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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01 Jan 1989
TL;DR: This article showed that most well-known classes of nondifferentiable functions are in some sense quite smooth, i.e., they are smooth in the sense that they can be approximated by smooth functions.
Abstract: Our aim is to show that most well-known classes of nondifferentiable functions are in some sense quite smooth. Nonsmooth analysis (for short, NSA) is one of most attractive and promising areas in modern mathematics. A lot of new profound results have been obtained and much more seem to come (see, e.g., [1–6] and References therein).
16 citations
TL;DR: In this paper, it was shown that diversification of a portfolio of convex functions of heavy-tailed risks increases the portfolio's riskiness if expectations of these risks are infinite.
Abstract: This paper analyzes portfolio diversification for nonlinear transformations of heavy-tailed risks. It is shown that diversification of a portfolio of convex functions of heavy-tailed risks increases the portfolio’s riskiness if expectations of these risks are infinite. In contrast, for concave functions of heavy-tailed risks with finite expectations, the stylized fact that diversification is preferable continues to hold. The framework of transformations of heavy-tailed risks includes many models with Pareto-type distributions that exhibit local or moderate deviations from power tails in the form of additional slowly varying or exponential factors. The class of distributions under study is therefore extended beyond the stable class.
16 citations
01 Jan 2001
TL;DR: In this paper, the authors consider the problem of unicast equation-based rate control in the Internet and show that the control is always conservative if the loss-event inter-arrival times are an exogeneous stationary random process.
Abstract: We consider one aspect of the general problem of unicast equation based rate control in the Internet, which we formulate as follows. When a so called ``loss-event" occurs, a data source updates its sending rate by setting it to $f(\hat{p_n})$, where $\hat{p}_n$ is an estimate of $\overline{p}$, the rate of loss-events. Function $f$ (the target loss-throughput function) defines the objective of the control method: we would like that the throughput $\overline{x}$, attained by the source, satisfies the equation $\overline{x}\leq f(\overline{p})$. If so, we say that the control is conservative. In the Internet, function $f$ is obtained by analyzing the dependency of throughput versus the rate of loss-events for a real TCP source. A non-TCP source which implements a control system as we describe is said to be TCP-friendly if the control is conservative. In this paper, we examine whether such a control system is conservative. We first consider a simple stochastic model which assumes that the intensity of the loss-events is proportional to the current sending rate. We show that, for this model, the control is always conservative if $f(p)$ is a concave function of $1/p$; otherwise this may not be true. Then we consider a second model where the loss-event inter-arrival times is an exogeneous stationary random process. We show that, for this second model, there exist statistics of the loss-event inter-arrival times such that the control is non-conservative, even if $f(p)$ is a concave function of $1/p$. We validate our analytical results with simulations. Another aspect of unicast equation-based rate control in the Internet is the influence of the variability of round-trip times, which is not analyzed in this paper. KEYWORDS: Equation-based, Rate control, TCP-friendliness, Internet, Stochastic recurrence, Autoregressive process, Markov modulated process, Non-linear system, Estimation, Palm expectation
16 citations
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TL;DR: In this paper, the authors studied real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables and proved a homogeneous decomposition theorem of McMullen type, and a representation formula for those valuations which are N-homogeneous.
Abstract: We study real-valued, continuous and translation invariant valuations defined on the space of quasi-concave functions of N variables. In particular, we prove a homogeneous decomposition theorem of McMullen type, and we find a representation formula for those valuations which are N-homogeneous. Moreover, we introduce the notion of Klain's functions for these type of valuations.
16 citations
TL;DR: It is shown that utility functions act as the Lagrange multipliers of the stochastic order constraints in this general setting, and that the dual problem is a search over utility functions.
Abstract: We study convex optimization problems with a class of multivariate integral stochastic order constraints defined in terms of parametrized families of increasing concave functions. We show that utility functions act as the Lagrange multipliers of the stochastic order constraints in this general setting, and that the dual problem is a search over utility functions. Practical implementation issues are discussed.
16 citations