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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, a finite element description of the Hessian was proposed and proved convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes.
Abstract: Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics.
In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given.
In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes.
Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.
3 citations
TL;DR: In this paper, a method for exactly calculating norm on the sum of the cones of nonincreasing or concave functions in Lorentz spaces is proposed, which makes it possible to prove new extrapolation theorems for cones in Lebesgue, and Marcinkiewicz spaces with exact constants.
Abstract: A method for exactly calculating norm on the sum of the cones of nonincreasing or concave functions in Lorentz spaces is proposed. The obtained result makes it possible to prove new extrapolation theorems for cones in Lorentz, Lebesgue, and Marcinkiewicz spaces with exact constants.
3 citations
TL;DR: Under standard assumptions on the energy potential (Lipschitz continuity), it is demonstrated rigorously that the method converges optimally for symmetric schemes, and suboptimally for nonsymmetric schemes.
3 citations
TL;DR: A new proof of optimality of the Bottom Up algorithm is given, considerably shorter and simpler than the original proof, based on the analysis of the greedy algorithm for this problem and properties of greedy solutions.
3 citations
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TL;DR: In this paper, the authors describe a method for solving concave numerical dynamic programming problems based on a pair of polyhederal approximations of concave functions, which is robust in that it is globally convergent, it produces exact error bounds on the computed value function which can in theory be made arbitrarily tight, and its implementation boils down to solving a sequence of linear programs.
Abstract: This paper describes a method for solving concave numerical dynamic programming problems which is based a pair of polyhederal approximations of concave functions. The method is robust in that (i) it is globally convergent, (ii) it produces exact error bounds on the computed value function which can in theory be made arbitrarily tight, and (iii) its implementation boils down to solving a sequence of linear programs. This is true regardless of the dimensionality of the state space, the pattern of binding constraints, and the smoothness of model primitives. Numerical examples suggest that the method is capable of producing accurate solutions in an ecient manner.
3 citations