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 2003TL;DR: This paper proposes a generalized algorithm that can handle base station assignment and hand-off, as well as power control, and study by extensive simulations its performance in a dynamic environment.
Abstract: In this paper, we present the power control problem in CDMA wireless data networks in the analytical setting of noncooperative game theory. User satisfaction is represented as a net utility function, which is the difference of a strictly concave function, based on signal to interference ratio, and a cost term on user's power. A detailed analysis for the existence and uniqueness of Nash equilibrium for the above noncooperative game is presented. Next, a decentralized power control algorithm is developed which converges to the Nash equilibrium, as demonstrated by both analytical and simulation methods. The framework is then extended to the multicell case, making user utilities depend on base-station assignment as well as powers. We propose a generalized algorithm that can handle base station assignment and hand-off, as well as power control, and study by extensive simulations its performance in a dynamic environment.
70 citations
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10 Apr 2016TL;DR: In this article, the authors propose a framework called DCCP, which combines the ideas of disciplined convex programming (DCP) with convex-concave programming (CCP).
Abstract: In this paper we introduce disciplined convex-concave programming (DCCP), which combines the ideas of disciplined convex programming (DCP) with convex-concave programming (CCP). Convex-concave programming is an organized heuristic for solving nonconvex problems that involve objective and constraint functions that are a sum of a convex and a concave term. DCP is a structured way to define convex optimization problems, based on a family of basic convex and concave functions and a few rules for combining them. Problems expressed using DCP can be automatically converted to standard form and solved by a generic solver; widely used implementations include YALMIP, CVX, CVXPY, and Convex. jl. In this paper we propose a framework that combines the two ideas, and includes two improvements over previously published work on convex-concave programming, specifically the handling of domains of the functions, and the issue of subdifferentiability on the boundary of the domains. We describe a Python implementation called DCCP, which extends CVXPY, and give examples.
69 citations
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01 Jan 2002TL;DR: In this article, it was shown that the Wigner lattice can exist in one dimension at least, and that the fundamental solution of the diffusion equation can be found in the same dimension.
Abstract: THE following is a preliminary report on some recent work, the full details of which will be published elsewhere. We have come across some inequalities about integrals and moments of log concave functions which hold in the multidimensional case and which are useful in obtaining estimates for multidimensional modified Gaussian measures. By making a small jump (we shall not go into the technical details) from the finite to the infinite dimensional case, upper and lower bounds to certain types of functional integrals can be obtained. As a non-trivial application of the latter we shall, for the first time, prove that the one-dimensional one-component quantummechanical plasma has long-range order when the interaction is strong enough. In other words, the Wigner lattice can exist, in one dimension at least. As another application we shall prove a log concavity theorem about the fundamental solution (Green’s function) of the diffusion equation.
69 citations
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TL;DR: It is shown that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines two kinds of geodesics and proves an analogue of the celebrated generalized Pythagorean theorem from classical information geometry.
Abstract: A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper, we showed that gradient maps of exponentially concave functions provide solutions to a Monge–Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines two kinds of geodesics. We show that the induced geometry is dually projectively flat but not flat. Nevertheless, we prove an analogue of the celebrated generalized Pythagorean theorem from classical information geometry. On the other hand, we consider displacement interpolation under a Lagrangian integral action that is consistent with the optimal transport problem and show that the action minimizing curves are dual geodesics. The Pythagorean theorem is also shown to have an interesting application of determining the optimal trading frequency in stochastic portfolio theory.
69 citations
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TL;DR: A new hybrid approach to interactive evolutionary multi-Objective optimization that uses a partial preference order to act as the fitness function in a customized genetic algorithm that demonstrates its performance on the multi-objective knapsack problem.
68 citations