Decomposition/Coordination Algorithms in Stochastic Optimization

Culioli, J.C.; Cohen, Guy

doi:10.1137/0328072

Cited by 40 publications

(25 citation statements)

References 14 publications

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“…Clearly, this question is highly problem dependent. It would be interesting to compare the empirical performance of SHAPE to the auxiliary function method of Culioli and Cohen (1990), which o ers the same structural properties as SHAPE. At the same time, comparisons should be undertaken against other popular solution methods such as stochastic linearization (Gupal andBazhenov 1972, Ermoliev 1983), stochastic decomposition (Higle and Sen 1991), and sample path optimization (Robinson 1996).…”

Section: Discussionmentioning

confidence: 99%

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Shape – A Stochastic Hybrid Approximation Procedure for Two-Stage Stochastic Programs

Cheung

Powell

2000

Operations Research

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We consider the problem of approximating the expected recourse function for two-stage stochastic programs. Our problem is motivated by applications that have special structure, such as an underlying network that allows reasonable approximations to the expected recourse function to be developed. In this paper, we show how these approximations can be improved by combining them with sample gradient information from the true recourse function. For the case of strictly convex nonlinear approximations, we prove convergence for this hybrid approximation. The method is attractive for practical reasons because it retains the structure of the approximation.A common problem in operations research is the challenge of making a decision now in a way that minimizes the expectation of costs in the future that depend on random events. For example, we may have the situation of determining how much product to ship from plants to warehouses from which we then satisfy demands at di erent retailers. We must decide how much product to ship to each warehouse before we know the retail demand. Once retail demands are known, we are able to optimize shipping patterns between warehouses and retailers.This problem, and many like it, can be posed as two-stage stochastic programs. The decision made now (in stage 1) determines what state we are in when we have to solve the problem in stage 2. If we could exactly capture the structure of the expected cost function (or recourse function) for stage 2, we would be able to make optimal decisions now. The di culty is that in most cases, the structure of this expected cost function is too complex.There is an extensive literature on two-stage stochastic programming problems, which is nicely summarized in several recent books (Infanger 1994 andKall andWallace 1994). General solution methods include scenario optimization (e.g., Rockafellar and Wets 1991), stochastic gradient techniques (e.g., Ermoliev 1983, Ruszczynski 1980, Benders decomposition and its variants (e.g., Van Slyke and Wets 1969, Birge 1985, and Higle and Sen 1991, sample path optimization (Robinson 1996), and other approximation techniques (e.g., Beale et al. 1980). These techniques are, for the most part, very general and are not designed speciÿcally to take advantage of approximations that may produce good but not optimal solutions.In this paper, we propose a new algorithm called SHAPE (successive hybrid approximation procedure) that combines an initial nonlinear approximation with iteratively sampled stochastic gradient information. The initial nonlinear approximation can exploit problem structure, while the stochastic gradient information, which is easy to obtain for most problems, tunes the approximation. We describe the algorithm, provide a small numerical illustration, and prove convergence.Section 1 provides a formal problem statement and introduces the basic idea behind the algorithm. Section 2 presents a method we call the stochastic hybrid approximation procedure (SHAPE) for solving two-stage stochastic programs with recourse. ...

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Section: Discussionmentioning

confidence: 99%

“…The algorithm SHAPE is similar in spirit to the auxiliary function method of Culioli and Cohen (1990). Their method solves sequences of problems with the form…”

Section: Cheung and Powell / 75mentioning

confidence: 99%

Shape – A Stochastic Hybrid Approximation Procedure for Two-Stage Stochastic Programs

Cheung

Powell

2000

Operations Research

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“…The Auxiliary Problem Principle (APP) (c.f., Cohen (1978), Cohen and Zhu (1984), Culioli and Cohen (1990), Zhu and Marcotte, (1995)) offers an elegant solution to the above problem. In the APP framework, an auxiliary function is introduced to the objective function while the coupling term is linearized.…”

Section: Lagrangian In Mathematical Decomposition Algorithms a Main mentioning

confidence: 99%

Decentralizing semiconductor capacity planning via internal market coordination

Karabük

2002

IIE Transactions

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Semiconductor capacity planning is a cross-functional decision that requires coordination between the marketing and manufacturing divisions. We examine main issues of a decentralized coordination scheme in a setting observed at a major US semiconductor manufacturer: marketing managers reserve capacity from manufacturing based on product demands, while attempting to maximize profit; manufacturing managers allocate capacity to competing marketing managers so as to minimize operating costs while ensuring efficient resource utilization. This cross-functional planning problem has two important characteristics: (1) both demands and capacity are subject to uncertainty, and (2) all decision entities own private information while being self-interested. To study the issues of coordination we first formulate the local marketing and the manufacturing decision problem as separate stochastic programs. We then formulate a centralized stochastic programming model (JCA), which maximizes the firm's overall profit. JCA establishes a theoretical benchmark for performance, but is only achievable when all planning information is public. If local decision entities are to keep their planning information private, we submit that the best achievable coordination corresponds to an alternative stochastic model (DCA). We analyze the relationship and the theoretical gap between JCA and DCA, thereby establishing the price of decentralization. Next, we examine two mechanisms that coordinate the marketing and manufacturing decisions to achieve DCA using different degrees of information exchange. Using insights from the Auxiliary Problem Principle (APP), we show that under both coordination mechanisms the divisional proposals converge to the global optimal solution of DCA. We illustrate the theoretic insights using numerical examples as well as a real-world case.2

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“…The Stochastic Dual Dynamic Programming method [2] is widely used for compagnies having large stocks of water to manage. When the company portfolio is composed of many stocks of water and many power plants a decomposition method can be used [3] and the bundle method may be used for coordination [4].…”

Section: Introduction and Objectivesmentioning

confidence: 99%