This paper addresses time consistency of risk-averse optimal stopping in stochastic optimization. It is demonstrated that time-consistent optimal stopping entails a specific structure of the functionals describing the...
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This paper addresses time consistency of risk-averse optimal stopping in stochastic optimization. It is demonstrated that time-consistent optimal stopping entails a specific structure of the functionals describing the transition between consecutive stages. The stopping risk measures capture this structural behavior and allow natural dynamic equations for risk-averse decision making over time. Consequently, associated optimal policies satisfy Bellman's principle of optimality, which characterizes optimal policies for optimization by stating that a decision maker should not reconsider previous decisions retrospectively. We also discuss numerical approaches to solving such problems.
Quadratic stochastic programming (QSP) in which each subproblem is a convex piecewise quadratic program with stochastic data, is a natural extension of stochastic linear programming. This allows the use of quadratic o...
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Quadratic stochastic programming (QSP) in which each subproblem is a convex piecewise quadratic program with stochastic data, is a natural extension of stochastic linear programming. This allows the use of quadratic or piecewise quadratic objective functions which are essential for controlling risk in financial and project planning. Two-stage QSP is a special case of extended linear-quadratic programming (ELQP). The recourse functions in QSP are piecewise quadratic convex and Lipschitz continuous. Moreover, they have Lipschitz gradients if each QP subproblem is strictly convex and differentiable. Using these properties, a generalized Newton algorithm exhibiting global and superlinear convergence has been proposed recently for the two stage case. We extend the generalized Newton algorithm to multistage QSP and show that it is globally and finitely convergent under suitable conditions. We present numerical results on randomly generated data and modified publicly available stochastic linear programming test sets. Efficiency schemes on different scenario tree structures are discussed. The large-scale deterministic equivalent of the multistage QSP is also generated and their accuracy compared. (C) 2001 Elsevier Science B.V. All rights reserved.
作者：
Chen, XinZhang, JiaweiUniv Illinois
Dept Ind & Enterprise Syst Engn Urbana IL 61801 USA NYU
Stern Sch Business IOMS Operat Management New York NY 10012 USA
In this paper, we present a unified approach to study a class of cooperative games arising from inventory centralization. The optimization problems corresponding to the inventory games are formulated as stochastic pro...
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In this paper, we present a unified approach to study a class of cooperative games arising from inventory centralization. The optimization problems corresponding to the inventory games are formulated as stochastic programs. We observe that the strong duality of stochastic linear programming not only directly leads to a series of recent results concerning the nonemptiness of the core of such games, but also suggests a way to find an element in the core. The proposed approach is also applied to inventory games with concave ordering cost. In particular, we show that the newsvendor game with concave ordering cost has a nonempty core. Finally, we prove that it is NP-hard to determine whether a given allocation is in the core of the inventory games even in a very simple setting.
The maintenance, repair and operation (MRO) spare parts that are vital to machine operations are playing an increasingly important role in manufacturing enterprises. MRO spare parts supply chain management planning mu...
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The maintenance, repair and operation (MRO) spare parts that are vital to machine operations are playing an increasingly important role in manufacturing enterprises. MRO spare parts supply chain management planning must be coordinated to ensure spare part availability while keeping the total cost to a minimum. Due to the specificity of MRO spare parts, randomness and uncertainties in production and storage should be quantified to formulate the problem in a mathematical model. Given these considerations, this paper proposes an improved stochastic programming model for the supply chain planning of MRO spare parts. In our stochastic programming model, the following improvements are made: First, we quantify the uncertain production time capacity as a random variable with a probability distribution. Second, the upper bound of the storage cost is modeled as a multi-choice variable in the constraint. To derive the equivalent deterministic model, the Lagrange interpolating polynomial approach is used. The results of the numerical examples validate the feasibility and efficiency of the proposed model. Finally, the model is tested in the supply chain planning of continuous caster (CC) bearings. (C) 2017 Published by Elsevier Inc.
The main part of the paper (Section 3) provides an overview of case studies based on stochastic programming models in which reality of incomplete information about distribution of random parameters is clearly accepted...
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The main part of the paper (Section 3) provides an overview of case studies based on stochastic programming models in which reality of incomplete information about distribution of random parameters is clearly accepted and treated. Also, several techniques of scenario generation are surveyed in this context. Basic ideas of various approaches to stochastic programming models under incomplete information are briefly explained on illustrative examples in Section 2.
This special issue addresses the advances in stochastic programming and robust optimization for supply chain planning by examining novel methods, practices, and opportunities. The articles present and analyze opportun...
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This special issue addresses the advances in stochastic programming and robust optimization for supply chain planning by examining novel methods, practices, and opportunities. The articles present and analyze opportunities to improve supply chain planning through exploring various uncertainty situations and problems, sustainability assessment, vendor selection, risk mitigation, retail supply chain planning, and supply chain coordination. This editorial note summarizes the discussions on the stochastic models, algorithms, and methodologies developed for the evaluation and effective implementation of supply chain planning under various concerns. A dominant finding is that supply chain planning through the advancement of stochastic programming and robust optimization should be explored in a variety of ways and within different fields of applications. (C) 2018 Elsevier Ltd. All rights reserved.
The nonlinear stochastic programming problem involving CVaR in the objective and constraints is considered. Solving the latter problem in a framework of bi-level stochastic programming, the extended Lagrangian is intr...
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The nonlinear stochastic programming problem involving CVaR in the objective and constraints is considered. Solving the latter problem in a framework of bi-level stochastic programming, the extended Lagrangian is introduced and the related KKT conditions are derived. Next, the sequential simulation-based approach has been developed to solve stochastic problems with CVaR by finite sequences of Monte Carlo samples. The approach considered is grounded by the rule for iterative regulation of the Monte Carlo sample size and the stochastic termination procedure, taking into account the stochastic model risk. The rule is introduced to regulate the size of the Monte Carlo sample inversely proportionally to the square of the stochastic gradient norm allows us to solve stochastic nonlinear problems in a rational way and ensures the convergence. The proposed termination procedure enables us to test the KKT conditions in a statistical way and to evaluate the confidence intervals of the objective and constraint functions in a statistical way as well. The results of the Monte Carlo simulation with test functions and solution of the practice sample of trade-offs of gas purchases, storage and service reliability, illustrate the convergence of the approach considered as well as the ability to solve in a rational way the nonlinear stochastic programming problems handling CVaR in the objective and constraints, with an admissible accuracy, treated in a statistical manner.
We discuss the incorporation of risk measures into multistage stochastic programs. While much attention has been recently devoted in the literature to this type of model, it appears that there is no consensus on the b...
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We discuss the incorporation of risk measures into multistage stochastic programs. While much attention has been recently devoted in the literature to this type of model, it appears that there is no consensus on the best way to accomplish that goal. In this paper, we discuss pros and cons of some of the existing approaches. A key notion that must be considered in the analysis is that of consistency, which roughly speaking means that decisions made today should agree with the planning made yesterday for the scenario that actually occurred. Several definitions of consistency have been proposed in the literature, with various levels of rigor;we provide our own definition and give conditions for a multi-period risk measure to be consistent. A popular way to ensure consistency is to nest the one-step risk measures calculated in each stage, but such an approach has drawbacks from the algorithmic viewpoint. We discuss a class of risk measures which we call expected conditional risk measures that address those shortcomings. We illustrate the ideas set forth in the paper with numerical results for a pension fund problem in which a company acts as the sponsor of the fund and the participants' plan is defined-benefit. (C) 2015 Elsevier B.V. and Association of European Operational Research Societies (EURO) within the International Federation of Operational Research Societies (IFORS). All rights reserved.
Integrals of optimal values of random optimization problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure. Under fairly broad conditions,...
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Integrals of optimal values of random optimization problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure. Under fairly broad conditions, it is proved that uniform convergence of empirical approximations of the right hand sides of the constraints implies uniform convergence of the optimal values in the linear and convex case.
This paper presents the evolution and design of the Swiss reserve market and describes its two-stage stochastic market-clearing model. In Switzerland, the reserve market comprises weekly and daily auctions. The decisi...
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This paper presents the evolution and design of the Swiss reserve market and describes its two-stage stochastic market-clearing model. In Switzerland, the reserve market comprises weekly and daily auctions. The decision-making problem is to determine the amount of reserves that should be procured in each market stage. The stochasticity stems from daily offers which are not available at the beginning of the week, when the first-stage decisions are made. The new market-clearing model minimizes expected procurement costs of reserves, while taking reserve dimensioning criteria and market properties into consideration. Since the last week of January 2014, this model has been clearing the reserve market in Switzerland. To our knowledge, this is the first real-world implementation of a stochastic market-clearing model in electricity markets.
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