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 article, it was shown that the least area required to enclose two volumes in ℝn orS n forn ≥ 3 is a strictly concave function of the two volumes.
Abstract: We show that the least area required to enclose two volumes in ℝn orS
n forn ≥ 3 is a strictly concave function of the two volumes. We deduce that minimal double bubbles in ℝn have no empty chambers, and we show that the enclosed regions are connected in some cases. We give consequences for the structure of minimal double bubbles in ℝn. We also prove a general symmetry theorem for minimal enclosures ofm volumes in ℝn, based on an idea due to Brian White.
43 citations
Proceedings Article•
06 Jun 2016
TL;DR: In this article, the authors considered a contextual version of the multi-armed bandit problem with global knapsack constraints and gave a computationally efficient algorithm with slightly better regret bounds, by generalizing the approach of Agarwal et al.
Abstract: We consider a contextual version of multi-armed bandit problem with global knapsack constraints. In each round, the outcome of pulling an arm is a scalar reward and a resource consumption vector, both dependent on the context, and the global knapsack constraints require the total consumption for each resource to be below some pre-fixed budget. The learning agent competes with an arbitrary set of context-dependent policies. This problem was introduced by Badanidiyuru et al. (2014), who gave a computationally inefficient algorithm with near-optimal regret bounds for it. We give a computationally efficient algorithm for this problem with slightly better regret bounds, by generalizing the approach of Agarwal et al. (2014) for the non-constrained version of the problem. The computational time of our algorithm scales logarithmically in the size of the policy space. This answers the main open question of Badanidiyuru et al. (2014). We also extend our results to a variant where there are no knapsack constraints but the objective is an arbitrary Lipschitz concave function of the sum of outcome vectors.
43 citations
TL;DR: A decomposition branch and bound approach is considered for the global minimization of an indefinite quadratic function over a polytope using an easily constructed convex underestimator of the objective function.
43 citations
TL;DR: In particular, for the Inada conditions to hold, a production function must be asymptotically Cobb-Douglas as discussed by the authors, and it is shown that every twice-continuously differentiable and strictly concave function f: R +→ R + can be bracketed between two CES functions at each open interval.
43 citations
TL;DR: A game-theoretic analysis of decode and forward cooperative communications is presented for additive white Gaussian noise (AWGN) and Rayleigh fading channels and shows a Nash Equilibrium in which users mutually cooperate can be obtained for AWGN channels when strict power control is enforced and users care about future payoff.
Abstract: A game-theoretic analysis of decode and forward cooperative communications is presented for additive white Gaussian noise (AWGN) and Rayleigh fading channels. Cooperative communications is modeled as a repeated game in which the two participating terminals are selfish and seek to maximize their own payoff, a general utility function that monotonically increases with signal-to-noise ratio. Results show a Nash Equilibrium in which users mutually cooperate can be obtained for AWGN channels when strict power control is enforced and users care about future payoff. However, such power control may not be necessary to achieve cooperative Nash Equilibrium when the game is played in Rayleigh fading channels. We study the Rayleigh fading channel as a two state Markov model in this paper. In this case, a mutually cooperative Nash Equilibrium 1) always exists when the utility function is convex and users care somewhat about future payoff; and 2) may not always exist when the utility function is concave, especially in adverse channel conditions. Examinations of several widely-used concave functions, however, demonstrate that mutual cooperation is more likely when users increase their value on future payoff. Additionally, it is shown that improving the effective uplink channel conditions of users, e.g., by using multiple transmit antennas, further encourages cooperation.
43 citations