Showing papers on "Concave function published in 1976"
TL;DR: In this article, the authors extend the Prekopa-leindler theorem to other types of convex combinations of two positive functions and strengthen it by introducing the notion of essential addition.
1,018 citations
TL;DR: In this paper, a man-machine interactive mathematical programming method is presented for solving the multiple criteria problem involving a single decision maker, where all decision-relevant criteria or objective functions are concave functions to be maximized, and the constraint set is convex.
Abstract: In this paper a man-machine interactive mathematical programming method is presented for solving the multiple criteria problem involving a single decision maker. It is assumed that all decision-relevant criteria or objective functions are concave functions to be maximized, and that the constraint set is convex. The overall utility function is assumed to be unknown explicitly to the decision maker, but is assumed to be implicitly a linear function, and more generally a concave function of the objective functions. To solve a problem involving multiple objectives the decision maker is requested to provide answers to yes and no questions regarding certain trade offs that he likes or dislikes. Convergence of the method is proved; a numerical example is presented. Tests of the method as well as an extension of the method for solving integer linear programming problems are also described.
732 citations
TL;DR: A new method designed to globally minimize concave functions over linear polyhedra is described, and an example problem is solved, and computational considerations are discussed.
Abstract: A new method designed to globally minimize concave functions over linear polyhedra is described. Properties of the method are discussed, an example problem is solved, and computational considerations are discussed.
169 citations
TL;DR: In this paper, the algebraic foundation of (1) was uncovered by giving a short proof of a generalization, which is simpler and more direct than the proof given by RotfeΓd, and is based on an interesting matrix valued triangle inequality.
Abstract: when / is an increasing concave function of a nonnegative real variable, with /(0) = 0. This inequality is of some interest as in previously published work a convexity (rather than concavity) hypothesis has usually been necessary to establish results of the general type of (1). See, for example, Gohberg and Krein [3], page 49, or Marcus and Mine [4], pages 103 and 116. In this paper we shall uncover the algebraic foundation of (1) by giving a short proof of a generalization. Our proof, which is simpler and more direct than the proof of (1) given by RotfeΓd, will be based on an interesting matrix valued triangle inequality, a special case of which was given by RotfeΓd. We note that the methods used by RotfeΓd are very much adapted to the inequality (1) that he wished to prove, and do not appear capable of proving the extensions of his results to be presented below.
80 citations
TL;DR: In this article, the minimization of a quasi-convex functional or a quasiconcave functional over a convex set of functions is studied. But the problem is not related to the minimisation of a functional function.
Abstract: Optimization problems are considered where ratios of functionals of several different types (convex/concave, concave/convex, convex/convex, and concave/concave) are to be minimized over a convex set. All these problems are related to the minimization of a quasi-convex functional or a quasi-concave functional.
45 citations
01 Jan 1976
TL;DR: The method is based on a well-known method for unconstrained optimization and assumes that the decision maker’s utility is not an explicitly known concave function of several objectives, which are linear functions of activities defined on a convex polyhedral set.
Abstract: The method is based on a well-known method for unconstrained optimization developed by Nelder and Mead. It is assumed that the decision maker’s utility is not an explicitly known concave function of several objectives, which are linear functions of activities defined on a convex polyhedral set. It has thus been possible to modify the simplex procedure to handle constraints. Furthermore, in the interaction with the decision maker, the method only relies on a simple ranking procedure. The method seems to have many good properties. However, its advantages still have to be empirically verified.
8 citations
TL;DR: A new procedure to compute a concave PLDF using Chebyshev polynominals, where the number of linear functions necessary for adequate results of the PLDF is determined adaptively during the procedure.
Abstract: In two-class pattern classification problems, the use of piecewise linear discriminant functions (PLDF's) is often encountered. Following a consideration of the relative advantages of a PLDF above an optimal?high degree?discriminant function, a new procedure to compute a concave PLDF is presented. The characteristic property of this procedure is that the number of linear functions necessary for adequate results of the PLDF is determined adaptively during the procedure. The linear functions are computed with Chebyshev polynominals.
5 citations
TL;DR: In this paper, a discrete-time Markov process on [0, ∞] is considered, where the process is generated by selecting at each time, in an independent and stationary way, a concave non-decreasing function.
1 citations