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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: In this paper, necessary and sufficient conditions for a function to be representable as a sum of an increasing convex and an increasing concave function are given, adding a complementary slackness requirement yields a uniquely determined representation.

8 citations

Journal ArticleDOI
TL;DR: The one-stage result and properties of concave functions lead to simple conditions for the existence of optimal policies, composed of increasing and Lipschitz continuous decision rules, for several dynamic programs with discrete state and action space, in which discrete concavity plays a dominant role.
Abstract: Conditions are presented for the existence of increasing and Lipschitz continuous maximizers in a general one-stage optimization problem. This property results in substantial numerical savings in case of a discrete parameter space. The one-stage result and properties of concave functions lead to simple conditions for the existence of optimal policies, composed of increasing and Lipschitz continuous decision rules, for several dynamic programs with discrete state and action space, in which case discrete concavity plays a dominant role. One of the examples, a general multi-stage allocation problem, is considered in detail. Finally, some known results in the case of a continuous state and action space are generalized.

8 citations

Posted Content
TL;DR: The optimal switching problem for one-dimensional diffusions is solved by directly using the dynamic programming principle and the excessive characterization of the value function using the properties of concave functions.
Abstract: We explicitly solve the optimal switching problem for one-dimensional diffusions by directly employing the dynamic programming principle and the excessive characterization of the value function. The shape of the value function and the smooth fit principle then can be proved using the properties of concave functions.

8 citations

Journal ArticleDOI
TL;DR: In this article, a truncated uniform distribution is defined for the worst-case miss distance in a concave function of the engagement duration on a certain interval, and the worst case miss distance can be determined from a singe Monte Carlo trial.
Abstract: performance” satisfaction, is characterized as a truncated uniform distribution.This result was applied to the performance analysis of engagementdurationand of several error sources.Linear theory enables the formulation of analytical solutions to the problem.When numericalMonteCarlo simulationsare used to search for theworstcase distribution,a limited number of simulation runs typically are neededfor the search.Moreover,when themiss distanceis a concave function of the con ict duration on a certain interval, the worstcase miss distance can be determined from a singe Monte Carlo trial.

8 citations

Posted Content
TL;DR: In this article, it was shown that for all convex or concave functions, the sum-product problem can be reduced to the convex concave expansion problem, and that the result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sumproduct problem.
Abstract: In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$ This result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sum-product problem. We also adjust our technique to also prove the three-variable expansion result \[ |AB+A|\gtrsim |A|^{\frac32 +\frac3{170}}\,. \] Our methods follow a series of recent developments in the sum-product literature, presenting a unified picture. Of particular interest is an adaptation of a regularisation technique of Xue, that enables us to find positive proportion subsets with certain desirable properties.

8 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202316
202240
202158
202049
201952
201860