algorithms which require less computational effort than CSD. These algorithms combine features of both CSD and SD and are referred to as hybrid algorithms. Following our exploration of these algorithms for two stage problems, we next explore an extension of the SD algorithm that can be used for multistage problems with stagewise independent random variables. For the sake of notational brevity, our technical development is centered around the three stage case, although the extension to multistage problems is straightforward. Under mild conditions, convergence results similar to those found in the two stage algorithms hold. Multistage stochastic decomposition is currently a largely uncharted area. Our research represents the first major effort in this direction.

Contributions to the theory of stochastic convexity and stochastic majorization.Li, Haijun.; Shaked, Moshe; Myers, Donald; Wright, Larry (The University of Arizona., 1994)This dissertation presents some contributions to the theory of stochastic convexity and stochastic majorization. In the first part of this dissertation, we develop an operatoranalytic approach to study temporal stochastic convexity and concavity of Markov processes. We obtain sufficient and necessary conditions for the process {X(t),t ∊ S} which imply that the expectation Ef(X(t)) is a monotone convex (concave) function of t whenever f is a monotone convex (concave) function. Our operatoranalytic approach is quite powerful, but not as intuitive as sample path approaches used in other works. However, using it, we can obtain results that we could not obtain otherwise. In particular, we show that a result of Shaked and Shanthikumar is incorrect and we prove two alternative versions of it. In the second part of this dissertation, we discuss some applications of stochastic convexity. Using the Fmonotonicity and Fconvexity developed in the first part of this dissertation, we obtain relations among the several notions of stochastic convexity and stochastic majorization, and characterize the relationship between stochastic submodularity and stochastic rearrangement. These results generalize and extend several known results in the literature. Applications of our results to stochastic allocation problems are also discussed.