Topic
Concave function
About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.
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20 Oct 2012
TL;DR: In this paper, a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, was considered, and a polynomial time algorithm for solving corresponding flow maximization problems was given.
Abstract: We consider a nonlinear extension of the generalized network flow model, with the flow leaving an arc being an increasing concave function of the flow entering it, as proposed by Truemper and Shigeno. We give a polynomial time combinatorial algorithm for solving corresponding flow maximization problems, finding an $\varepsilon$-approximate solution in $O(m(m+\log n)\log(MUm/\varepsilon))$ arithmetic operations and value oracle queries, where $M$ and $U$ are upper bounds on simple parameters. This also gives a new algorithm for linear generalized flows, an efficient, purely scaling variant of the Fat-Path algorithm by Goldberg, Plot kin and Tardos, not using any cycle cancellations. We show that this general convex programming model serves as a common framework for several market equilibrium problems, including the linear Fisher market model and its various extensions. Our result immediately provides combinatorial algorithms for various extensions of these market models. This includes nonsymmetric Arrow-Debreu Nash bargaining, settling an open question by Vazirani [4].
9 citations
TL;DR: In this article, an algorithm was developed for finding the global minimum of a continuously differentiable function on a compact interval in R 1, where the function is assumed to be the sum of a convex and a concave function, each of which belongs to C 1 [a, b ].
9 citations
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TL;DR: This paper considers ambiguity in choice functions over a multi-attribute prospect space and proposes two approaches based respectively on the support functions and level functions of quasi-concave functions to develop tractable formulations of the maximin preference robust optimization model.
Abstract: Decision maker's preferences are often captured by some choice functions which are used to rank prospects. In this paper, we consider ambiguity in choice functions over a multi-attribute prospect space. Our main result is a robust preference model where the optimal decision is based on the worst-case choice function from an ambiguity set constructed through preference elicitation with pairwise comparisons of prospects. Differing from existing works in the area, our focus is on quasi-concave choice functions rather than concave functions and this enables us to cover a wide range of utility/risk preference problems including multi-attribute expected utility and $S$-shaped aspirational risk preferences. The robust choice function is increasing and quasi-concave but not necessarily translation invariant, a key property of monetary risk measures. We propose two approaches based respectively on the support functions and level functions of quasi-concave functions to develop tractable formulations of the maximin preference robust optimization model. The former gives rise to a mixed integer linear programming problem whereas the latter is equivalent to solving a sequence of convex risk minimization problems. To assess the effectiveness of the proposed robust preference optimization model and numerical schemes, we apply them to a security budget allocation problem and report some preliminary results from experiments.
9 citations
TL;DR: In this paper, the authors consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle pa, a? (1, 2), at infinity, and show that every such function is close-to-convex of order (a - 1).
Abstract: We consider functions that map the open unit disc conformally onto the complement of an unbounded convex set with opening angle pa, a ? (1, 2], at infinity. In this paper, we show that every such function is close-to-convex of order (a - 1) and is included in the set of univalent functions of bounded boundary rotation. Many interesting consequences of this result are obtained. We also determine the extreme points of the set of concave functions with respect to the linear structure of the Hornich space.
9 citations
TL;DR: It is demonstrated the existence of a tree shape for which a single incorrect tree topology will be guaranteed to be preferred if the corrected distance function is convex, and the standard practice of treating gaps in sequence alignments as missing data is sufficient to produce non-linear corrected distance functions.
9 citations