Showing papers in "Optimization in 1978"
TL;DR: In this paper the method of dynamic programming is carried to recurrence computations of sets of optimal values for discrete dynamic systems and vector-valued objective functions.
Abstract: In this paper the method of dynamic programming is carried to recurrence computations of sets of optimal values for discrete dynamic systems and vector-valued objective functions.
32 citations
TL;DR: In this paper, a class of stochastic processes (PMP) with an imbedded stationary point process of "basic points" which must not be renewal points is defined and a relation between stationary distribution of PMP at arbitrary points in time and distributions and intensities connected with the basic points is proved.
Abstract: In generalization of special cases of the literature a class of stochastic processes (PMP) is defined with an imbedded stochastic marked point process of “basic points” which must not be renewal points. A theorem (“intensity conservation principle”) has been proved concerning a relation between stationary distribution of PMP at arbitrary points in time and distributions and intensities connected with the basic points. This relationship simultaneously yields a general method for determination of stationary quantities at arbitrary points in time by means of the corresponding “imbedded” quantities. Some applications to concrete queueing systems have been demonstrated, where arrival or departure epochs of customers are used as basic points. Under weaker independence assumptions as till now done in the literature, new relations are given.
30 citations
TL;DR: In this article, a new extremal approach to deriving dual optimization problems with proper duality inequality is presented, which simplifies, and generalizes, the Fenchel-Rockafellar scheme.
Abstract: This paper presents a new extremal approach to deriving dual optimization problems with proper duality inequality which simplifies, and generalizes the Fenchel-Rockafellar scheme. Our derivation proceeds in two stages, (i) inequality attainment,(ii) decoupling primal and dual variables, The power and convenience of this approach are exhibited through a new, much simpler derivation of the Charnes-Cooper results for Khinchin-Kullback-Leibler statistical estimation [1], the immediate establishment of the C2 duality for general distributions and its extensions to general linear inequality constraints, plus the development of a new two-person zero-sum game connection.
25 citations
TL;DR: In this paper, the authors combine primal and dual decomposition procedures where both a vertical and a horizontal decomposition approach can be applied to the initializing the procedure, which has been shown to be crucial for the convergence of the procedures, Additionally mixed procedures have important interpretations as mixed price and budget planning procedures for decentralized organizations.
Abstract: This paper shows how to combine primal and dual decomposition procedures where both a vertical and a horizontal decomposition approach can be applied. The motivation for using a mixed procedure is that it improves the possibility of efficiently initializing the procedure, which has been shown to be crucial for the convergence of the procedures, Additionally mixed procedures have important interpretations as mixed price and budget planning procedures for decentralized organizations.
21 citations
TL;DR: In this paper, the authors deal with necessary conditions for optimization problems with infinitely many inequality constraints assuming various differentiability conditions, such as Gâteaux-, Hadamaed-and weak differentiability with corresponding tangential cones.
Abstract: This paper deals with necessary conditions for optimization problems with infinitely many inequality constraints assuming various differentiability conditions. By introducing a second topology N on a topological vector space we define generalized versions of differentiability and tangential cones. Different choices of N lead to Gâteaux-, Hadamaed- and weak differentiability with corresponding tangential cones. The general concept is used to derive necessary conditions for local optimal points in form of inequalities and generalized multiplier rules, Special versions of these theorems are obtained for different differentiability assumptions by choosing properly. An application to approximation theory is given.
16 citations
TL;DR: In this paper, a survey about the following directions of generalizations for separation theorems involving some new results are given: Separation of finite families of convex sets, separation of sets in product spaces, separation in projective spaces and separation in convexity spaces; as a basis for all these considerations vector spaces without topology are used.
Abstract: In this paper a survey about the following directions of generalizations for separation theorems involving some new results are given: Separation of finite families of convex sets; separation of sets in product spaces; separation of convex sets in projective spaces; separation of convex sets in convexity spaces. As a basis for all these considerations vector spaces without topology are used.
12 citations
11 citations
TL;DR: A modification is proposed for Scarf's algorithm for computing fixed points, which allows restarting, and hence overcoming a serious difficulty of this well known algorithm.
Abstract: A modification is proposed for Scarf's algorithm for computing fixed points, which allows restarting, and hence overcoming a serious difficulty of this well known algorithm. A particular feature of the new algorithm is that it uses for restarting an idea quite different from the sandwiching procedure of Merrill-Mackinnon, and operates with a non regular grid, in contrast with most other algorithms which use regular grids. From the computational point of view, it is very simple and seems to be quite competitive with other existing methods.
11 citations
TL;DR: In this article, the interpretation of stationery and PALM distribution for stochastic point processes as a description of statistical equilibrium of a system in continuous time or from the point of view of certain events is discussed.
Abstract: In this paper the investigation concerns the interpretation of stationery and PALM distribution for stochastic point processes as a description of statistical equilibrium of a system in continuous time or from the point of view of certain events, which frequently used in the applications of point processes in the queneign theory.
11 citations
TL;DR: In this paper, the optimal policies and minimal value functions of the so-constructed Markovian Decision Process are calculated' by dynamic programming, and convergence theorems are proved and bounds for the minimal value function are obtained.
Abstract: In this note we investigate a stochastic dynamic programming problem with countable state space and finite action space. An approximation procedure is given for the numerical solution by a decomposition algorithm of the state space. The optimal policies and minimal value functions of the so-constructed Markovian Decision Process are calculated' by dynamic programming. Convergence theorems are proved and bounds for the minimal value function are obtained.
9 citations
TL;DR: It is shown that it is possible under certain assumptions to obtain some or even all £-efficient solutions of the stochastic problem by solving the parametric problem with respect to a certain parameter set.
Abstract: In this paper we study the relation between the general concept for an optimal solution for stochastic programming problems with a random objective function-the concept of an £-efficient solution-and the associated parametric problem, We show that it is possible under certain assumptions to obtain some or even all £-efficient solutions of the stochastic problem by solving the parametric problem with respect to a certain parameter set.
TL;DR: In this paper, the authors reformulated three popular feasible direction methods for solving convex programming problems so that they now work in the absence of Slater's condition or any other constraint qualification.
Abstract: Three popular feasible direction methods for solving convex programming problems are reformulated so that they now work in the absence of Slater’s condition or any other constraint qualification.
TL;DR: In this paper, duality theorems for a special class of nonconvex optimization problems were proved in terms of the separation of the sets epif and hypog.
Abstract: The paper is dealt with duality theorems for a special class of nonconvex optimization problems. Thereby the Φ- conjugate functions (introduced by the authors in earlier papers) are used. The primal objective function contains a pair of functions f g - but in contrast to the problem of Fenchel not in terms of a difference. In proving duality theorems the separation of the sets epif and hypog is used essentially
TL;DR: In this paper, a general formula for stochastic processes with imbedded marked point processes (PMP) is given for the validity of a relationship between stationary time and customer state probabilities for loss systems G/GI/s/O (Theorem 3).
Abstract: By means of a general formula for stochastic processes with imbedded marked point processes (PMP) some necessary and sufficient condition is given for the validity of a relationship, which is well-known in the case of exponentially distributed service times, between stationary time and customer state probabilities for loss systems G/GI/s/O (Theorem 3). A result of Miyazawa for the GI/GI/l/∞ queue is generalized to the case of non-recurrent interarrival times (Theorem 4)-. Furthermore, bounds are derived for the mean increment of the waiting time process at arrival epochs and for the mean actual waiting time in multi-server queues.
TL;DR: In this article, bounds for the mean stationary waiting time and the stationary probability of wait in GI/G/1 were derived for queues with interarrival times having an Erlang distribution.
Abstract: Using methods similar to those of Marshall [3], bounds for the mean stationary waiting time and the stationary probability of wait in GI/G/1 are obtained. In particular, two-sided bounds for queues with interarrival times having an Erlang distribution are given, and bounds which are basing on special properties of the distribution of difference of interarrival and service times.
TL;DR: For optimal control problems with parameters depending on the time, the extended Bellmanean functional equations are in general only sufficient optimality conditions as mentioned in this paper, and it is shown by examples that the optimal control problem may have a greater number of solutions than the functional equation.
Abstract: For optimal control problems with parameters depending on the time, the extended Bellmanean functional equations are in general only sufficient optimality conditions. It is shown by examples that the optimal control problem may have a greater number of solutions than the functional equation of this problem. However, a sufficient condition of the equality of the solution sets of the optimal control problem and the functional equation is given on the basis of rigorous concavity, too.
TL;DR: In this article, the authors extend the results in the literature to programming problems over arbitrary convex cone domains (not necessarily polyhedral) without differentiability assumptions, and show that the Kuhn-Tucker type duality associated with saddlepoint problems can be established over convex polyhedral cone domains without the need for differentiability.
Abstract: In earlier results by Sposito and David, Kuhn-Tucker type duality associated with a certain saddlepoint problem was established over convex cone domains (not necessarily polyhedral) without differentiability. Recent variants have established necessary optimality conditions of the Fritz John and Kuhn-Tucker type for programming problems over" convex polyhedral cone domains which require differentiability assumptions. This paper extends the present results in the literature to programming problems over arbitrary convex cone domains (not necessarily polyhedral).
TL;DR: In this article, error estimations for solutions of quadratic programming problems are given by solution of a linear system of equations by means of the methods of interval arithmetic, and it is quite certain that the round off errors are also included in these estimations.
Abstract: This paper is dealt with error estimations for solutions of quadratic programming problems. These estimations are given by solution of a linear system of equations by means of the methods of interval arithmetic. Therefore it is quite certain that the round off errors are also included in these estimations.
TL;DR: In this article, an extension of feasible direction methods is proposed permitting nonfeasible points to be used as starting points, and a new technique adapted to the given optimization problem is presented.
Abstract: An extension of feasible direction methods is proposed permitting nonfeasible points to be used as starting points. The principle is described at a P1-method. Furthermore to reduce the computational effort a new “ϵ-technique” adapted t the given optimization problem is presented.
TL;DR: In this article, the authors considered multistep decision processes with a vector-valued objective function, where the information about the step-dependent parameters is either complete or incomplete, and the well-known necessary and sufficient functional equations can be extended with respect to the parameter dependence.
Abstract: Multistep decision processes with a vector-valued objective function are considered whereby the information about the step-dependent parameters is either complete or incomplete In case of complete information the well-known necessary and sufficient functional equations can be extended with respect to the parameter dependence, but in case of incomplete information only the sufficiency of certain functional equations can be proved
TL;DR: Extending parameter free penalty methods, a general method of exterior centers for solving nonlinear programming problems is introduced in this paper, where the main purpose is to estimate the rate of convergence in certain methods of exterior center.
Abstract: Extending parameter-free penalty methods a general method of exterior centers for solving nonlinear programming problems is introduced. The main purpose of the paper is to estimate the rate of convergence in certain methods of exterior centers.
TL;DR: In this paper, an extension of a method of L.R. Bittner for deriving maximum principles to general control processes in Banach spaces is given by means of the known multiplier rule of M.Hestenes.
Abstract: By means of the known multiplier rule of M.R. Hestenes there is given an extension of a method of L. Bittner for deriving maximum principles to general control processes in Banach spaces. This extension is applied to a class of optimal control problems for processes governed by Hammerstein integral equations with weakly singular kernels.
TL;DR: The formula for calculating the number of feasible solutions to a knapsack problem given by O. Achou is converted so that only rational numbers appear and all complex roots of unity are removed as discussed by the authors.
Abstract: The formula for calculating the number of feasible solutions to a knapsack problem given by O. Achou is converted so that only rational numbers appear and all complex roots of unity are removed.
TL;DR: In this paper, a semi-ordering for the set of n-dimensional 0-1 vectors with at least ϱ positive components is introduced, where each real valued function f(x) is defined on, with f (x) ≦ f(y) for x < y < y.
Abstract: On the set of the n-dimensional 0 – 1 vectors with at least ϱ positive components a semi-ordering is introduced. Each real valued function f(x), defined on , with f(x) ≦ f(y) for x < y is called 0 – 1 – ρ-monoton increasing. Combinatorial algorithms, presented by the authors for linear 0 – 1-programming, are extend to 0 – 1-programming problems with 0 – 1 – ρ-monoton increasing objective function and arbitrary restrictions.
TL;DR: In this paper, the authors derive connections between the asymptotic behaviours of the distribution functions which appear in the Wiener-Hopf factorization (7) and derive conservation theorems in the waiting model GI/G/1 with and without warming-up time.
Abstract: We derive connections between the asymptotic behaviours of the distribution functions which appear in the Wiener-Hopf-factorization (7). Similar conservation theorems in the waiting model GI/G/1 with and without warming-up time are given by means of the same methods.
TL;DR: In this paper, an ordering over the set G(n, ϱ, τ) of all 0-1-vectors of dimension n with at least ϱ and at most τ positive components is defined.
Abstract: We define an ordering over the set G(n, ϱ, τ) of all 0-1-vectors of dimension n with at least ϱ and at most τ positive components. A rule is given to calculate the follower x' ∈ G (n, ϱ, τ) of an element x ∈ G (n, ϱ, τ) according to this ordering. The theorems 2-7 contain properties of the ordered set G (n, ϱ, τ) and construction methods for special elements of this set.
TL;DR: In this paper, a stochastic multicommodity periodic review inventory system with proportional costs plus a single set-u cost is considered, and the replenishment policy is of the (σS)-type (either nothing is ordered or all commodities are ordered simultaneously).
Abstract: We consider a stochastic multicommodity periodic review inventory system with proportional costs plus a single set-u cost. The replenishment policy is of the (σS)-type (either nothing is ordered or all commodities are ordered simultaneously). For two classes of (σS)-policies and Erlang demand distributions of the order one and two we obtain approximations to optimal ordering parameters. Numerical examples are given.
TL;DR: In this paper, a theory of necessary conditions' for optimal control problems with Noetherian operator equations in Banach spaces developed by M. Goebel and the author is applied to two problems of parameter control in nonlinear boundary value problems for an ordinary differential equation and for the plane Laplace equation.
Abstract: A theory of necessary conditions'for optimal control problems with Noether-ian operator equations in Banach spaces developed by M. Goebel and the author is applied to two problems of parameter control in nonlinear boundary value problems for an ordinary differential equation and for the plane Laplace equation.
TL;DR: A general cutting plane algorithm of Kelley type for convex programs and a general column generation algorithm of Dantzig-Wolfe type for Wolfe's generalized program are shown to be dual.
Abstract: Cutting plane and column generation algorithms are known to be dual in a certain sense. In this paper, a general cutting plane algorithm of Kelley type for convex programs and a general column generation algorithm of Dantzig-Wolfe type for Wolfe's generalized program are described. These two algorithms are shown to be dual. An estimation of the optimal value known for Dantzig-Wolfe methods is carried over to cutting plane methods.