Showing papers on "Stochastic programming published in 1992"
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TL;DR: Convergence with probability one is proved for a variety of classical optimization and identification problems and it is demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence.
Abstract: A new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated. Convergence with probability one is proved for a variety of classical optimization and identification problems. It is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence.
1,970 citations
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28 Aug 1992TL;DR: In this article, the authors considered the problem of optimal control of linear stochastic systems with partial information with an exponential-of-integral performance index (EoIPI).
Abstract: Preface 1. Linear filtering theory 2. Optimal stochastic control for linear dynamic systems with quadratic payoff 3. Optimal control of linear stochastic systems with an exponential-of-integral performance index 4. Non linear filtering theory 5. Perturbation methods in non linear filtering 6. Some explicit solutions of the Zakai equation 7. Some explicit controls for systems with partial observation 8. Stochastic maximum principle and dynamic programming for systems with partial observation 9. Existence results for stochastic control problems with partial information References Index.
514 citations
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TL;DR: In this paper, the existence and stability of invariant distributions for stochastically monotone Markov processes are studied. And the existence of fixed points of mappings on compact sets of measures that are increasing with respect to a stochastic ordering is established.
Abstract: The existence and stability of invariant distributions for stochastically monotone processes is studied. The Knaster-Tarski fixed point theorem is applied to establish existence of fixed points of mappings on compact sets of measures that are increasing with respect to a stochastic ordering. Global convergence of a monotone Markov process to its unique invariant distribution is established under an easily verified assumption. Topkis' theory of supermodular functions is applied to stochastic dynamic optimization, providing conditions under which optimal stationary decisions are monotone functions of the state and induce a monotone Markov process. Applications of these results to investment theory, stochastic growth, and industry equilibrium dynamics are given.
314 citations
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TL;DR: In this paper, several financial planning problems are posed as dynamic generalized network models with stochastic parameters, including asset allocation for portfolio selection, international cash management, and programmed-trading arbitrage.
Abstract: Several financial planning problems are posed as dynamic generalized network models with stochastic parameters. Examples include: asset allocation for portfolio selection, international cash management, and programmed-trading arbitrage. Despite the large size of the resulting stochastic programs, the network structure can be exploited within the solution strategy giving rise to efficient implementations. Empirical results are presented indicating the benefits of the stochastic network approach for the asset allocation case.
272 citations
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01 Dec 1992
TL;DR: It is shown how large-scale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo sampling techniques.
Abstract: For many practical problems, solutions obtained from deterministic models are unsatisfactory because they fail to hedge against certain contingencies that may occur in the future. Stochastic models address this shortcoming, but up to recently seemed to be intractable due to their size. Recent advances both in solution algorithms and in computer technology now allow us to solve important and general classes of practical stochastic problems. We show how large-scale stochastic linear programs can be efficiently solved by combining classical decomposition and Monte Carlo (importance) sampling techniques. We discuss the methodology for solving two-stage stochastic linear programs with recourse, present numerical results of large problems with numerous stochastic parameters, show how to efficiently implement the methodology on a parallel multi-computer and derive the theory for solving a general class of multi-stage problems with dependency of the stochastic parameters within a stage and between different stages.
250 citations
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TL;DR: In this paper, a stochastic search method is proposed for finding a global solution to the discrete optimization problem in which the objective function must be estimated by Monte Carlo simulation, and it is shown under mild conditions that the Markov chain is strongly ergodic.
Abstract: In this paper a stochastic search method is proposed for finding a global solution to the stochastic discrete optimization problem in which the objective function must be estimated by Monte Carlo simulation. Although there are many practical problems of this type in the fields of manufacturing engineering, operations research, and management science, there have not been any nonheuristic methods proposed for such discrete problems with stochastic infrastructure. The proposed method is very simple, yet it finds a global optimum solution. The method exploits the randomness of Monte Carlo simulation and generates a sequence of solution estimates. This generated sequence turns out to be a nonstationary Markov chain, and it is shown under mild conditions that the Markov chain is strongly ergodic and that the probability that the current solution estimate is global optimum converges to one. Furthermore, the speed of convergence is also analyzed.
204 citations
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TL;DR: In this paper, a Bayesian stochastic dynamic programming (BSDP) model is proposed to generate optimal reservoir operating rules, which includes inflow, storage, and forecast as state variables, and describes streamflows with a discrete lag 1 Markov process.
Abstract: Operation of reservoir systems using stochastic dynamic programming (SDP) and Bayesian decision theory (BDT) is investigated in this study. The proposed model, called Bayesian stochastic dynamic programming (BSDP), which includes inflow, storage, and forecast as state variables, describes streamflows with a discrete lag 1 Markov process, and uses BDT to incorporate new information by updating the prior probabilities to posterior probabilities, is used to generate optimal reservoir operating rules. This continuous updating can significantly reduce the effects of natural and forecast uncertainties in the model. In order to test the value of the BSDP model for generating optimal operating rules, real-time reservoir operation simulation models are constructed using 95 years of monthly historical inflows of the Gunpowder River to Loch Raven reservoir in Maryland. The rules generated by the BSDP model are applied in an operation simulation model and their performance is compared with an alternative stochastic dynamic programming (ASDP) model and a classical stochastic dynamic programming (SDP) model. BSDP differs from the other two models in the selection of state variables and the way the transition probabilities are formed and updated.
147 citations
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TL;DR: This work considers the problem of optimally meeting a stochastically growing demand for capacity over an infinite horizon and shows that this stochastic problem can be transformed into an equivalent deterministic problem.
Abstract: We consider the problem of optimally meeting a stochastically growing demand for capacity over an infinite horizon. Under the assumption that demand for product follows either a nonlinear Brownian motion or a non-Markovian birth and death process, we show that this stochastic problem can be transformed into an equivalent deterministic problem. Consistent with earlier work by A. Manne, the equivalent problem is formed by replacing the stochastic demand by its deterministic trend and discounting all costs by a new interest rate that is smaller than the original, in approximate proportion to the uncertainty in the demand.
139 citations
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TL;DR: A decomposition algorithm for solving the finite horizon problems and a heuristic procedure that is based on the structure of the optimal policy for two-period problems, which parallels the decision rules used by managers in practice.
Abstract: In this paper, we model production problems where yields are stochastic, demands are substitutable, and several items are jointly produced. We formulate this problem as a profit maximizing convex program, and study two approximation procedures. The first method solves finite horizon stochastic programs on a rolling horizon basis. We develop a decomposition algorithm for solving the finite horizon problems. The finite horizon problems are linear programs. Our algorithm utilizes the network-like structure of the coefficient matrix of the linear programs. The second method is a heuristic procedure that is based on the structure of the optimal policy for two-period problems. The heuristic parallels the decision rules used by managers in practice. The computational results suggest that the performance of this heuristic is comparable to that of the rolling horizon approach.
128 citations
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TL;DR: A dual-based procedure is presented and it is indicated how the dual-descent and primal-dual adjustment procedures proposed by D. Erlenkotter in the static case can be made monotonically improving in the stochastic case.
Abstract: In this paper, we study how the uncapacitated facility location problem is transformed into a two-stage stochastic program with recourse when uncertainty on demand, selling prices, production and transportation costs are introduced. We then present a dual-based procedure and indicate how the dual-descent and primal-dual adjustment procedures proposed by D. Erlenkotter (1978) in the static case can be made monotonically improving in the stochastic case. Results of computer experiments are reported.
124 citations
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TL;DR: A review of the state of the art of systems analysis and optimization techniques developed in the field of water resources for the planning and management of a ground-water system can be found in this paper.
Abstract: The objective of this paper is to review the state of the art of systems analysis and optimization techniques developed in the field of water resources for the planning and management of a ground-water system. The areas reviewed include the following: ground-water management models, inverse solution techniques for parameter identification, and optimal experimental design methods. Emphasis is placed upon ground-water supply management models, as opposed to models used for ground-water quality management. The techniques that have been used in the optimization of ground-water management include: linear programming, mixed-integer and quadratic programming, differential dynamic programming, nonlinear programming, and simulation. The inverse problem of parameter identification pertains the optimal determination of model parameters using historical input and output observations. Because of data limitation in both quantity and quality, the inverse problem is inherently ill posed. This paper summarizes recent advances made in the inverse procedures and methods developed to alleviate the problems of instability and nonuniqueness of the identified parameters. The optimal experimental design problem addresses the issue of data requirements and optimal sampling strategies for the purpose of parameter identification. A criterion must be established for the optimal design of a pumping test. The fundamental concept of optimal experimental design and various criteria used for optimization are reviewed.
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TL;DR: In this paper, a stochastic optimization model for containment of a plume of groundwater contamination through the installation and operation of pumping wells is developed, which considers explicitly uncertainty about hydraulic conductivity in the aquifer and seeks to minimize the expected total cost of operating the pumping wells plus the recourse cost incurred when containment of the contaminant plume is not achieved.
Abstract: A stochastic optimization model for containment of a plume of groundwater contamination through the installation and operation of pumping wells is developed. It considers explicitly uncertainty about hydraulic conductivity in the aquifer and seeks to minimize the expected total cost of operating the pumping wells plus the recourse cost incurred when containment of the contaminant plume is not achieved. Four different formulations of the model are examined, ranging from simply replacing all uncertain parameters by their expected values to a full stochastic programming with recourse model involving nonsymmetric linear quadratic penalty functions. The full stochastic programming with recourse model, which minimizes the expected total costs over a number of realizations of outcomes of the random parameters, is nonlinear and possibly nonconvex and is solved by an extension of the finite generation algorithm. The value of information about the uncertain parameters is defined through the differences between the values of the optimal solutions to the different formulations. A sample problem is solved using all four formulations. The results indicate that the explicit incorporation of uncertainty does make a difference in the solutions obtained. The work indicates that stochastic programming with recourse is a useful tool in management under uncertainty, and that it can be used with reasonable computational resources for problems of moderate size.
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TL;DR: The combined augmented Lagrangian/barrier method applies in a natural way to stochastic programming and multicommodity networks.
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TL;DR: The test showed that dual dynamic programming has reasonable computing times and can be a useful tool in stochastic scheduling in a hydro-dominated system.
Abstract: The aim is to show an application of stochastic dual dynamic programming to seasonal planning in a part of the Norwegian hydro-dominated power system. The subsystem under study has 35 reservoirs on 28 watercourses. It is found that for the study system the new procedure is entirely feasible and gives good results. Two implementation details are studied more closely: use of relaxation in the solution of the subproblems, and a starting technique, called pre-segment, to save iterations in the overall problem. Both are found to have a significant effect on computer time. The test showed that dual dynamic programming has reasonable computing times and can be a useful tool in stochastic scheduling in a hydro-dominated system. >
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TL;DR: In this paper, a necessary and sufficient condition for local optimal solutions of bilevel programming problems is developed using differential stability results for parametric optimization problems, and verification of these conditions reduces to the solution of some auxiliary combinatorial optimization problems.
Abstract: A necessary and a sufficient condition for local optimal solutions of bilevel programming problems are developed using differential stability results for parametric optimization problems. Verification of these conditions reduces to the solution of some auxiliary combinatorial optimization problems.
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01 Jan 1992
TL;DR: Mixtures comprising a low molecular weight diol, a liquid modifier compound having a boiling point above about 150 DEG C, such as a fatty acid or fatty oil, a polyisocyanate having a functionality of at least about 2.5 and an organotin catalyst instantly set to a solid, dense, rigid polymeric product which can be demolded within a period of from less than 1 minute to about 5 minutes.
Abstract: Mixtures comprising a low molecular weight diol, a liquid modifier compound having a boiling point above about 150 DEG C, such as a fatty acid or fatty oil, a polyisocyanate having a functionality of at least about 2.5 and an organotin catalyst instantly set, after a brief induction period, to a solid, dense, rigid polymeric product which can be demolded within a period of from less than 1 minute to about 5 minutes.
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TL;DR: Rationality of the search for a global minimum is formulated axiomatically and the features of the corresponding algorithm are derived from the axioms.
Abstract: A review of statistical models for global optimization is presented. Rationality of the search for a global minimum is formulated axiomatically and the features of the corresponding algorithm are derived from the axioms. Furthermore the results of some applications of the proposed algorithm are presented and the perspectives of the approach are discussed.
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TL;DR: In this paper, an approach to the vehicle routing problem for the case of stochastic demand is presented based on the simulated annealing technique, which is illustrated with a numerical example.
Abstract: Research work dealing with the vehicle routing problem has not paid adequate attention to the cases where the demand for services at certain nodes is a random variable. This paper develops an approach to the vehicle routing problem for the case of stochastic demand. The approach is based on the simulated annealing technique. It is illustrated with a numerical example.
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TL;DR: Methods of disjunctive programming are used to approximate the convex hull of the feasible region of the possible region of mathematical programs with probabilistic constraints in which the random variables are discrete.
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TL;DR: In this paper, explicit factorizations of the search direction matrices are proposed to reduce the computational complexity of interior point algorithms by reducing the density of these matrices, but at the expense of increased problem size.
Abstract: Solving deterministic equivalent formulations of two-stage stochastic linear programs using interior point methods may be computationally diflicult due to the need to factorize quite dense search direction matrices (e.g., AAT). Several methods for improving the algorithmic efficiency of interior point algorithms by reducing the density of these matrices have been proposed in the literature. Reformulating the program decreases the effort required to find a search direction, but at the expense of increased problem size. Using transpose product formulations (e.g., A*A) works well but is highly problem dependent, Schur complements may require solutions with potentially near singular matrices. Explicit factorizations of the search direction matrices eliminate these problems while only requiring the solution to several small, independent linear systems. These systems may be distributed across multiple processors. Computational experience with these methods suggests that substantial performance improvements are possible with each method and that, generally, explicit factorizations require the least computational effort.
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TL;DR: In this paper, a two-stage stochastic programming approach with MINLP recourse was proposed for the synthesis problem of heat integrated distillation sequences that can handle prespecified changes in the composition and/or flowrate of the multicomponent feed to the separation system for a finite number of periods of operation.
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TL;DR: The application of multiple objective optimization techniques based on the methods of nonlinear goal programming to perform optimal synthesis of general planar mechanisms is presented and the results are discussed.
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TL;DR: A fuzzy nonlinear goal programming approach is presented for solving multiobjective optimization problems involving vague and imprecise information and aid in the preliminary design of structural systems involving imprecising and vague information about the goals and/or constraints.
Abstract: A fuzzy nonlinear goal programming approach is presented for solving multiobjective optimization problems involving vague and imprecise information. Several computational models, including simple additive, weighted additive, and preemptive priority models, are given for the numerical solution of the problem. The methodolo- gies are illustrated with the help of two structural optimization problems involving multiple goals. The solution of the first example is obtained using a graphical procedure whereas the second example is solved using nonlinear programming techniques. Linear membership functions are used in the numerical work for simplicity. The methodologies presented in this work aid in the preliminary design of structural systems involving imprecise and vague information about the goals and/or constraints. INEAR goal programming has been extensively used in solving decision-making problems involving linear equa- tions and multiple conflicting goals. The goals can be rank ordered depending on their importance to the decision maker. Goal programming attempts to achieve as man}7 of these goals as possible by minimizing deviational variables from the goal levels depending on their relative weights. Linear goal pro- gramming algorithms were developed by Charnes et al., 1 Ig- nizio,2 and Zanakis and Gupta.3 The extension of goal pro- gramming to nonlinear optimization problems has also been considered by several authors.3'4 A fuzzy programming ap- proach for linear programming problems with several objec- tives was suggested by Zimmerman. 5 Subsequently, several authors proposed different fuzzy goal programming ap- proaches for solving linear goal programming problems in- volving imprecise statements and information.6"10 Narasimhan6 suggested a method for solving fuzzy linear goal programming problems under the assumption of linear membership functions. His method involves solving a set of 2k linear programming problems, each containing 3k constraints where k denotes the number of goals in the original problem. Hannan7 indicated a procedure for formulating a fuzzy goal programming problem as an equivalent single linear program- ming problem with 2k goal-related constraints. The problems associated with the definition of fuzzy priorities were dis- cussed in Ref. 7. A brief review of the history and the state of the art in fuzzy multicriteria programming as of 1982 was presented by Ignizio.8 The distinction between fuzzy goal pro- gramming and fuzzy multicriteria formulations was given in Ref. 9. Models are presented by Hannan7 for the use of fuzzy goal programming with preemptive priorities, with Archime- dian weights, and with the maximization of the membership function corresponding to the minimum goal. A methodology based on the use of a nested hierarchy of priorities for each goal was presented by Rubin and Narasimhan.10 The impor- tance of multiple objectives in the design of practical engineer-
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TL;DR: In this paper, a stochastic dynamic programming based algorithm is employed to determine the optimal short-term generation scheduling and battery storage policy which minimize the fuel consumption for the next 24-hour period.
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03 Jan 1992TL;DR: This paper deals with a special class of nonlinear discrete design optimization problems which involve nonlinear separable objective functions and bilinear constraints and identifies two special cases for which advantage can be taken of the discrete nature of the design variables to reformulate these problems as MILP models which can be solved to global optimality.
Abstract: This paper deals with a special class of nonlinear discrete design optimization problems which involve nonlinear separable objective functions and bilinear constraints. These constraints involve products of design and state variables in which the former are restricted to take discrete values Two special cases are identified for which advantage can be taken of the discrete nature of the design variables to reformulate these problems as MILP models which can be solved to global optimality. The computational expense can be reduced with SOS 1 sets and a simple solution strategy that is proposed. The application of the MILP reformulations is applied to multiproduct batch plant problems in chemical engineering and 1 Engineering Design Research Center, Carnegie Mellon University, Pittsburgh, PA 15213. The authors gratefully acknowledge financial support from the Engineering Design Research Center. University Ubraritt Carnegie Msiion y | Pittsburgh PA l to structural design problems in civil engineering. Numerical results and comparisons with other methods are also presented.
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TL;DR: In this paper, the authors considered the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier (LMM), and showed that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman (HJB SPDE) that appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of
Abstract: Using the decomposition of solution of SDE, we consider the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process (nonanticipativity equality constraint) It is shown that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman stochastic partial differential equation (HJB SPDE) This appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of the Wiener process is used Our approach extends the Wong-Zakai type results [20] from SDE to the stochastic dynamic programming equation by showing how this arises as average of the limit of a sequence of deterministic dynamic programming equations The stochastic characteristics method of Kunita [13] is used to represent the
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TL;DR: The authors evaluated the ability of linear programming, LP with a margin of safety (LPMS), and stochastic programming (SP) models to formulate poultry rations at least cost, with a given probability to meet nutrient requirements as set by the National Research Council in 1984.
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TL;DR: Variable and row aggregation as a technique of simplifying a mathematical program is utilized to develop bounds for two-stage stochastic convex programs with random right-hand sides and the Gassmann-Ziemba inequality is used for first moment upper bounds.
Abstract: Variable and row aggregation as a technique of simplifying a mathematical program is utilized to develop bounds for two-stage stochastic convex programs with random right-hand sides. If one is able to utilize the problem structure along with only first moment information, a tighter bound than the usual mean model bound based on Jensen's inequality may be obtained. Moreover, it is possible to construct examples for which the mean model bound will be arbitrarily poor. Consequently, one can tighten Jensen's bound for stochastic programs when the distribution has a compact support. This bound may be improved further by partitioning the support using conditional first moments. With regard to first moment upper bounds, the Gassmann-Ziemba inequality is used for the stochastic convex program to seek a model which can be solved using standard convex programming techniques. Moreover, it allows one to easily construct upper bounds using the solution of the lower bounding problem. Finally, the results are extended to multistage stochastic convex programming problems.
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TL;DR: A group of optimization models for real-time operation of a hydropower system of reservoirs is presented in this paper, where the dimensionality problems usually found in dynamic programming formulations are solved by a space-time aggregation/disaggregation procedure that combines stochastic dynamic programming and linear programming techniques.
Abstract: A group of optimization models for the real-time operation of a hydropower system of reservoirs is presented in this paper. The dimensionality problems usually found in dynamic programming formulations are solved by a space-time aggregation/disaggregation procedure that combines stochastic dynamic programming and linear programming techniques. The reservoirs in a hydropower system are aggregated in power units rather than in water units, and an optimal operating policy for the equivalent aggregated reservoir is found in the first part of this work. The objective function in this first part is to minimize the total cost or energy production for a hydrothermal system. The aggregated policy obtained is used in the real-time operation of the system to determine the recommended daily releases and power production from each reservoir of the system. The proposed methodology is applied to a case study, the Lower Caroni system in Venezuela, which is composed of four reservoirs in series and a total installed capacity of 17,000 MW, with satisfactory results.
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TL;DR: The paper studies the linear semi-infinite programming problems and its dual problems and develops an applicable algorithm to solve such kinds of problems.
Abstract: The paper studies the linear semi-infinite programming problems and its dual problems. The main purpose is to develop an applicable algorithm to solve such kinds of problems.