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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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TL;DR: It is shown that the rate-distortion bound (R(d) \leq C) remains true when -\log x in the definition of mutual information is replaced by an arbitrary concave nonincreasing function satisfying some technical conditions.
Abstract: It is shown that the rate-distortion bound (R(d) \leq C) remains true when -\log x in the definition of mutual information is replaced by an arbitrary concave (\cup) nonincreasing function satisfying some technical conditions. Examples are given showing that for certain choices of the concave functions, the bounds obtained are better than the classical rate-distortion bounds.
93 citations
TL;DR: In this paper, it was shown that weakly upper semicontinuous concave Schur concave functions coincide with concave Fenchel transform and Hardy and Littlewood's inequality.
Abstract: A representation result is provided for concave Schur concave functions on L∞(Ω). In particular, it is proven that any monotone concave Schur concave weakly upper semicontinuous function is the infinimum of a family of nonnegative affine combinations of Choquet integrals with respect to a convex continuous distortion of the underlying probability. The method of proof is based on the concave Fenchel transform and on Hardy and Littlewood's inequality. Under the assumption that the probability space is nonatomic, concave, weakly upper semicontinuous, law-invariant functions are shown to coincide with weakly upper semicontinuous concave Schur concave functions. A representation result is, thus, obtained for weakly upper semicontinuous concave law-invariant functions.
93 citations
TL;DR: It is shown that the limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density and its derivative are, under comparable smoothness assumptions, the same (up to sign) as in the convex density estimation problem.
Abstract: We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form $f_0=\exp\varphi_0$ where $\varphi_0$ is a concave function on $\mathbb{R}$. The pointwise limiting distributions depend on the second and third derivatives at 0 of $H_k$, the "lower invelope" of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of $\varphi_0=\log f_0$ at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode $M(f_0)$ and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.
93 citations
TL;DR: A model for a serial supply chain in which production, inventory, and transportation decisions are integrated in the presence of production capacities and concave cost functions is considered, generalizing the uncapacitated serial single-item multilevel economic lot-sizing model by adding stationary production capacities at the manufacturer level.
Abstract: We consider a model for a serial supply chain in which production, inventory, and transportation decisions are integrated in the presence of production capacities and concave cost functions. The model we study generalizes the uncapacitated serial single-item multilevel economic lot-sizing model by adding stationary production capacities at the manufacturer level. We present algorithms with a running time that is polynomial in the planning horizon when all cost functions are concave. In addition, we consider different transportation and inventory holding cost structures that yield improved running times: inventory holding cost functions that are linear and transportation cost functions that are either linear, or are concave with a fixed-charge structure. In the latter case, we make the additional common and reasonable assumption that the variable transportation and inventory costs are such that holding inventories at higher levels in the supply chain is more attractive from a variable cost perspective. While the running times of the algorithms are exponential in the number of levels in the supply chain in the general concave cost case, the running times are remarkably insensitive to the number of levels for the other two cost structures.
92 citations
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TL;DR: In this article, a general notion of transport cost is introduced, which encompasses many costs used in the literature, including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's.
Abstract: We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.
91 citations