Showing papers by "Abraham Charnes published in 1962"
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TL;DR: In this paper, a ship-chartering problem is considered in which independent normally distributed deviates from the (known) average demands may occur in any of the periods to be considered, and the use of the resulting duality relations for evaluating risk and quality levels of planned performance is examined.
Abstract: Linear programming as a problem in optimizing a linear functional subject to linear inequality constraints is first discussed along with possible uses of the resulting duality relations. Various approaches to dealing with such problems when parts of the data are subject to error are briefly reviewed. Chance constrained programming refers to the class of such cases in which contraint violations are admissible up to pre- assigned probability levels. This topic is elaborated in the context of a ship-chartering problem in which independent normally distributed deviates from the (known) average demands may occur in any of the periods to be considered. Certain additional assumptions (including the use of a specified class of linear decision rules) make it possible to effect reductions first to a nonlinear (mathematical) programming and then to a linear programming problem. The use of the resulting duality relations for evaluating risk and quality levels of planned performance are then briefly examined.
190 citations
TL;DR: Using Theorem 0 and the fact that H*(X, E*) may be written in terms of Lie algebra cohomology, Theorem 1 is completed using several spectral sequences in Lie algebra Cohomology.
Abstract: THEOREM 0. Let En -En X be a homogeneous line bundle over a C-space of complex dimension n. Suppose that the first Chern class ci(Eo) is given by a negative semi-definite quadratic form of index k < n. Then HI(X, E*) = 0 for q < k. (If k = n, we have again Kodaira's theorem.) To apply Theorem 0 to C-spaces, a fairly extensive study of the differential geometry of homogeneous vector bundles is useful; these results may be of independent interest. The reason is that the Atiyah construction of the Chern classes in terms of forms does not work in the non-Kahler case and so one must use a curvature tensor in order to construct the forms. Using Theorem 0 and the fact that H*(X, E*) may be written in terms of Lie algebra cohomology, Theorem 1 is completed using several spectral sequences in Lie algebra cohomology. The Leray spectral sequence used by Bott does not seem to give the complete information here. As mentioned above, the details of the proofs together with other results and applications will appear later.
82 citations
TL;DR: The July, I960, issue of Management Science contains an English translation of an important original article by L. V. Kantorovich as discussed by the authors, "Mathematical Methods of Organizing and Planning Production".
Abstract: The July, I960, issue of Management Science contains an English translation of an important original article by L. V. Kantorovich [Kantorovich, L. V. Mathematical Methods of Organizing and Planning Production. Leningrad University, 1939, with a Foreword by A. R. Marchenko (Russian). An English translation, prepared by R. W. Campbell and W. H. Marlow, appears under this same title in Management Science Vol. 6, No. 4, (July, 1960), pp. 366–422.] and an introductory note by T. C. Koopmans [Koopmans, T. C. 1960. A note about Kantorovich's paper, ‘Mathematical Methods of Organizing and Planning Production'. Management Sci. 6 (4, July) 363–365.] which is both evaluative and explanatory. In his penultimate paragraph, Professor Koopmans accords a tribute to this writing of Kantorovich which is at least partly deserved, although it is also paid for, we think, at too high a price in modesty for Professor Koopman's own work as well as the work of others. There is a certain lack of precision at critical points in the...
22 citations
TL;DR: In this article, it is shown that a bounded solution set in one problem implies an unbounded solution set for another problem, unless both are one-point sets, and the ideas of projection equivalence are then developed to suggest a possible route for utilizing these onepoint solution properties for analyzing or solving linear programming problems.
Abstract: Model equivalences may sometimes be used to replace “realistic” but unwieldy initial formulations with simpler counterparts. This can involve sophisticated uses of prototypes, quasi models, etc., or it may involve only simpler ideas of redundancy elimination, removal of extraneous variables, etc. In either case questions can arise concerning the properties of these models when further analyses are to be conducted via parameterizations, duality, etc. These topics are examined in the general context of direct and dual linear programming problems with special reference to boundedness properties of the associated solution sets. It is shown that a bounded solution set in one problem implies an unbounded solution set in the dual problem, unless both are one-point sets. The ideas of projection equivalence are then developed to suggest a possible route for utilizing these one-point solution properties for analyzing or solving linear programming problems. These possibilities might prove useful when, for example, i...
18 citations
TL;DR: In this article, a dual theory of convex programming with maximal finite algebra and minimal topology is presented, which includes a dual theorem covering the most general convex program situation (e.g. no differentiability conditions qualifying the convex function or constraints, or homogeneity).
Abstract: The existence of a solution to the problem of minimizing a convex function subject to restriction of the variables to a closed convex set in w-space (\"convex programming\") has been characterized (for suitable differentiability conditions) by the Kuhn-Tucker theorem [5]. In general, no dual programming problem (not involving the variables of the direct problem) has been associated with this situation except in the linear programming case, and very recently by E. Eisenberg in [3], for homogeneity of order one in the function and linear inequality constraints, and by R. J. Duffin [2] in an inverse manner for a highly specialized problem. Starting with a little known paper of A. Haar [4] in the light of current linear programming constructs (e.g., \"regularization\" [ l ] ) , we effect a generalization of these ideas (with maximal finite algebra and minimal topology) so that a dual theory practically as straightforward as linear programming theory is obtained, and which includes a dual theorem covering the most general convex programming situation (e.g. no differentiability conditions qualifying the convex function or constraints, or homogeneity, etc.). This general theorem is made possible by associating a suitably restricted, usually infinite-dimensional space problem with the minimization problem in w-space instead of the usual association of another finite w-space problem. The space we use is a \"generalized finite sequence space\" (g.f.s.s.), defined with respect to an index set / of arbitrary cardinality as the vector space, S> of all vectors X— [Xt: i G / ] over an ordered field F with only finitely many nonzero entries. Such spaces possess the following key characteristics for linear programming of ordinary w-spaces. Let F be a vector space over F and consider a collection of vectors: PQ, P*: i £ I in V. Let R be the subspace spanned by these vectors, and let
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