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Showing papers on "Stochastic programming published in 1972"


Book
01 Jan 1972

258 citations


Proceedings ArticleDOI
01 Dec 1972
TL;DR: In this paper, a closed-loop control of a discrete-time linear system with possibly time-varying random parameters in the presence of input and output noise is presented.
Abstract: A new method is presented for controlling a discrete-time linear system with possibly time-varying random parameters in the presence of input and output noise. The cost is assumed to be quadratic in the state and control. Previous algorithms for the above problem when the system had both zeros and poles unknown were of the open-loop feedback type, i.e., they did not take into account that future observations will be made. Therefore, even though these schemes were adaptive, their learning was "accidental." In contrast to this, the new approach uses an expression of the optimal cost-to-go that exhibits the dual purpose of the control, i.e., learning and control. The effect of the present control on the future estimation ("learning") appears explicitly in the cost used in the stochastic dynamic programming equation. The resulting sequence of controls, which is of the closed-loop type, is shown via simulations to appropriately divide its energy between the learning and the control purposes. Therefore, this control is called actively adaptive because it regulates the speed and amount of learning as required by the performance index. The simulations on a third-order system with six unknown parameters also demonstrate the computational feasibility of the proposed algorithm.

129 citations


Book
01 Jan 1972

117 citations



Journal ArticleDOI
TL;DR: In this article, a framework for stochastic unconstrained optimization theory is provided, with which convergence of stochastically versions of conjugate gradient, partan, etc., can be discussed and proved.
Abstract: The aim of this paper is the provision of a framework for a practical stochastic unconstrained optimization theory. The results are based on certain concepts of stochastic approximation, although not restricted to those procedures, and aim at incorporating the great flexibility of currently available deterministic optimization ideas into the stochastic problem, whenever optimization must be done by Monte Carlo or sampling methods. Hills with nonunique stationary points are treated. A framework has been provided, with which convergence of stochastic versions of conjugate gradient, partan, etc., can be discussed and proved.

32 citations


Journal ArticleDOI
TL;DR: In this paper, the methods of stochastic control processes are combined with considerations regarding convexity, which are characteristic for the deterministic models of a developing economy, to obtain a deterministic theory which can also take into account the influence of convex factors.
Abstract: In this paper the methods of stochastic control processes are combined with considerations regarding convexity, which are characteristic for the deterministic models of a developing economy. As a result a stochastic theory is obtained, heavily resembling the deterministic one, but which can also take into account the influence of stochastic factors. Bibliography: 8 items.

31 citations


Book ChapterDOI
01 Jan 1972
TL;DR: In this paper, the authors discuss the application of stochastic analysis in economic models based on a few fundamental considerations, such as the need for operational presentation of the mathematical results of the results in stochastically processes and control, and the potential usefulness of such methods in economic model analysis.
Abstract: This chapter discusses the application of stochastic analysis in economic models based on a few fundamental considerations. It focuses on the need for operational presentation of the mathematical results in stochastic processes, stochastic control, and stochastic programming that have actual or potential usefulness in economic models. It discusses empirical applications primarily for illustrative purposes to highlight the complexities of calculation involved and the nature of pay-offs expected. In realistic applications, the researcher should consider a few stochastic methods as broad guidelines, although in near future, the computational difficulties would be considerably reduced as a result of the recent trend of developments of computer algorithms in stochastic control and mathematical programming. The chapter highlights the applications of economic models primarily in the macrodynamic fields of economic growth, development, and investment planning. In the fields of operations research, stochastic methods, for example, stochastic programming and control, are most frequently applied to microeconomic fields such as the firm, the industry, and the projects.

30 citations


Journal ArticleDOI
TL;DR: A planning problem relating to spatial diversification of beef production in the Clarence region of N.S.W. is investigated using a model comprising both simulation and linear programming components, and it is concluded that such composite models are valuable for the analysis of sequential stochastic decision processes not presently amenable to solution by Stochastic programming alone.
Abstract: Methods of whole-farm planning under risk are briefly reviewed, noting especially associated operational problems. A planning problem relating to spatial diversification of beef production in the Clarence region of N.S.W. is investigated using a model comprising both simulation and linear programming components. It is concluded that such composite models are valuable for the analysis of sequential stochastic decision processes not presently amenable to solution by stochastic programming alone.

26 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered a class of linear programming problems: maximize cx subject to Ax = b with the additional constraint that x must also be an extreme point of a second convex polyhedron Dx = d, x ≧ 0.
Abstract: The paper considers a class of optimization problems. The problems are linear programming problems: maximize cx subject to Ax = b with the additional constraint that x must also be an extreme point of a second convex polyhedron Dx = d, x ≧ 0. A cutting-plane algorithm for solving such problems is presented. Two numerical examples are also included.

26 citations


Journal ArticleDOI
TL;DR: In this article, interior parametric sequential unconstrained maximization and generalized programming methods are used to solve portfolio selection problems with stochastic constraints, and an example of a portfolio selection problem is given.
Abstract: In many nonlinear programming applications the objective function has an inherent uncertainty that depends upon a set of random variables that have a known distribution. If one wishes to optimize the expectation of the objective, as suggested by the expected utility theorem, then as is shown here one can often solve such problems by modifying standard nonlinear programming algorithms. To illustrate what is involved, the details and justification for the application of the interior parametric sequential unconstrained maximization technique and the generalized programming method for the solution of such problems are given. Some related problems with stochastic constraints for which the solution method applies are mentioned and an example of a portfolio selection problem is given.

23 citations


Journal ArticleDOI
TL;DR: In this article, the authors adopt a Bayesian approach to the problem of analysis and make certain assumptions, which can utilize the vast amount of information in a stochastic dynamic prediction along with the information contained in observations.
Abstract: The result of a stochastic dynamic prediction is the expected values of the model parameters and the covariances among all the parameters. By adopting a Bayesian approach to the problem of analysis and making certain assumptions, one can utilize the vast amount of information in a stochastic dynamic prediction along with the information contained in observations. By making simulated observations of a pre-defined atmosphere, it is shown that the uncertainty in the analyzed values is substantially less than either the uncertainty in the forecast or in the observation. In addition, the results indicate that the effects of the limiting assumptions are minimal. Further experiments are performed in which only heights or only temperatures are actually observed, and in each case it is possible to obtain an analysis for all the parameters in the model. The method is particularly useful for assessing the value and impact of different amounts or types of data.

Journal ArticleDOI
TL;DR: In this article, a class of probabilistic constrained programming problems with concave functions is considered, and it is shown that the function of the left function is the same as that of the right function.
Abstract: A class of probabilistic constrained programming problems are considered where the probabilistic constraint is of the form and the functions are concave. I t is shown that the function of the left ...


Journal ArticleDOI
TL;DR: The purpose of this paper is to provide the algorithm builder with tests which would verify if the problem is well formulated, i.e. feasible, bounded, solvable, ⋯, and the theorems which are obtained indicate that such test can be performed at the very beginning of the algorithm and at negligible cost as far as running time is concerned.
Abstract: Stochastic programs deals with optimization problems whose parameters are only known up to a given probability distribution. This naturally implies that the decision is to be selected before the actual value of these random variables is known. When the predicted outcome does not match the realizations the decision maker is allowed to select a corrective action, we say that he is given a recourse decision. An algorithm to solve the “linear” case is described in papers by Dantzig and Madansky, Kall and Van Slyke and Wets. See [3] for appropriate references. The purpose of this paper is to provide the algorithm builder with tests which would verify if the problem is well formulated, i.e. feasible, bounded, solvable, ⋯. The theorems which we obtain indicate that such test can be performed at the very beginning of the algorithm and at negligible cost as far as running time is concerned.

Proceedings ArticleDOI
01 Dec 1972
TL;DR: The performance of some suboptimal controllers in relation to the performance of the optimal feedback controller and the optimal open-loop controller is studied.
Abstract: In dynamic minimax and stochastic optimization problems frequently one is forced to use a suboptimal controller since the computation and implementation of the optimal controller based on dynamic programming is impractical in many cases. In this paper we study the performance of some suboptimal controllers in relation to the performance of the optimal feedback controller and the optimal open-loop controller. Attention is focused on some classes of, so called, open-loop-feed-back controllers. It is shown under quite general assumptions that these open-loop-feedback controllers perform at least as well as the optimal open-loop controller. The results are developed for general minimax problems with perfect and imperfect state information. In the latter case the open-loop-feedback controller mkes use of an estimator which is required to perform at least as well as a pure predictor in order for the results to hold. Some of the results presented have stochastic counterparts.


Proceedings ArticleDOI
01 Aug 1972
TL;DR: An algorithm is given for the maximization of any function of many variables which may be described as the envelope of a family of linear functions.
Abstract: An algorithm is given for the maximization of any function of many variables which may be described as the envelope of a family of linear functions. It is shown how the large-scale problems of linear programming to which various decomposition schemes apply can be posed in this way. Computational experience with some of these problems is reviewed.

Journal ArticleDOI
TL;DR: Three of the most promising algorithms in the class of methods of feasible directions for solving constrained optimization problems in Rn are reviewed: an extension of the Frank-Wolfe method, a dual method due to Pironneau and Polak, and a methodDue to Zoutendijk.
Abstract: Although the class of methods of feasible directions, which can be used for solving constrained optimization problems in Rn, is quite large, only a few of these methods can be extended for the solution of optimal control problems. This paper reviews three of the most promising algorithms in this class: an extension of the Frank-Wolfe method due to Barnes, a dual method due to Pironneau and Polak, and a method due to Zoutendijk.

Book ChapterDOI
01 Jan 1972
TL;DR: In this article, the authors discuss stochastic programming and the relationship between the feasibility of vectors x and the parameters, and discuss the decision regions for optimality and convexity of the decision region.
Abstract: Publisher Summary This chapter discusses stochastic programming. The chapter reviews the relationship between the feasibility of vectors x and the parameters. Kall's theorem, optimality, and convexity are reviewed. Decision regions for optimality are reviewed. If there are no degenerate solutions, then decision regions do not overlap, but two adjacent regions contain both their common boundary. The chapter also discusses probability distributions.

Journal ArticleDOI
TL;DR: This paper presents a computational procedure for the solution of the “minimum-risk problem” associated with a stochastic linear program where costs are random variables with normal multidimensional joint distribution for the nonlinear program maxxϵX(c′x − t)/(x′Vx)1/2.
Abstract: This paper presents a computational procedure for the solution—via reduction to a parametric quadratic program—of the “minimum-risk problem” associated with a stochastic linear program where costs are random variables with normal multidimensional joint distribution, i.e., for the nonlinear program maxxϵX(c′x − t)/(x′Vx)1/2 in where t is a given number, V a positive-definite matrix, and X a given convex polyhedron in n-dimensional Euclidean space Rn.

Proceedings ArticleDOI
01 Dec 1972
TL;DR: It is shown that in many cases the expected value of the objective function is differentiable and thus the resulting optimization problem can be analyzed and solved by using classical analytical or numerical methods.
Abstract: In this paper we examine a class of stochastic optimization problems characterized by nondifferentiability of the objective function. It is shown that in many cases the expected value of the objective function is differentiable and thus the resulting optimization problem can be analyzed and solved by using classical analytical or numerical methods. The results are subsequently applied to the solution of a class of stochastic programming problems.


01 Jan 1972
TL;DR: Discrete parameter stochastic optimization problems necessary conditions, deriving maximum principle and how to solve these problems.
Abstract: Discrete parameter stochastic optimization problems necessary conditions, deriving maximum principle


01 Apr 1972
TL;DR: An introduction is given to those results in mathematical programming which appear to be most important for the development and analysis of practical algorithms, and that subclass of descent methods which requires the evaluation of first derivatives of the objective function.
Abstract: : These notes are based on a course of lectures given at Stanford, and cover three major topics relevant to optimization theory. First an introduction is given to those results in mathematical programming which appear to be most important for the development and analysis of practical algorithms. Next unconstrained optimization problems are considered. The main emphasis is on that subclass of descent methods which (a) requires the evaluation of first derivatives of the objective function, and (b) has a family connection with the conjugate direction methods. Numerical results obtained using a program based on this material are discussed in an Appendix. In the third section, penalty and barrier function methods for mathematical programming problems are studied in some detail, and possible methods for accelerating their convergence indicated.

Journal ArticleDOI
TL;DR: In this article, the problem of probabilistic minimum weight limit design is formulated as a stochastic programming problem and an algorithm is proposed and illustrated for the case of pin-jointed structures with two random continuous variables.
Abstract: The problem of probabilistic minimum weight limit design is formulated as a stochastic programming problem. The theoretical treatment considers monodimensional structures, geometrical parameters and integer design variables. An algorithm is proposed and illustrated for the case of pin-jointed structures with two random continuous variables. The paper follows previous researches of the Authors in the field of probabilistic approach to structural safety [29 ÷ 33].

01 Apr 1972
TL;DR: Recent developments in chance-constrained programming both published, unpublished and new are presented in a framework which unifies the many variants of probabilistic programming under the rubric of chance- constrained programming.
Abstract: : Selected recent developments in chance-constrained programming both published, unpublished and new, are presented in a framework which unifies the many variants of probabilistic programming under the rubric of chance- constrained programming. Developments presented range from two stage linear programming under uncertainty, through acceptance region theory, and cross chance-constrained games.

Journal ArticleDOI
01 Feb 1972-Infor
TL;DR: In this article, the problem of making inventory decisions when faced with a highly seasonal demand pattern is an important managerial problem and most existing theoretical formulations cannot be easily adapted to handle this kind of situation in practice.
Abstract: The problem of making inventory decisions when faced with a highly seasonal demand pattern is an important managerial problem Most existing theoretical formulations cannot be easily adapted to handle this kind of situation in practice While it is possible to compute optimal (Ss, S) type policies using stochastic dynamic programming, such an approach is generally too costly in practice because of the large amount of computer time required per product In addition, such a dynamic program must be rerun in its entirety each time the seasonal pattern, cost structure, or forecasts change In this paper, we propose and compare heuristic methods that are both operationally practicable and computationally feasible and which at the same time yield near-optimal results

Journal Article
TL;DR: The use of discrete conditional dependency matrices as input to stochastic decision models is examined and a new transform is introduced that should produce reasonable results when the record of stream flow (data) has a highly skewed distribution.
Abstract: The use of discrete conditional dependency matrices as input to stochastic decision models is examined. Some of the problems and initial assumptions involved with the construction of the above mentioned matrices are discussed. Covered in considerable detail is the transform used to relate the gamma space with the normal space. A new transform is introduced that should produce reasonable results when the record of stream flow (data) has a highly skewed distribution. Finally, the possibility of using the matrices to provide realistic inputs to a stochastic dynamic program is discussed.

Journal ArticleDOI
TL;DR: In this paper, the authors consider an irrigation planning problem and illustrate how more and more refined variants of this problem are successively cast into stochastic programming with recourse forms, and present an outline of the state of the art with method limitations and demands on model formulation clearly indicated.
Abstract: Recent progress in operations research has refined stochastic programming with recourse sufficiently to significantly increase its potential for use in water resource planning. To demonstrate its strengths and weaknesses this paper considers an irrigation planning problem and illustrates how more and more refined variants of this problem are successively cast into stochastic programming with recourse forms. The result is an outline of the state of the art with method limitations and demands on model formulation clearly indicated.