Showing papers in "Siam Journal on Applied Mathematics in 1967"
TL;DR: An algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints) is given.
Abstract: This paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.
1,264 citations
679 citations
TL;DR: In this paper, the authors extend the theory of stochastic programs with recourse to the general case when essentially all the parameters involved are random, and show that without restriction, the equivalent deterministic form of a deterministic 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.
Abstract: : 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 right-hand 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 W-condition. 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)
176 citations
TL;DR: A formalism within which the application of dynamic programming to discrete, deterministic problems is rigorously studied is developed, and the representations of discrete decision processes by monotone sequential decision processes are characterized.
Abstract: This paper develops a formalism within which the application of dynamic programming to discrete, deterministic problems is rigorously studied. The two central concepts underlying this development are discrete decision process and sequential decision process. Discrete decision processes provide a convenient means of problem statement, while monotone sequential decision processes (which are finite automata with a certain cost structure superimposed) correspond naturally to dynamic programming algorithms. The representations of discrete decision processes by monotone sequential decision processes are characterized, and this characterization is used in the deviation of dynamic programming algorithms for a variety of problems.
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TL;DR: In this paper, the Lagrange multipliers and Kuhn-Tucker conditions are extended to the case where the problem functions are required to have continuous first derivatives, and the necessary and sufficient conditions are established for mixed inequality and equality problems.
Abstract: This paper establishes two sets of "second order" conditions-one which is necessary, the other which is sufficient-in order that a vector x* be a local minimum to the constrained optimization problem: minimize f(x) subject to the constraints \( g_{i}(x)\geqq 0,i=1,\cdots ,m,\; \rm{and} \; h_{i}(x)=0,j=1,\cdots,p, \) where the problem functions are twice continuously differentiable. The necessary conditions extend the well-known results, obtained with Lagrange multipliers, which apply to equality constrained optimization problems, and the Kuhn-Tucker conditions, which apply to mixed inequality and equality problems when the problem functions are required only to have continuous first derivatives. The sufficient conditions extend similar conditions which have been developed only for equality constrained problems. Examples of the applications of these sets of conditions are given.
87 citations
81 citations
TL;DR: The authors derive a procedure for the players which, if abided by, leads to an outcome in the kernel of a cooperative game, and the results that are obtained reduce considerably the amount of computation which is needed to compute the kernels of such games.
Abstract: : A study of the kernel of a cooperative game. In this paper the authors derive a procedure for the players which, if abided by, leads to an outcome in the kernel. Moreover, each outcome in the kernel can be reached by this procedure. The procedure consists of a set of three rules and involves the formation of 'intermediate coalitions' which play 'intermediate games,' after which the members of each intermediate coalition play a 'reduced game' to decide the share of their spoils. The procedure is further analyzed in the case of monotonic games and in the case of simple games, and the results that are obtained reduce considerably the amount of computation which is needed to compute the kernels of such games. In particular, they compute the kernel of the 7-person projective game (for the grand coalition), which is a star consisting of seven straight-line segments connecting the center to the points of the main simple solution. Finally, conditions under which modifications of the characteristic function do not change its kernel are presented. (Author)
TL;DR: In this article, instead of the distributions of incremental crack extension having increasing failure rate, only the expected residual damage increment either in crack initiation or extension, given the damage exceeds a preassigned amount, is less than the damage increment which was expected for that load fluctuation before it was imposed.
Abstract: Miner’s rule for the cumulative damage due to fatigue, a deterministic formula which is well known in engineering practice, was examined earlier from a probabilistic point of view with Birnbaum in [2]. Here the assumptions of that model are weakened. Previously the basic assumptions were that crack growth was stochastic in nature with incremental extensions having a distribution with increasing failure rate, and that the spectrum of load fluctuations was fixed and then repeated under program. We now assume, instead of the distributions of incremental crack extension having increasing failure rate, only that for a given load fluctuation, the expected residual damage increment either in crack initiation or extension, given the damage exceeds a preassigned amount, is less than the damage increment which was expected for that load fluctuation before it was imposed. We also weaken the assumptions concerning the type of loading spectra which are admitted, considering the case of random load fluctuations which a...
TL;DR: The method is a generalization of the “selected points” method described by C. Lanczos for solving ordinary differential equations, and it is applied in detail to a particular example, the eigenvalue problem for a vibrating L-shaped membrane.
Abstract: In this paper a new method is proposed for the solution of partial differential equations in terms of two-dimensional Chebyshev series. The method is a generalization of the “selected points” method described by C. Lanczos [3] for solving ordinary differential equations. Some indications are given of the possible applications of the method, and it is applied in detail to a particular example, the eigenvalue problem for a vibrating L-shaped membrane. The work involved in computation is much the same as for finite difference methods of solving the problem, but our method seems to be simpler to apply and eigenfunctions are obtained in a convenient form.
TL;DR: In this article, the authors studied the problem of optimal control of a non-stochastic process over time in a convex space of infinite sequences of real numbers, and proposed a generalization of the discount factor to stationary utility functions.
Abstract: : This paper is concerned with a problem in the optimal control of a nonstochastic process over time. It can also be looked on as a problem in convex programming in a space of infinite sequences of real numbers. The literature on optimal economic growth contains several papers in which a utility function of the form (1) U(x1,x2,...) = Summation, t=1 to t=infinity, of alpha (superscript(t-1)) u(x sub t), Oalpha1, is maximized under given conditions of technology and population growth. Here xt is per capita consumption in period t, and u(x) is a strictly concave, increasing, single-period utility function. Alpha is called a discount factor. A generalization of (1) has been proposed under the name stationary utility, and is definable by a recursive relation (2) U(x1, x2, x3,...) = V(x1, U(x2, x3,...)). One obtains (1) by V(x, U) = u(x) + alpha U. The natural generalization of alpha in (1) to stationary utility is the function (2a) alpha(x) = (the partial derivative of V(x,U) with respect to U) subscript U = U(x,x,x,...). In this paper we study the maximization of (2) under production assumptions.
TL;DR: In this paper, an algorithm for solving the problem of minimizing a convex function subject to the constraint that the function is a concave function, and that the convex functions can be minimized over all x, nonnegative t, for a strictly decreasing null sequence of k > 0, is presented.
Abstract: An algorithm for solving the problem: minimize $f( x )$ (a convex function) subject to $g_i ( x )\geqq 0$, $i = 1, \cdots ,m$, each $g_i $ a concave function, is presented. Specifically, the function \[ P\left[ {x,t,r_k } \right] \equiv f( x ) + r_k^{ - 1} \sum {\left[ {g_i ( x ) - t_i } \right]} ^2 \] is minimized over all x, nonnegative t, for a strictly decreasing null sequence $\{ {r_k } \}$. This extends the work of T. Pietrzykowski [5]. It is proved that for every $r_k > 0$, there exists a finite point $[ {x( {r_k } ),t( {r_k } )} ]$ which minimizes P, and which solves the convex programming problem as $r_k \to 0$. This algorithm is similar to the Sequential Unconstrained Minimization Technique (SUMT) [1] in that it solves the (Wolfe) dual programming problem [6]. It differs from SUMT in that (1) it approaches the optimum from the region of infeasibility (i.e., it is a relaxation technique), (2) it does not require a nonempty interior to the nonlinearly constrained region, (3) no separate feasibilit...
TL;DR: The Laplace transform of the characteristic function of the integral is obtained in a general form by use of matrix notation in this article for a stationary semi-Markov process and in the case of a stationary S.M.P.
Abstract: Consider an m-state, irreducible, recurrent semi-Markov process (S.M.P.) and a step function $f( \cdot )$ which takes on the value $v_i $, $i = 1, \cdots $, m, when the S.M.P. is in state i. We study the integral of $f( \cdot )$ between 0 and t.The Laplace transform of the characteristic function of the integral is obtained in a general form by use of matrix notation. In the case of a stationary semi-Markov process the transform of the expected value of the integral is inverted in closed form. Asymptotic properties of the expected value of the integral are derived by applying “Smith’s Key Renewal Theorem”.
TL;DR: In this article, a numerical approach for the solution of the Orr-Sommerfeld equations is presented, where the form in which the results are to be presented makes the actual computation into an inner-outer type iteration scheme, and the appropriate balancing of these iterations leads to a considerable saving in computer time.
Abstract: Numerical methods for the solution of the Orr-Sommerfeld equations are considered. The problem is stated in §1, and the basic numerical tools (the setting up of the finite difference approximations, and the algorithm for solving the resulting algebraic eigenvalue problem) are developed in §§2 and 3. In §4 it is shown that the form in which the results are to be presented makes the actual computation into an inner-outer type iteration scheme, and that the appropriate balancing of these iterations leads to a considerable saving in computer time. Some questions bearing on the stability of the calculations are considered in §5.
TL;DR: Bremermann as mentioned in this paper showed that the Cauchy representation of a complex variable can be represented by analytic functions of one complex variable, and that analytic functions can be used to represent a distribution in the topology of O'L2.
Abstract: H. J. BREMERMANNt 1. Representation of distributions by analytic functions of one complex variable. This section is a brief summary; most of the proofs may be found in Bremermann [5]. Compare also Beltrami-Wohlers [1]. Terminology and notations are consistent with L. Schwartz [12]. Let T be a distribution in (&'(E1)), where E1 denotes the reals, or T C (O'L2); then the function T'(z) = (1/2ri)(T, 1/t -z) is an analytic function for Im z # 0 (Im z imaginary part of z). T(z) represents T in the following sense: T(x + iE) T(x iE) converges to T for E -> 0+ in the topology of (O'). We call T(z) the Cauchy representation of T. For T C (D'(El)) the function (t z)f1 is not a test function, T(z), in general, does not exist, but the following result holds: there exists a pair of functions f+(z) and f_(z), analytic in the upper and lower half-plane, respectively, such thatf+(x + iE) f(x ie) converges to T for e -> 0+ in the topology of (OD'). We call f+ the forward function, f_ the backward function and the pair an analytic representation of T. Analytic representations of the same distribution differ by at most an eintire function [14], [24]. If a pair of analytic functions represents T, then their complex derivatives represent T'. If the complement of the support of a distribution is nonempty, then for any analytic representation of T the forward and backward functions f+ and f_ are analytic continuations of each other and remain analytic on the complement of the support of T. Not every pair of functions, analytic in the upper and lower half-planes, respectively, represents a distribution: f+(z) = f_(z) = exp (1/z2) is a counter-example. A tempered function is a continuous complex-valued function oni El that grows at infinity no faster than a polynomial. For a tempered function f
TL;DR: In this article, the number of ways that an art judge can rank n paintings, each having d numerical attributes, by forming weighted averages of the attributes was found, where nSk iS is the sum of the n-2Ck = (n 2)!/(n 2 k)! k! possible products of numbers taken k at a time without repetition from the set {2, 3, *,n-1}.
Abstract: where nSk iS the sum of the n-2Ck = (n 2) !/(n 2 k)! k! possible products of numbers taken k at a time without repetition from the set {2, 3, *,n-1}. Thus we have found Q(n, d), the number of ways that an art judge can rank n paintings, each having d numerical attributes, by forming weighted averages of the attributes. Our interest in this problem stems from work [1], [2], [3] on classification of vector-valued patterns by means of linear discriminants. Notice that the number of linearly inducible orderings is independent of configuration (up to general position). Two examples, however, will show
TL;DR: In this paper, a conjecture made in [8] regarding the relationship between the p-dimensional biased Rayleigh distribution and the $( {p + 1} )$-dimensional unbiased Rayleigh distributions is proved.
Abstract: A conjecture made in [8] regarding the relationship between the p-dimensional biased Rayleigh distribution and the $( {p + 1} )$-dimensional unbiased Rayleigh distribution is proved. Other results relating biased and unbiased Rayleigh distributions of various dimensions are also proved. Some new explicit formulas (in terms of higher transcendental functions) are obtained for the four-dimensional Rayleigh frequency function.