Showing papers on "Concave function published in 1974"

Optimal Facility Location with Concave Costs

Abstract: The following problem is considered: select plant sites from a given set of sites and choose their production and distribution levels to meet known demand at discrete points at minimum cost. The construction and operating cost of each plant is assumed to be a concave function of the total production at that plant, and the distribution cost between each plant and demand point is assumed to be a concave function of the amount shipped. There may be capacity restrictions on the plants. A branch-and-bound algorithm for identifying an optimal solution is described; it is equivalent to the solution of a finite sequence of transportation problems. The algorithm is developed as a particular case of a simplified algorithm for minimizing separable concave functions over linear polyhedra. Computational results are cited for a computer code implementing the algorithm.

Information, Investment Behavior, and Efficient Portfolios

Abstract: The purpose of this paper is to indicate that the opportunity to obtain information regarding the probability distribution of the return on a risky asset, such as a portfolio or a mutual fund, may cause a risk-averse decision maker to accept a single-period actuarially unfair gamble. This behavior is the same as that implied by utility functions that have convex segments, as originally considered by Friedman and Savage [2] and by Markowitz [12], but the utility function derived is not convex on any interval, since it is the envelope of a finite set of strictly increasing, strictly concave functions. Similar utility functions have been obtained, by Fleming [1] because of transactions costs, by Hakansson [4] by imposing a borrowing restriction on an investment-consumption model, and by Masson [14] in the context of an imperfect capital market. In this paper acceptance of single-period actuarially unfair gambles by an individual risk averse with respect to future wealth levels results from the opportunity to acquire information. The acquisition of information creates a set of conditional decisions each of which the individual may treat in an optimal manner, and that set of conditional decisions may induce risk-taking behavior.

A Lagrangian approach to optimal design

Abstract: where u is a column vector of k components and prime denotes transpose. The D-optimal design problem is to determine a A which maximizes log det M(A). Now log det is a concave function on the set of nonnegative-definite symmetric matrices and so the D-optimal problem can be regarded as a particular case of the following one, which we shall call the qs-optimal design problem: given a concave function q6 on the nonnegativedefinite symmetric k x k matrices, determine A to maximize q{M(A)}. Recently, one line of research has shown that much of the highly developed theory of D-optimal design can be generalized to qs-optimal design (Fedorov, 1972, Chapter 2; Kiefer, 1974; Whittle, 1973). Another has shown how Lagrangian theory can be exploited in the D-optimal design area (Silvey, 1972; Sibson, 1972a, b; Silvey & Titterington, 1973). It is the object of the present note to develop the Lagrangian approach to the q6optimal problem.

On Comparative Dynamics

Abstract: Lately, there has been an increased interest in stability of growth paths, see e.g., Brock and Scheinkman [1974]. The problem has been stated in terms of properties of stationary paths. In order to appreciate the difficulty of the general stability problem, one must realize that there are two types of "time" involved in the analysis: stability "time” and path “time.” Thus, the appropriate mathematical field is that of differential equations defined on a space of functions rather than a finite dimensional space. Naturally, if one restricts one’s attention to stationary paths, then the usual stability analysis is appropriate. However, we would be then discussing the asymptotic behavior of the asymptotic state of the economy. This note strives to put the problem of path stability in the proper perspective by discussing the much simpler problem of comparative dynamics. Unfortunately this term has been used in the economic growth literature to discuss the basically comparative statics problem of comparing stationary growth paths. By comparative dynamics, we mean the determination of the “direction” of change in the optimal path of decision variables due to a change in the exogenous variables. The traditional method of deriving comparative statics results has been to use second order conditions for optimality. However, if one is willing to assume concavity, these results could be derived in a more direct way by utilizing the fact that a differentiable concave function lies below its tangent plane. We shall use this concept in deriving the main inequalities of this note. By way of motivation, we first derive two inequalities of comparative statics. Then we derive the comparative dynamics results and finally we discuss some economic theoretical examples.

On Rational Approximations of Functions with a Convex Derivative

Abstract: Let be the least uniform deviation of a continuous function () from the rational functions of degree not greater than ().Theorem. Suppose a function is given on an interval and is times differentiable , its th derivative being convex. Then where is a constant depending on and .The estimate is sharp for any and any modulus of continuity of the function if the factors of form are neglected.Bibliography: 7 items.

A note on column generation in Dantzig--Wolfe decomposition algorithms

Abstract: A modification of the column generation operation in Dantzig--Wolfe decomposition is suggested. Instead of the usual procedure of solving one or more subproblems at each major iteration, it is shown how the subproblems may be solved parametrically in such a way as to maximize the immediate improvement in the value of objective in the "master problem", rather than to maximize the "reduced profit" of the entering column. The parametric problem is shown to involve the maximization of a piece-wise linear concave function of a single variable. It is hoped that in some cases the use of the suggested procedure may improve the slow rates of convergence common in decomposition algorithms.

On indefinite programming problems

Abstract: The aim of this paper is to show that the product of nonlinear concave nondifferentiable functions which take nonnegative values, is strictly quasi concave. If the functions are concave differentiable and positive the product function is shown to be pseudo-concave.