Showing papers by "Abraham Charnes published in 1965"

On representations of semi-infinite programs which have no duality gaps.

Abstract: Duality gaps which may occur in semi-infinite programs are shown to be interpretable as a phenomenon of an improper representation of the constraint set, uTPi ≧ ci, i e I. Thus, any semi-infinite system of linear inequalities has a canonically closed equivalent (with interior points) which has no duality gap. With respect to the original system of inequalities, duality gaps may be closed by adjoining additional linear inequalities to the original system. Also, for consistent, but not necessarily canonically closed programs, a partial regularization of original data removes duality gaps that may occur. In contrast, a new “weakly consistent” duality theorem without duality gap may have a value determined by an inequality which is strictly redundant with respect to the constraint set defined by the total inequality system.

Constrained Generalized Medians and Hypermedians as Deterministic Equivalents for Two-Stage Linear Programs under Uncertainty

Abstract: In linear programming under uncertainty the two-stage problem is handled by assuming that one chooses a first set of constrained decision variables; this is followed by observations of certain random variables after which another set of decisions must be made to adjust for any constraint violations. The objective is to optimize an expected value functional defined relative to the indicated choices. This paper shows how such problems may always be replaced with either constrained generalized medians or hypermedians in which all random elements appear only in the functional. The resulting problem is called a deterministic equivalent for the original problem since (a) the originally defined objective replaces all random variables by corresponding expected values and (b) the remaining constraints do not contain any random terms. Significant classes of cases are singled out and special attention is devoted to the structure of the constraint matrices for these purposes. Numerical examples are supplied and relat...

Some Special P-Models in Chance-Constrained Programming

Abstract: This paper establishes sufficient conditions for decision rules to be optimal for two n-penod P-models of chance-constrained programming. The models considered are the triangular model with total probability constraints and the block triangular model with conditional probability constraints. It is shown that, for both these models, a sufficient condition for optimality is that the decision rule be the optimal piecewise linear rule. Proofs of theorems are based on results contained in earlier papers by the authors, on n-penod E-models of chance-constrained programming.

Structural sensitivity analysis in linear programming and an exact product from left inverse

Abstract: : This paper shows how to obtain an exact inverse in the form of a product modification to a given inverse when the matrix has been altered additively by a matrix of a certain class. Explicit formulae are derived for sensitivity analyses, e.g., in linear programming, wherein the elements of the structural matrix are to be varied.

Optimal decision rules for the e model of chance-constrained programming

Abstract: : The first five sections of this paper contain an introduction to the topic of chance-constrained programming. Then the general n-period expectation- objective model of chance-constrained programming is presented and certain necessary conditions are established for decision rules to be optimal for such a model. The question of the consistency of the constraints and the finiteness of the optimal value of the objective function for such problems is discussed and several methods of resolving these questions are presented. The simplification that results when the chance-constrained problem is treated as a problem of linear programming under uncertainty is also discussed. The paper is concluded by solving two two-stage problems.

Modular Design, Generalized Inverses, and Convex Programming

Abstract: It is shown that the modular design problem \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\begin{array}{[email protected]{\quad}[email protected]{\quad}l}\mbox{minimize} & \sum^{i=m}_{i=1}y_{i} \sum^{j=n}_{j=1} z_{j},\\ \mbox{subject to} & y_{i}z_{j}\geq r_{ij},\\ & y_{i}z_{j} > 0, & i=1, \ldots, m,\ j=1, \ldots, n\end{array}$$ \end{document} can be transformed into a problem of minimizing a separable convex function subject to linear equality constraints and nonnegativities. This transformation is effected by using a generalized inverse of the constraint matrix. Moreover the nature of the functional and the constraints of the separable problem are such that a good starting point for its solution can be obtained by solving a particular transportation problem. Several possible methods for solving the separable problem are discussed, and the results of our computational experience with these methods are given. It is also shown that the modular design problem can be viewed as a special case of a large class of general engineering design problems that have been discussed in the literature.

Extreme point solutions in mathematical programming: an opposite sign algorithm

Abstract: : Several important and efficient methods of solution of specific types of linear programming problems have the feature of sometimes providing optimal solutions which are not extreme-point (or basic) solutions, so that important and useful analyses provided by knowledge of the optimal dual evaluators are not available. It is also often desirable to be able to begin with a solution suggested by knowledgeable persons with experience in the field (or other considerations) and to proceed immediately to a basic solution at least as good as the suggested one. In this paper it is shown how part of the technique of proof of the opposite sign theorem can be employed in a simple algorithm to achieve this end. This method is equally valid when maximizing a nonlinear but convex objective function. A tested ALGOL code is provided for executing the algorithm in a manner compatible (as a procedure) with other programs.

Simulation and Analyses of a Refuse Collection System

Abstract: A mathematical simulation analysis of refuse collection and disposal systems is presented. A computer program capable of considering stochastic variations in the system parameters has been written for the daily route method of refuse collection. Initial computations serve to delineate the relationships of the several performance parameters (over-all collection efficiency, length of workday, etc.) to the characteristics of the system (average daily quantity of refuse, daily variability in the quantity of refuse, truck capacity, etc.). Results are presented in a series of nineteen graphs. Useful extensions of this work are briefly considered.

Application of chance-constrained programming to solution of the so- called 'savings and loan association' type of problem

Abstract: : This paper contains an application of chance-constrained programming to a problem in financial planning. In particular the problem is one of planning for liquidity in a savings and loan association first discussed by Charnes and Thore. The optimal rules for this problem are found and compared with the optimal linear rules given by Charnes and Thore. The discontinuous nature of the optimal rules is discussed from economic and control theory viewpoints.

Letter to the Editor-A Note on the Fail-Safe Properties of the Generalized Lagrange Multiplier Method

Abstract: This paper shows that the Lagrange multiplier method suggested by H. Everett does not have "Fail-Safe" properties when applied to simple situations with nonconvexity only in the functional. Although each Lagrange multiplier problem for these situations has immediate solutions no means of combining them to obtain a solution to any reasonable approximation appears to be visible.

Properties of a generalized inverse with applications to linear programming theory

Abstract: : In the first five sections of this paper various properties of a Rao generalized inverse of a matrix are established. A method of computing such an inverse is also given. In order to illustrate the differences between the Rao and other generalized inverses, a survey of results on Penrose-Moore inverses is included. The last three sections are devoted to showing how a generalized inverse can be used in the theoretical development of the simplex and modified simplex methods of linear programming. In particular, it is shown that the fundamental equations and iteration formulas of these methods can be derived using matrix notation without requiring the assumption that the linear programming problem has no redundant constraints.