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Showing papers on "Stochastic programming published in 1986"


Book
01 Jan 1986
TL;DR: The solution of Large-scale Programming Problems: Generalized Linear Programming and Decomposition Techniques Dynamic Programming Optimization in Infinite Dimension and Applications is presented.
Abstract: Preface Foreword Notation Fundamental Concepts Linear Programming One-dimensional Optimization Nonlinear, Unconstrained Optimization Nonlinear Optimization with Constraints Nonlinear Constrained Optimization Integer Programming Solution of Large-scale Programming Problems: Generalized Linear Programming and Decomposition Techniques Dynamic Programming Optimization in Infinite Dimension and Applications References Appendices Index.

564 citations


Journal ArticleDOI
TL;DR: A multiperiod stochastic linear programming model ALM is developed that includes the essential institutional, legal, financial, and bank-related policy considerations, and their uncertainties, yet is computationally tractable for realistically sized problems and generates superior policies.
Abstract: In managing its assets and liabilities in light of uncertainties in cash flows, cost of funds and return on investments, a bank must determine its optimal trade-off between risk, return and liquidity. In this paper we develop a multiperiod stochastic linear programming model ALM that includes the essential institutional, legal, financial, and bank-related policy considerations, and their uncertainties, yet is computationally tractable for realistically sized problems. A version of the model was developed for the Vancouver City Savings Credit Union for a 5-year planning period. The results indicate that ALM is theoretically and operationally superior to a corresponding deterministic linear programming model, and that the effort required for the implementation of ALM, and its computational requirements, are comparable to those of the deterministic model. Moreover, the qualitative and quantitative characteristics of the solutions are sensitive to the model's stochastic elements, such as the asymmetry of cash flow distributions. We also compare ALM with the stochastic decision tree SDT model developed by S. P. Bradley and D. B. Crane. ALM is computationally more tractable on realistically sized problems than SDT, and simulation results indicate that ALM generates superior policies.

324 citations


Proceedings ArticleDOI
01 Dec 1986
TL;DR: This paper gives a short survey of Monte Carlo algorithms for stochastic optimization, with emphasis on the analysis of convergence rate.
Abstract: This paper gives a short survey of Monte Carlo algorithms for stochastic optimization. Both discrete and continuous parameter stochastic optimization are discussed, with emphasis on the analysis of convergence rate. Some future research directions for the area are also indicated.

323 citations


Journal ArticleDOI
TL;DR: A new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component and is finitely convergent without any nondegeneracy assumptions is proposed.
Abstract: A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.

301 citations


Book ChapterDOI
01 Jan 1986
TL;DR: Various approximation schemes for stochastic optimization problems involving either approximating of the probability measures and/or approximates of the objective functional, are investigated and their potential implementation as part of general procedures for solving Stochastic programs with recourse is discussed.
Abstract: Various approximation schemes for stochastic optimization problems involving either approximates of the probability measures and/or approximates of the objective functional, are investigated. We discuss their potential implementation as part of general procedures for solving stochastic programs with recourse.

253 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered nonlinear programming problems with stochastic constraints and showed that the deterministic surrogate problem CE-P contains a penalty function which penalizes violation of the constraints in the mean.
Abstract: We consider nonlinear programming problem P with stochastic constraints. The Lagrangean corresponding to such problems has a stochastic part, which in this work is replaced by its certainty equivalent in the sense of expected utility theory. It is shown that the deterministic surrogate problem CE-P thus obtained, contains a penalty function which penalizes violation of the constraints in the mean. The approach is related to several known methods in stochastic programming such as: chance constraints, stochastic goal programming, reliability programming and mean-variance analysis. The dual problem of CE-P is studied for problems with stochastic righthand sides in the constraints and a comprehensive duality theory is developed by introducing a new certainty equivalent NCE concept. Motivation for the NCE and its potential role in Decision Theory are discussed, as well as mean-variance approximations.

178 citations


Journal ArticleDOI
TL;DR: In this article, an interactive procedure is described to obtain a best compromise for such a MOSLP problem, which involves in particular, the concepts of stochastic programming with recourse, and the efficiency projection techniques are used to provide the decisionmaker with detailed graphical information on efficient solution families.

150 citations


Journal ArticleDOI
TL;DR: This paper studies a problem, common to a wide variety of manufacturing companies, of determining the production schedule of style goods, such as clothing and consumer durables, under capacity constraints, by exploiting the problem's two-level hierarchical structure.
Abstract: In this paper we study a problem, common to a wide variety of manufacturing companies, of determining the production schedule of style goods, such as clothing and consumer durables, under capacity constraints. Demand for items is stochastic and occurs in the last season of the planning horizon. Demand estimates are revised in each period. We exploit the problem's two-level hierarchical structure, which is characterized by families and items. Production changeover costs from one family to another are high, compared to other costs. However, changeover costs between items in the same family are negligible. We first formulate this problem as a difficult-to-solve stochastic mixed integer programming problem. Then, exploiting the problem's hierarchical structure, we formulate a deterministic, mixed integer programming problem and solve it by means of an algorithm that provides an approximate solution. A lower bound is obtained by applying generalized linear programming to the approximate problem. We illustrate the procedure using the disguised data of a consumer electronics company. The computational results demonstrate the effectiveness of the proposed approach in a practical setting.

122 citations


Book ChapterDOI
01 Jan 1986
TL;DR: A new method is proposed for solving two-stage problems in linear and quadratic stochastic programming that approximates the dual objective over the convex hull of finitely many dual feasible solutions.
Abstract: A new method is proposed for solving two-stage problems in linear and quadratic stochastic programming. Such problems are dualized, and the dual, althought itself of high dimension, is approximated by a sequence of quadratic programming subproblems whose dimensionality can be kept low. These subproblems correspond to maximizing the dual objective over the convex hull of finitely many dual feasible solutions. An optimizing sequence is produced for the primal problem that converges at a linear rate in the strongly quadratic case. An outer algorithm of augmented Lagrangian type can be used to introduce strongly quadratic terms, if desired.

119 citations


Book
01 Aug 1986
TL;DR: In this article, the authors present a dual of a dynamic inventory control model: the deterministic case and the stochastic case, and present a list of optimization problems for both cases.
Abstract: 1 Introduction and Summary.- 2 Mathematical Programming and Duality Theory.- 3 Stochastic Linear Programming Models.- 4 Some Linear Programs in Probabilities and Their Duals.- 5 On Integrated Chance Constraints.- 6 On The Behaviour of the Optimal Value Operator of Dynamic Programming.- 7 Robustness against Dependence in Pert.- 8 A Dual of a Dynamic Inventory Control Model: The Deterministic and the Stochastic Case.- List of Optimization Problems.

99 citations


Journal ArticleDOI
TL;DR: It is shown that in a static model the stocks of product-specific components always increase when other components are combined, and optimal policy for the dynamic version is shown to be myopic.

Journal ArticleDOI
TL;DR: In this article, a stochastic programming model of the monopolistic competition banking firm is developed, where bank decisions are made within a two-stage framework where realized disturbances that violate constraints can be rectified ex post at a cost.

Book ChapterDOI
TL;DR: A general approach to global optimization based on a solution method for d.c. c. programming problems is presented.
Abstract: A d.c. function is a function which can be represented as a difference of two convex functions. A d.c. programming problem is a mathematical programming problem involving a d.c. objective function and (or) d.c. constraints. We present a general approach to global optimization based on a solution method for d.c. programming problems.

Journal ArticleDOI
TL;DR: In this article, a general dynamic programming algorithm for the solution of optimal stochastic control problems concerning a class of discrete event systems is presented, where the emphasis is put on the numerical technique used for the approximation of the dynamic programming equation.
Abstract: This paper presents a general dynamic programming algorithm for the solution of optimal stochastic control problems concerning a class of discrete event systems. The emphasis is put on the numerical technique used for the approximation of the solution of the dynamic programming equation. This approach can be efficiently used for the solution of optimal control problems concerning Markov renewal processes. This is illustrated on a group preventive replacement model generalizing an earlier work of the authors.

Journal ArticleDOI
TL;DR: A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained.
Abstract: This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.

Book ChapterDOI
01 Jan 1986
TL;DR: A new stochastic subgradient algorithm for solving convex Stochastic programming problems is described and Convergence with probability one is proved and numerical examples are described.
Abstract: A new stochastic subgradient algorithm for solving convex stochastic programming problems is described. It uses an auxiliary filter to average stochastic subgradients observed and is provided with on-line rules for determining stepsizes and filter gains in the course of computation. Convergence with probability one is proved and numerical examples are described.

Journal ArticleDOI
TL;DR: In this paper, a simple model is presented which allows us to determine the optimal size, fillup, and drawdown rates for a Strategic Petroleum Reserve SPR under a variety of supply and demand conditions.
Abstract: A simple model is presented which allows us to determine the optimal size, fillup, and drawdown rates for a Strategic Petroleum Reserve SPR under a variety of supply and demand conditions. The optimal policy variables are determined by minimizing an analytic expression which we derive for the expected insecurity cost rate to the U.S. due to uncertainty in supply of imported oil. The oil market is modeled in terms of an elastic demand curve and two levels of supply which alternate according to a stationary, continuous time Markov process. The SPR policy is characterized by a fixed fillup rate up to the reserve's capacity, when supply is at its normal level, and a fixed drawdown rate during shortages. The insecurity cost rate being minimized includes consumer welfare loss due to price hikes, reserve holding cost and capital appreciation of the reserve. Base case results and sensitivity analysis are presented and compared to results obtained by previous approaches. These comparisons suggest that the proposed model can reasonably approximate the more computationally demanding stochastic dynamic programming formulation. The main advantage of the new approach is that it permits extensive sensitivity analysis which is important given the quality of the data.

Book ChapterDOI
01 Jan 1986
TL;DR: The stability of the optimal solution of a stochastic program with recourse with respect to small changes of the underlying distribution of random coefficients is considered in this paper, where contamination of the given distribution by another one is suggested and the original stability problem is thus reduced to that with linearly perturbed objective function.
Abstract: In the paper, stability of the optimal solution of a stochastic program with recourse with respect to small changes of the underlying distribution of random coefficients is considered. As a tool, contamination of the given distribution by another one is suggested and the original stability problem is thus reduced to that with linearly perturbed objective function. The theory of perturbed Kuhn-Tucker points and strongly regular equations is used to get explicit formulas for Gâteaux differentials of optimal solutions under different assumptions. Possible exploitation of the results for further robustness studies is indicated.

Book ChapterDOI
01 Apr 1986
TL;DR: In this article, a PERT-type project planning problem is considered, under the assumption (to be relaxed in Section 4) that the marginal distributions of the durations of the activities are known.
Abstract: A PERT-type project planning problem is considered, under the assumption (to be relaxed in Section 4) that the marginal distributions of the durations of the activities are known. Instead of the assumption of independence a minimax approach is proposed. A complete characterization of worst-case joint distributions, which by definition maximize the mean delay of the project completion time over a fixed target time T, is given. In the same framework also an optimal value for T is determined: it balances the costs of delay with the costs for large values of T in a two-stage stochastic program.

Journal ArticleDOI
TL;DR: A sequential stochastic assignment problem in a stationary Markov chain, where the states are not known explicitly, is considered, and an optimal policy and the total expected reward under this policy are obtained.
Abstract: A sequential stochastic assignment problem in a stationary Markov chain, where the states are not known explicitly, is considered. This is an optimization problem in a partially observable Markov chain, and an optimal policy and the total expected reward under this policy are obtained. Here we specify the learning procedure by the Bayes' theorem, and the optimal policy is not always a critical number policy. As a special case of this problem, a problem of optimal selections is considered, and a relation to former results of a sequential stochastic assignment problem is observed.

01 Jul 1986
TL;DR: Two new approaches to optimization of the complex stochastic systems that arise in a manufacturing context are presented; both are Monte Carlo simulation-oriented, and are therefore broadly applicable.
Abstract: : The design of modern manufacturing systems presents a number of challenges. In particular, the stochastic nature of machine failures in combination with the large number of decision variables makes optimization of such systems difficult. This paper presents two new approaches to optimization of the complex stochastic systems that arise in a manufacturing context; both are Monte Carlo simulation-oriented, and are therefore broadly applicable. The first technique involves using a likelihood ratio gradient estimate to drive a Robbins-Monro algorithm, and is relevant to problems in which the decision variables are continuous. The second idea employs homotopy methods to follow an optimal path in decision variable space, and can be used for both discrete and continuous optimization.

Book ChapterDOI
01 Jan 1986
TL;DR: Since an evident framework for the quantitative analysis of uncertainty is provided by probability theory it seems only natural to interpret the uncertain coefficient values as realizations of random variables, which characterizes stochastic linear programming.
Abstract: Linear programming has proven to be a suitable framework for the quantitative analysis of many decision problems. The reasons for its popularity are obvious: many practical problems can be modeled, at least approximately, as linear programs, and powerful software is available. Nevertheless, even if the problem has the necessary linear structure it is not sure that the linear programming approach works. One of the reasons is that the model builder must be able to provide numerical values for each of the coefficients. But in practical situations one often is not sure about the “true” values of all coefficients. Usually the uncertainty is exorcized by taking reasonable guesses or maybe by making careful estimates. In combination with a sensitivity analysis with respect to the most inaccurate coefficients this approach is satisfactory in many cases. However, if it appears that the optimal solution depends heavily on the value of some inaccurate data, it might be sensible to take the uncertainty of the coefficients into consideration in a more fundamental way. Since an evident framework for the quantitative analysis of uncertainty is provided by probability theory it seems only natural to interpret the uncertain coefficient values as realizations of random variables. This approach characterizes stochastic linear programming.

Journal ArticleDOI
TL;DR: In this paper, the authors present an application of the Dantzig-Wolfe decomposition principle to the problem of investment planning in the electric power sector, and the formulation of the capacity planning problem incorporates uncertainties in long-term load growth and in fuel supply availability.
Abstract: This paper presents an application of the Dantzig-Wolfe decomposition principle to the problem of investment planning in the electric power sector. The formulation of the capacity planning problem incorporates uncertainties in long-term load growth and in fuel supply availability. In addition, the formulation permits the inclusion of such demand-side investment decisions as conservation as well as conventional and renewable supply investments, and it allows flexibility in modeling system reliability. Reliability targets can be incorporated as constraints or reliability can be optimized by minimizing customer outage costs in addition to investment costs and operating costs. Results of an application to the Pacific Northwest, involving problem sizes up to 30,000 rows and 54,000 columns, are reported.

Journal ArticleDOI
TL;DR: A recent simulation technique presented by the author for computing values of the distribution function can be modified to yield appropriate procedure for computing probabilities of rectangles.
Abstract: In several stochastic programming models and statistical problems the computation of probabilities of n-dimensional rectangles is required in case of n-dimensional normal distribution. A recent simulation technique presented by the author for computing values of the distribution function can be modified to yield appropriate procedure for computing probabilities of rectangles. Some numerical work is provided to illustrate the use of the new algorithm.

Book ChapterDOI
01 Jan 1986
TL;DR: Of special interest to physicists are the Ising model in a random field and spin glasses, which are known to lead to difficulties in conventional Monte-Carlo algorithms.
Abstract: Fast multi-level techniques are developed for large-scale problems whose variables may assume only discrete values (such as spins with only “up” and “down” states), and/or where the relations between variables is probabilistic. Motivation and examples are taken from statistical mechanics and field theory. Detailed procedures are developed for the fast global minimization of discretestate functionals, or other functionals with many local minima, using new principles of multilevel interactions. Tests with Ising spin models are reported. Of special interest to physicists are the Ising model in a random field and spin glasses, which are known to lead to difficulties in conventional Monte-Carlo algorithms.


Journal ArticleDOI
TL;DR: In this paper, a stochastic subgradient method for solving convex convex Stochastic Programming problems is considered and on-line rules for determining stepsizes are derived from the concept of local regularized improvement functions.

Journal ArticleDOI
TL;DR: Results from applying the model to the Queensland cattle slaughtering industry demonstrate the inappropriateness of using traditional deterministic plant location models to analyse problems with major stochastic elements.
Abstract: A plant location model with two major aspects is outlined. First, discrete stochastic programming is used to handle variabilty in supplies and demands. Second, the cost structure of plants is modelled in more detail and with more realism than is usual. Results from applying the model to the Queensland cattle slaughtering industry demonstrate the inappropriateness of using traditional deterministic plant location models to analyse problems with major stochastic elements.-from Author

Book ChapterDOI
TL;DR: The present chapter analyze the mathematical properties of integrated chance constraints as a modeling tool for here-and-now stochastic programming problems in some detail and considers “penalty cost” models as recourse models.
Abstract: In Chapter 3 we introduced integrated chance constraints (ICCs) as a modeling tool for here-and-now stochastic programming problems; see (3.28). In the present chapter we analyze the mathematical properties of this new concept in some detail. Let us review its rationale. As indicated in Section 3.2, if in the constraints of a linear programming problem random coefficients occur with unknown realizations, then in order to have a unequivocal meaning of “feasibility” one has to make additional specifications. There are two well-known modeling techniques for this: in chance-constrained programming (CCP) the probability of infeasibility is restricted, and in stochastic programming with recourse (SPR) the effects of infeasibility are penalized. For convenience, we here consider “penalty cost” models as recourse models, see Remark 3.6. Several authors [12,10,11,3,2] established certain equivalences between CCP and SPR. Their results are not completely convincing, however; for example, CCP problems may be nonconvex whereas SPR problems are always convex [5 page 90]. Even if mathematical equivalence can be established there still are differences between CCP and SPR models which might be important for the model builder.

Book ChapterDOI
01 Jan 1986
TL;DR: In this article, a strategy for computing first-order corrections to allow for the uncertainty in a deterministic stochastic programming problem is proposed, and the problem of implementing this strategy is studied by considering some simple examples.
Abstract: There are many types of multi-time-period stochastic programming problems. In particular, there are problems where activities in one time period provide inventories or new capacities of uncertain magnitude for use in the next time period. One approach is then to ignore the uncertainties and solve a deterministic model using mean values. A slightly more sophisticated approach is to make first-order corrections to allow for the uncertainty. This paper suggests a strategy for computing such corrections. The problem of implementing this strategy is then studied by considering some very simple examples. These examples suggest that it may be seriously misleading to assume that all the relevant random variables are normally distributed unless the variance is small compared with the mean. This is because in reality the random variables are nonnegative. Fortunately the approach also works if the variables are assumed to have Gamma distributions.