Showing papers on "Extreme point published in 1973"

A revised simplex method for linear multiple objective programs

Abstract: For linear multiple-objective problems, a necessary and sufficient condition for a point to be efficient is employed in the development of a revised simplex algorithm for the enumeration of the set of efficient extreme points. Five options within this algorithm were tested on a variety of problems. Results of these tests provide indications for effective use of the algorithm.

Global Controllability of Linear Systems with Positive Controls

Abstract: A linear autonomous differential control system is considered, where the zero control is an extreme point of the restraint set. Necessary and sufficient conditions are given for global controllability in the case of bounded or unbounded scalar control.

Majority Rule Under Transitivity Constraints

Abstract: In this paper we are concerned with imposing constraints directly on the admissible majority decisions so as to insure transitivity without restricting individual preference orderings. We demonstrate that this corresponds to requiring that majority decisions be confined to the extreme points of a convex polyhedron. Thus, transitive majority decisions can be characterized as basic solutions of a set of linear inequalities. Through the use of a majority decision function (which is not restricted to be linear) it is shown that constrained majority rule is equivalent to an integer programming problem. Some special forms of majority decision functions are studied including the generalized lp norm and an indicator function. Implications of an integer programming solution, including alternate optima and post optimality analysis, are also discussed.

A Class of Deterministic Production Planning Problems

Abstract: A deterministic production planning problem with limited backlogging, inventory and production capacity constraints is considered. The model also includes a certain type of production cost function which is neither convex nor concave. A characterization of the extreme points is provided and an algorithm is suggested for the problem with equal production capacities. It is also shown how initial inventory can easily be incorporated into the model.

The Generalized Lattice-Point Problem

Abstract: The generalized lattice-point problem, posed by Charnes and studied by M. J. L. Kirby, H. Love, and others, is a linear program whose solutions are constrained to be extreme points of a specified polytope. We show how to exploit this and more general problems by convexity or intersection cut strategies without resorting to standard problem-augmenting techniques such as introducing 0-1 variables. In addition, we show how to circumvent "degeneracy" difficulties inherent in this problem without relying on perturbation which provides uselessly shallow cuts by identifying nondegenerate subregions relative to which cuts may be defined effectively. Finally, we give results that make it possible to obtain strengthening cuts for problems with special structures.

Extensions of the Augmented Predecessor Index Method to Generalized Network Problems

Abstract: : The augmented predecessor indexing method is a procedure for efficiently updating the basis representation, flows and node potentials in an adjacent extreme point (or 'simplex' type) method for network problems, using ideas due to Ellis Johnson in his proposed application of a triple-label representation to networks. The procedure is extended here to accommodate the more complex basis structures and updating processes of the generalized network problem, specifying rules for expediting the calculations.

Strong-Cut Enumerative procedure for Extreme point Mathematical Programming Problems

Abstract: In this paper a “Strong-Cut Enumerative Procedure” for solving Extreme Point Mathematical Programming Problem:

On exposed points of the range of a vector measure

Abstract: Publisher Summary This chapter discusses the extremal structure of the range of a controlled vector measure v with values in a Hausdorff locally convex space X over the field of reals. The chapter also presents the determination of extreme points of a closed convex null of the range of v in terms of levels of the weak integral map. These points are strongly extreme (denting) when Hausdorff locally convex space is quasi-complete. The chapter presents a theorem that relates the existence of an exposed point of the range with the existence of a continuous linear functional on Hausdorff locally convex space for which the signed measure is equivalent to controlled vector measure. The range of v, its weak closure, and the closed convex hull have the same exposed points, which in turn are strongly exposed.

Attainable sets in infinite-dimensional spaces

Abstract: For every t ~ T, a set ~(t) is given in the product of countably many copies of the real line representing the restriction on choice of controls at the instant t, i.e., fi are to be chosen so that (fi(t)) belongs to ~-(t), for every t e T. A question arises about conditions to be imposed on measures m, and on t ~ ~(t), t e T, to assure that (1) has sense for every (reasonable) choice of (fi(t)) in ~(t), t e T. Moreover, considering the set of all values of (1) subject to a given restriction, what can one say about its extreme points, and about the controls for which the extreme points are reached. Here we give one answer to these questions posed to us by C. Otech. In the paper [5] C. Olech introduced the concept of extremal solution of a control system and has shown its relevance to problems of optimal control, especially problems of uniqueness. The results of the present note include contributions to the programme of extending Olech's results to systems in infinite-dimensional spaces, say to systems governed by partial differential equations. 1. Results. Let X be a quasi-complete locally convex topological vector space (1.c.t.v.s.) with dual X'. For a set W = X, coW and ~'-6W denote the convex hull and the closed convex hull of W, respectively. If W is convex, exW

A characterization of convex surfaces which are $L$-sets

Abstract: If the surface of a bounded three dimensional convex body has the property that each pair of its points can see some third point via the surface, then with a single exception the body must be a finite cone with a convex base. The exceptional shape is that of a solid hexahedron with six triangular plane faces formed as the union of two tetrahedra having a congruent face in common. A set S in a linear space is an L-set if each pair of its points can see some point of S via S. In other words, if x E S, y 0 S, there exists a point z E S (which may vary with x and y) such that xzc S, yzc S. Although much is known about L-sets in the plane E2 ([1], [2], [3], [4], [5], [6]), relatively little is known about L-sets in Euclidean n-space El when n_3. However, here we have succeeded in characterizing L-surfaces (surfaces which are L-sets) which are boundaries of compact convex bodies in E3. In order to observe intuitively the simple geometric nature of this situation, duality and elementary graph theory reveal that the following concepts are useful. DEFINITION 1. A face of a convex body B is the intersection of B with one of its planes of support. An exposed point x of B is a point which is also a face of B. (Hence, there exists a plane of support H of B such that H r)B=x.) Let exp B be the set of exposed points of B. REMARK. In the following treatment we prefer to use "exposed points" instead of "extreme points" since an "exposed point" is always a minimal face whereas an "extreme point" e of B (i.e., B-e is convex) is not necessarily even a face, let alone minimal. A minimal face is, of course, a face which does not contain a smaller face, and a corresponding definition holds for "maximal face". The following elementary theorem shows the graph theoretic nature of L-surfaces. THEOREM 1. Suppose S is the surface of a compact convex set B with nonempty interior in Euclidean n-space E'. Received by the editors July 16, 1971 and, in revised form, September 13, 1972. AMS (MOS) subject classifications (1970). Primary 52A15, 52A25.

The Equivalence of Integer Programs to Constrained Recursive Systems

Abstract: It has been established that integer programs with integral, recursive constraint matrices have integral extreme points for integral, nonnegative requirement vectors. In this paper, it is first shown that the assignment problem can be transformed to such a program, but with upper bounds on some variables. This equivalent program leads to additional results for solving the assignment problem. Algorithms are then delineated to transform matching, covering and travelling salesman problems to equivalent recursive programs of the aforementioned type, but with bounds on some variables. These results are extended to programs with special structure in their constraint matrices.

Convexity in Phase Behavior

Abstract: The study of laminar flow of multicomponent hydrocarbon mixtures in porous media requires an understanding of the qualitative properties of phase behavior relationships in a mathe- matical sense not required in ordinary chemical calculations. If $K \in E^n $ represents a proposed equilibrium ratio vector for an n-component mixture, let $\Omega ( K ) \subset E^n $ represent the set of possible $Z \in E^n $ such that Z and K describe a mathematically feasible chemical system in the sense that Z may be a mixture mole fraction vector consistent with K. $\Omega ( K )$ is shown to be a convex set in $E^n $ and its extreme points are calculated in terms of all the possible binary systems described by K.