Showing papers on "Concave function published in 1975"

A Unified Approach to Stochastic Dominance

Abstract: The fundamental problem facing an investor may be viewed as that of choice between two investment alternatives with random returns X and Y . For an expected utility maximizer with known utility function u , X is preferred to Y if Eu ( X ) Eu ( Y ). However, it is difficult in practice to find an investor's utility function. Accordingly, it would be most useful to know whether X is preferred to Y by all investors with utility functions u in some general class U . This approach to the problem of choice seeks to characterize the investor's utility function only in terms of its general properties such as monotonicity and concavity rather than its specific numerical values. For purposes of portfolio theory, the class of increasing concave utility functions is of special importance, as it characterizes risk-averse investors. Given a class U of utility functions, X is said to stochastically dominate Y if Eu ( X ) Eu(Y ) for all u Є U. The stochastic dominance relation over U , thus, partially orders the set of random variables. This chapter presents a unified approach to stochastic dominance. The chapter explores stochastic dominance over the class of concave functions to bring out its probabilistic content.

A Stochastic Sequential Allocation Model

Abstract: This paper considers the following model, described in terms of an investment problem. We have D units available for investment. During each of N time periods an opportunity to invest will occur with probability p. As soon as an opportunity presents itself, we must decide how much of our available resources to invest. If we invest y, then we obtain an expected profit P(y), where P is a nondecreasing continuous function. The amount y then becomes unavailable for future investment. The problem is to decide how much to invest at each opportunity so as to maximize total expected profit. When P(y) is a concave function, the structure of the optimal policy is obtained (§1). Bounds on the optimal value function and asymptotic results are presented in §2. A closed-form expression for the optimal value to invest is found in §3 for the special cases of P(y) = log y and P(y) = yα, for 0 < α < 1. §4 presents a continuous-time version of the model, i.e., we assume that opportunities occur in accordance with a Poisson ...

Reducing project duration at minimum cost: A time-cost tradeoff algorithm

Abstract: The problem of reducing project duration efficiently arises frequently, routinely, and repetitively in government and industry. Siemens [1] has presented an inherently simple time-cost tradeoff algorithm (SAM—for Siemens Approximation Method) for determining which activities in a project network must be shortened to meet an externally imposed (scheduled) completion date (which occurs prior to the current expected completion date). In that paper the network activities of the example problem all have constant cost-slopes. Siemens mentions that the algorithm can be used where the activities have (convex) nonlinear cost-slopes—instead of just one cost-slope and one supply (time available for shortening) for each activity, there can be multiple cost-slope/supply pairs for each activity. This technique is illustrated in this paper. Also illustrated here is an improvement suggested by Goyal [2]. In step 12 of the original algorithm Siemens suggests a review of the solution obtained by the first eleven steps to eliminate any unnecessary shortening. Goyal's modification does this systematically during application of the algorithm by de-shortening (partially or totally) selected activities which were shortened in a prior iteration. He claims that, empirically at least, the technique always yields an optimal solution. Our experience verifies this claim (given the assumption of convex cost functions). The authors have modified the original algorithm so that the requirement for convex cost functions can now be relaxed. Unfortunately, this modification is made only at the expense of simplicity. To further complicate matters we found that Goyal's technique does not always yield an optimal solution when concave functions are involved and thus still another modification was required. These are discussed in detail below. Finally, we discuss the applicability of the algorithm to situations involving discrete time-cost functions.

Mathematical Models for Optimal Timing of Drilling on Multilayer Oil and Gas Fields

Abstract: The following problem of optimal control of multilayer gas and oil fields is formulated: for a given planning horizon find an optimal number of wells to be drilled on each layer and to be transferred between layers as a function of time to meet technological constraints and requirements, and to provide minimum total reduced cost per unit output. The optimal control problem is presented in the form of a mathematical programming problem with a nonlinear fractional objective function containing separable concave functions, and with separable concave constraints. Constraints describing specific features of gas and oil fields are considered. Solution procedures and applications are discussed.

Bounds for nearly best approximations

Abstract: Let X be a uniformly convex space and k be the inverse function of the modulus of convexity 8(.). Assume here that q' is a concave function. Let V be a linear subspace of X and let f in X be such that j = 1 = minjIf -vll: v e VI. Then for 0 0 such that x, y eX, ||x|| E imply ||(x + y)/211 0. Obviously, qf is monotone nondecreasing. From 8(() 0 for 80 > 0. One can replace IX.) by a monotone increasing convex function 81( .) such that 0 80 for >/0((d). So, ||x|| 1 -(8 imply jjx yjI < Vi(3). Received by the editors July 2, 1974. AMS (MOS) subject classifications (1970). Primary 41A50; Secondary 41A65, 41A10, 30A82. .Copyright (3 1975, American Mathematical Society

Explicit Solution for a Class of Allocation Problems

Abstract: In this paper, we deal with the classical allocation problem of optimizing the function Open image in new window under the following constraints Open image in new window that can be solved by applying the dynamic programming algorithm 1. Under particular assumptions on fi(xi), the explicit solution can be obtained for the problem above 2.