Walkup, David W.
Overview
Works:  19 works in 20 publications in 1 language and 22 library holdings 

Genres:  Short stories Fiction 
Roles:  Author 
Publication Timeline
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Most widely held works by
David W Walkup
Thanksgiving again by David W Walkup(
Book
)
1 edition published in 2005 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2005 in English and held by 3 WorldCat member libraries worldwide
STOCHASTIC PROGRAMS WITH RECOURSE by David W Walkup(
Book
)
2 editions published between 1966 and 1967 in English and held by 2 WorldCat member libraries worldwide
So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the righthand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious Wcondition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)
2 editions published between 1966 and 1967 in English and held by 2 WorldCat member libraries worldwide
So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the righthand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious Wcondition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)
A MISSILE TARGETING PROBLEM(
Book
)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
The following missile assignment problem is considered. Missiles are to be assigned to targets in two distinct steps. First, each missile is programmed so that it can be fired at any one of a small number of targets, the number of targets being the missile capability. The programming of the missiles is represented by a qualification matrix Q. Second, if battle occurs, all missiles are to be assigned to targets and launched. Each missile must be assigned to a target for which it is programmed. It is assumed that only a random subset X of the missiles will actually be available for battle, and so the assignment must be made for a reduced qualification matrix Q(X). The questions considered are 'what is an optimal assignment given the reduced qualification matrix Q(X)., ' and 'what can be expected from this assignment.' Use of a damage function is proposed. An optimal assignment is one which maximizes the value of the damage function. The damage function may be chosen to represent a wide variety of optimization requirements. The main part of the paper describes Monte Carlo procedures for estimating the expected damage and the probability that the damage will be at least c for any number c. (Author)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
The following missile assignment problem is considered. Missiles are to be assigned to targets in two distinct steps. First, each missile is programmed so that it can be fired at any one of a small number of targets, the number of targets being the missile capability. The programming of the missiles is represented by a qualification matrix Q. Second, if battle occurs, all missiles are to be assigned to targets and launched. Each missile must be assigned to a target for which it is programmed. It is assumed that only a random subset X of the missiles will actually be available for battle, and so the assignment must be made for a reduced qualification matrix Q(X). The questions considered are 'what is an optimal assignment given the reduced qualification matrix Q(X)., ' and 'what can be expected from this assignment.' Use of a damage function is proposed. An optimal assignment is one which maximizes the value of the damage function. The damage function may be chosen to represent a wide variety of optimization requirements. The main part of the paper describes Monte Carlo procedures for estimating the expected damage and the probability that the damage will be at least c for any number c. (Author)
A note on decision rules for stochastic programs(
Book
)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
It is shown that a twostage stochastic program with recourse with righthand sides random (i.e., a twostage programming under uncertainty problem) has optimal decision rules which are continuous and piecewise linear. However, this result does not extend to programs with three or more stages. An example is given of a simple inventorytype threestage stochastic program with recourse for which the the optimal secondstage decision rule is not piecewise linear. The example is also recast in the framework of the conditional probability Emodel of chanceconstrained programming showing that the CharnesKirby theorem on the existence of piecewise linear decision rules for such programs is invalid for more than two stages. (Author)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
It is shown that a twostage stochastic program with recourse with righthand sides random (i.e., a twostage programming under uncertainty problem) has optimal decision rules which are continuous and piecewise linear. However, this result does not extend to programs with three or more stages. An example is given of a simple inventorytype threestage stochastic program with recourse for which the the optimal secondstage decision rule is not piecewise linear. The example is also recast in the framework of the conditional probability Emodel of chanceconstrained programming showing that the CharnesKirby theorem on the existence of piecewise linear decision rules for such programs is invalid for more than two stages. (Author)
Stochastic programs with recourse: on the continuity of the objective(
Book
)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
In an earlier paper, 'Stochastic Programs with Recourse, ' the authors introduced a general class of stochastic (linear) programs and showed, among other things, that the objective of any such program is convex when considered as a function of the firststage decision variables. In this paper it is shown that the objective is also lower semicontinuous. In the process of proving this result, a lemma of general interest in the theory of convex functions is established
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
In an earlier paper, 'Stochastic Programs with Recourse, ' the authors introduced a general class of stochastic (linear) programs and showed, among other things, that the objective of any such program is convex when considered as a function of the firststage decision variables. In this paper it is shown that the objective is also lower semicontinuous. In the process of proving this result, a lemma of general interest in the theory of convex functions is established
Some practical regularity conditions for nonlinear programs(
Book
)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
Some practical sufficient conditions are given for a program with linear constraints and nonlinear objective to have wellbehaved duality properties. (Author)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
Some practical sufficient conditions are given for a program with linear constraints and nonlinear objective to have wellbehaved duality properties. (Author)
Minimal interchanges of (0,1)matrices and disjoint circuits in a graph(
Book
)
1 edition published in 1963 in English and held by 1 WorldCat member library worldwide
It is shown that the minimal number of interchanges necessary to transform one (0,1)matrix into another equivalent one may be computed from the maximal number of edgedisjoint circuits in a bipartite graph derived from the difference of the matrices. This partially answers a question raised by Ryser. Two (0,1)matrices are said to be equivalent if their difference has zero row and column sums. They are said to differ by an interchange if they are equivalent and their difference is zero except for a 2x2 minor. (Author)
1 edition published in 1963 in English and held by 1 WorldCat member library worldwide
It is shown that the minimal number of interchanges necessary to transform one (0,1)matrix into another equivalent one may be computed from the maximal number of edgedisjoint circuits in a bipartite graph derived from the difference of the matrices. This partially answers a question raised by Ryser. Two (0,1)matrices are said to be equivalent if their difference has zero row and column sums. They are said to differ by an interchange if they are equivalent and their difference is zero except for a 2x2 minor. (Author)
Continuity of some convexconevalued mappings(
Book
)
1 edition published in 1966 in English and held by 1 WorldCat member library worldwide
The authors consider the class C of closed convex cones in a Hilbert space as a topological space and investigate the resulting topological properties of certain mappings into C. They show that with the proper choice of a Hausdorff metric for C the operation of taking the polar cone is an involutory isometry. The operation is considered of taking the positive hull of a finite set of points as a mapping into C and obtain some sufficient conditions for the continuity of this mapping. (Author)
1 edition published in 1966 in English and held by 1 WorldCat member library worldwide
The authors consider the class C of closed convex cones in a Hilbert space as a topological space and investigate the resulting topological properties of certain mappings into C. They show that with the proper choice of a Hausdorff metric for C the operation of taking the polar cone is an involutory isometry. The operation is considered of taking the positive hull of a finite set of points as a mapping into C and obtain some sufficient conditions for the continuity of this mapping. (Author)
THE NUMBER OF PLANAR TREES(
Book
)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
By a planar tree is meant a realization of a tree in the plane, and by an isomorphism between two planar trees is meant a mapping which is not only an isomorphism in the usual sense of trees but which also preserves the clockwise cyclic order of edges about each node. Explicit formulae are given for each of the following: (1) the number of nonisomorphic unrooted planar trees with n edges, (2) the number of nonisomorphic rooted planar trees with n edges, and (3) the number of nonisomorphic rooted planar trees with n edges such that the root is incident on exactly k edges, of which one is distinguished. (Author)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
By a planar tree is meant a realization of a tree in the plane, and by an isomorphism between two planar trees is meant a mapping which is not only an isomorphism in the usual sense of trees but which also preserves the clockwise cyclic order of edges about each node. Explicit formulae are given for each of the following: (1) the number of nonisomorphic unrooted planar trees with n edges, (2) the number of nonisomorphic rooted planar trees with n edges, and (3) the number of nonisomorphic rooted planar trees with n edges such that the root is incident on exactly k edges, of which one is distinguished. (Author)
A MULTIPLEASSIGNMENT PROBLEM(
Book
)
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
A generalization of Kuhn's simple assignment problem is considered: There are m men and n tasks given with each man qualified for certain of the tasks. The output from each task is given as a concave function of the number of qualified men assigned to it. Find an assignment of men to task, perhaps more than one man to a task, so as to maximize total output. An algorithm for solving this general problem is given in which transfers like those used by Kuhn on the simple problem are selected using a nodelabeling procedure on a related network. The algorithm yields for every k, 1 <k <m, an optimal assignment of the first k men only, employing a single transfer to increase k by one. Several special forms of the generalized problem are considered including a target assignment problem which A.S. Manne has formulated as a linear program
1 edition published in 1964 in English and held by 1 WorldCat member library worldwide
A generalization of Kuhn's simple assignment problem is considered: There are m men and n tasks given with each man qualified for certain of the tasks. The output from each task is given as a concave function of the number of qualified men assigned to it. Find an assignment of men to task, perhaps more than one man to a task, so as to maximize total output. An algorithm for solving this general problem is given in which transfers like those used by Kuhn on the simple problem are selected using a nodelabeling procedure on a related network. The algorithm yields for every k, 1 <k <m, an optimal assignment of the first k men only, employing a single transfer to increase k by one. Several special forms of the generalized problem are considered including a target assignment problem which A.S. Manne has formulated as a linear program
The lower bound conjecture for 3 and 4manifolds(
Book
)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
The socalled lower bound conjecture for simplicial polytopes asserts that e(P) = or> d.v(P)  d(d+1)/2, where e(P) and v(P) denote respectively the number of edges and vertices of any simplicial dpolytope P, i.e., any closed bounded convex polyhedron of dimension d, all of whose faces are simplices. This paper establishes analogous lower bounds for arbitrary triangulations of closed topological 3 and 4manifolds, including sharp lower bounds for the 3sphere, the 3dimensional analogues of the torus and Klein bottle, projective 3space, and the 4sphere with any number of handles. The results for the 3sphere and 4sphere immediately imply the previously unproven lower bound conjecture for simplicial 4 and 5polytopes. The result for projective 3space has similar implications for centrally symmetric simplicial 4polytopes. (Author)
1 edition published in 1969 in English and held by 1 WorldCat member library worldwide
The socalled lower bound conjecture for simplicial polytopes asserts that e(P) = or> d.v(P)  d(d+1)/2, where e(P) and v(P) denote respectively the number of edges and vertices of any simplicial dpolytope P, i.e., any closed bounded convex polyhedron of dimension d, all of whose faces are simplices. This paper establishes analogous lower bounds for arbitrary triangulations of closed topological 3 and 4manifolds, including sharp lower bounds for the 3sphere, the 3dimensional analogues of the torus and Klein bottle, projective 3space, and the 4sphere with any number of handles. The results for the 3sphere and 4sphere immediately imply the previously unproven lower bound conjecture for simplicial 4 and 5polytopes. The result for projective 3space has similar implications for centrally symmetric simplicial 4polytopes. (Author)
Binomial Convolution Preserves LogConcavity(
Book
)
1 edition published in 1974 in English and held by 1 WorldCat member library worldwide
A sequence f = (f0, f1 ...) of nonnegative numbers is logconcave if ln fi is a concave function of i. If two sequences f and g are both logconcave, then so is their binomial convolution. This complements the similar, wellknown result for ordinary convolution. (Modified author abstract)
1 edition published in 1974 in English and held by 1 WorldCat member library worldwide
A sequence f = (f0, f1 ...) of nonnegative numbers is logconcave if ln fi is a concave function of i. If two sequences f and g are both logconcave, then so is their binomial convolution. This complements the similar, wellknown result for ordinary convolution. (Modified author abstract)
A Probability Bound for the Union of Random Sets(
Book
)
1 edition published in 1974 in English and held by 1 WorldCat member library worldwide
Given a collection of independent random subsets of a finite set N satisfying certain symmetry conditions, there is an easilycomputed estimate for the probability their union is N. It is shown this estimate is a valid upper bound, provided certain functions associated with the distribution of the random size of the subsets have concave logarithms. Examples are given. The results complement known results in the theory of reliability on sums of random variables with increasing failure rates
1 edition published in 1974 in English and held by 1 WorldCat member library worldwide
Given a collection of independent random subsets of a finite set N satisfying certain symmetry conditions, there is an easilycomputed estimate for the probability their union is N. It is shown this estimate is a valid upper bound, provided certain functions associated with the distribution of the random size of the subsets have concave logarithms. Examples are given. The results complement known results in the theory of reliability on sums of random variables with increasing failure rates
Stochastic programs with recourse: special forms(
Book
)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
This paper is a sequel to AD637 139 in which stochastic programs with recourse were formulated as a generalization of the twostage programming under uncertainty model, and some of the theoretical properties of stochastic programs with recourse were developed. In this paper some special forms of stochastic programs with recourse are considered, which, because they are less general, may prove to be more amenable to computational solution. It is also shown that certain problems studied by other authors, including the active approach of G. Tintner and the conditional probability model of chance constrained programming treated by A. Charnes and M. Kirby, can be represented as stochastic programs with recourse. (Author)
1 edition published in 1967 in English and held by 1 WorldCat member library worldwide
This paper is a sequel to AD637 139 in which stochastic programs with recourse were formulated as a generalization of the twostage programming under uncertainty model, and some of the theoretical properties of stochastic programs with recourse were developed. In this paper some special forms of stochastic programs with recourse are considered, which, because they are less general, may prove to be more amenable to computational solution. It is also shown that certain problems studied by other authors, including the active approach of G. Tintner and the conditional probability model of chance constrained programming treated by A. Charnes and M. Kirby, can be represented as stochastic programs with recourse. (Author)
Polya Frequency Sequences and the Exponential Generating Function(
Book
)
1 edition published in 1974 in English and held by 1 WorldCat member library worldwide
A basic result in the theory of total positivity is that the convolution of any two P(F sub 2) sequences (sequences such that the logarithms of their elements are concave functions of the index) is again a P(F sub 2) sequence. This result, with its obvious interpretation in terms of ordinary generating functions is shown to carry over to binomial convolution and the exponential generating function
1 edition published in 1974 in English and held by 1 WorldCat member library worldwide
A basic result in the theory of total positivity is that the convolution of any two P(F sub 2) sequences (sequences such that the logarithms of their elements are concave functions of the index) is again a P(F sub 2) sequence. This result, with its obvious interpretation in terms of ordinary generating functions is shown to carry over to binomial convolution and the exponential generating function
LIFTING PROJECTIONS OF CONVEX POLYHEDRA(
Book
)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
Briefly, if T is a projection of a closed polyhedron P onto a polyhedron Q, then a lifting of Q into P is defined to be a singlevalued inverse T* of T such that T*(Q) is the union of closed faces of P. The main result of this paper, called the Lifting Theorem, asserts that there always exists a lifting T*, provided only that there exists at least one face of P on which T acts onetoone. The Lifting Theorem is seen as a unifying generalization of a number of results in the theory of convex polyhedra and has important applications in the theory of mathematical programming. In the course of proving the Lifting Theorem a result on linear programs of interest in its own right is proven, namely, that the optimal solution of a linear program can be chosen so that it is a continuous function of the righthand sides. (Author)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
Briefly, if T is a projection of a closed polyhedron P onto a polyhedron Q, then a lifting of Q into P is defined to be a singlevalued inverse T* of T such that T*(Q) is the union of closed faces of P. The main result of this paper, called the Lifting Theorem, asserts that there always exists a lifting T*, provided only that there exists at least one face of P on which T acts onetoone. The Lifting Theorem is seen as a unifying generalization of a number of results in the theory of convex polyhedra and has important applications in the theory of mathematical programming. In the course of proving the Lifting Theorem a result on linear programs of interest in its own right is proven, namely, that the optimal solution of a linear program can be chosen so that it is a continuous function of the righthand sides. (Author)
A lipschitzian characterization of convex polyhedra(
Book
)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
The Hausdorff distance between parallel crosssections of a closed convex polyhedron (whether bounded or not) possesses a Lipschitzian property. Moreover, this property characterizes convex polyhedra among the class of closed convex sets. (Author)
1 edition published in 1968 in English and held by 1 WorldCat member library worldwide
The Hausdorff distance between parallel crosssections of a closed convex polyhedron (whether bounded or not) possesses a Lipschitzian property. Moreover, this property characterizes convex polyhedra among the class of closed convex sets. (Author)
Matchings in Random Regular Bipartite Digraphs(
Book
)
1 edition published in 1974 in English and held by 1 WorldCat member library worldwide
Let G be a random directed bipartite graph with n nodes in each class and outward degree d at each node. The probability G contains a matching is shown to approach one for large n if d> or = 2, but to approach zero if d = 1. This result complements and contrasts with a result of Erdos and Renyi which implies the probability of a matching does to zero if the number of arcs (chosen at random without regard to regularity) grows more slowly than n log n
1 edition published in 1974 in English and held by 1 WorldCat member library worldwide
Let G be a random directed bipartite graph with n nodes in each class and outward degree d at each node. The probability G contains a matching is shown to approach one for large n if d> or = 2, but to approach zero if d = 1. This result complements and contrasts with a result of Erdos and Renyi which implies the probability of a matching does to zero if the number of arcs (chosen at random without regard to regularity) grows more slowly than n log n
The dstep conjecture for polyhedra of dimension d<6(
Book
)
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
Two functions A and B, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. A(d, n) is the maximum diameter of convex polyhedra of dimension d with n faces of dimension d1; similarly, B(d, n) is the maximum diameter of bounded polyhedra of dimension d with n faces of dimension d1. The diameter of a polyhedron P is the smallest integer k such that any two vertices of P can be joined by a path of k or fewer edges of P. It is shown that the bounded dstep conjecture, i.e. B(d,2d) = d, is true for d <or 5. It is also shown that the general dstep conjecture, i.e. A(d,2d) <or = d, of significance in linear programming, is false for d equal to or greater than 4. A number of other specific values and bounds for A and B are presented
1 edition published in 1965 in English and held by 1 WorldCat member library worldwide
Two functions A and B, of interest in combinatorial geometry and the theory of linear programming, are defined and studied. A(d, n) is the maximum diameter of convex polyhedra of dimension d with n faces of dimension d1; similarly, B(d, n) is the maximum diameter of bounded polyhedra of dimension d with n faces of dimension d1. The diameter of a polyhedron P is the smallest integer k such that any two vertices of P can be joined by a path of k or fewer edges of P. It is shown that the bounded dstep conjecture, i.e. B(d,2d) = d, is true for d <or 5. It is also shown that the general dstep conjecture, i.e. A(d,2d) <or = d, of significance in linear programming, is false for d equal to or greater than 4. A number of other specific values and bounds for A and B are presented
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