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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TL;DR: In this paper, the authors presented a method for minimizing the sum of a possibly nonsmooth convex function and a continuously differentiable function, which is a descent method which generates successive search directions by solving quadratic programming subproblems.
Abstract: This paper presents a method for minimizing the sum of a possibly nonsmooth convex function and a continuously differentiable function. As in the convex case developed by the author, the algorithm is a descent method which generates successive search directions by solving quadratic programming subproblems. An inexact line search ensures global convergence of the method to stationary points.
16 citations
TL;DR: It is shown that every code picked from an ensemble of regular low-density parity-check codes with minimum Hamming distance growing (with high probability) linearly with the block length has a Bethe entropy function that is convex in certain regions of its domain.
Abstract: We present a combinatorial characterization of the Bethe entropy function of a factor graph, such a characterization being in contrast to the original, analytical, definition of this function. We achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph. Analogously, we give a combinatorial characterization of the Bethe partition function, whose original definition was also of an analytical nature. As we point out, our approach has similarities to the replica method, but also stark differences. The above findings are a natural backdrop for introducing a decoder for graph-based codes that we will call symbolwise graph-cover decoding, a decoder that extends our earlier work on blockwise graph-cover decoding. Both graph-cover decoders are theoretical tools that help towards a better understanding of message-passing iterative decoding, namely blockwise graph-cover decoding links max-product (min-sum) algorithm decoding with linear programming decoding, and symbolwise graph-cover decoding links sum-product algorithm decoding with Bethe free energy function minimization at temperature one. In contrast to the Gibbs entropy function, which is a concave function, the Bethe entropy function is in general not concave everywhere. In particular, we show that every code picked from an ensemble of regular low-density parity-check codes with minimum Hamming distance growing (with high probability) linearly with the block length has a Bethe entropy function that is convex in certain regions of its domain.
16 citations
TL;DR: In this article, the authors used Frechet-derivative approach to obtain monotonicity, convexity/concavity and comparison results concerning strictly unmixed solutions of continuous and discrete-time algebraic Riccati equations.
Abstract: Using a Frechet-derivative-based approach some monotonicity, convexity/concavity and comparison results concerning strictly unmixed solutions of continuous- and discrete-time algebraic Riccati equations are obtained; it turns out that these solutions are isolated and smooth functions of the input data. Similarly, it is proved that the solutions of initial value problems for both Riccati differential and difference equations are smooth and monotonic functions of the input data and of the initial value. They are also convex or concave functions with respect to certain matrix coefficients.
16 citations
11 Feb 1999
TL;DR: In this article, the extremal points of a functional f f(Vu), for a convex or concave function f, were investigated and shown to be exactly u1 and u2 if these functions are convex and coincide on the boundary 9Q.
Abstract: We investigate the extremal points of a functional f f(Vu), for a convex or concave function f. The admissible functions u : Q C RN ->F R are convex themselves and satisfy a condition u2 < u < ul. We show that the extremal points are exactly u1 and u2 if these functions are convex and coincide on the boundary 9Q. No explicit regularity condition is imposed on f, u1, or U2. Subsequently we discuss a number of extensions, such as the case when ul or u2 are non-convex or do not coincide on the boundary, when the function f also depends on u, etc.
16 citations
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TL;DR: This work considers the contextual version of a multi-armed bandit problem with global convex constraints and concave objective function, and gives computationally efficient algorithms with near-optimal regret, generalizing the approach of Agarwal et al. (2014).
Abstract: We consider the contextual version of a multi-armed bandit problem with global convex constraints and concave objective function. In each round, the outcome of pulling an arm is a context-dependent vector, and the global constraints require the average of these vectors to lie in a certain convex set. The objective is a concave function of this average vector. The learning agent competes with an arbitrary set of context-dependent policies. This problem is a common generalization of problems considered by Badanidiyuru et al. (2014) and Agrawal and Devanur (2014), with important applications. We give computationally efficient algorithms with near-optimal regret, generalizing the approach of Agarwal et al. (2014) for the non-constrained version of the problem. For the special case of budget constraints our regret bounds match those of Badanidiyuru et al. (2014), answering their main open question of obtaining a computationally efficient algorithm.
16 citations