Concave function

About: Concave function is a research topic. Over the lifetime, 1415 publications have been published within this topic receiving 33278 citations.

Convex and concave functions

Abstract: This chapter introduces convex and concave functions defined on convex sets in Rn and gives some of their basic properties and provides some fundamental theorems involving these functions. These theorems are very important in deriving optimality conditions for nonlinear programming problems and developing suitable computational schemes. A function of f defined on a convex set S in Rn, is said to be a convex function on S, if it satisfies the equation f [(X1 + (1 – (,)X2] ≤ (f(X1) + (1 - () f(X2). The function f is said to be strictly convex on S if the above inequality is strict for X1≠ X 2, and 0 < ( < 1. A function f is said to be concave (strictly concave) if –f is convex (strictly convex). It is clear that a linear function is convex as well as concave but neither strictly convex nor strictly concave. Alternatively, a function f defined on a convex set S in Rn is convex (concave) if linear interpolation between the values of the function never underestimates the actual value at the interpolated point.

End-to-End Distributed Flow Control for Networks with Nonconcave Utilities

Abstract: This paper proposes near-optimal decentralized allocation for traffic generated by real-time applications in communication networks. The quality of experience perceived by users in practical applications cannot be accurately modeled using concave functions. Therefore, we tackle the problem of optimizing general nonconcave network utilities. The approach for solving the resulting nonconvex network utility maximization problem relies on designing a sequence of convex relaxations whose solutions converge to a point that characterizes an optimal solution of the original problem. Three different algorithms are designed for solving the proposed convex relaxation, and their theoretical convergence guarantees are studied. All proposed algorithms are distributed in nature, where each user independently controls its traffic in a way that drives the overall network traffic allocation to an optimal operating point subject to resource constraints. All computations required by the algorithms are performed independently and locally at each user using local information available to that user. We highlight the tradeoff between the convergence speed and the network overhead required by each algorithm. Furthermore, we demonstrate the robustness and scalability of these algorithms by showing that traffic is automatically rerouted in case of a link failure or having new users joining the network. Numerical results are presented to validate our findings.

Nonuniform principal component filter banks: definitions, existence, and optimality

Abstract: The optimality of principal component filter banks (PCFBs) for data compression has been observed in many works to varying extents. Recent work by the authors has made explicit the precise connection between the optimality of uniform orthonormal filter banks (FBs) and the principal component property: The PCFB is optimal whenever the minimization objective is a concave function of the subband variances of the FB. This gives a unified explanation of PCFB optimality for compression, denoising and progressive transmission. However not much is known for the case when the optimization is over a class of nonuniform Fbs. In this paper we first define the notion of a PCFB for a class of nonuniform orthonormal Fbs. We then show how it generalizes the uniform PCFBs by being optimal for a certain family of concave objectives. Lastly, we show that existence of nonuniform PCFBs could imply severe restrictions on the input power spectrum. For example, for the class of unconstrained orthonormal nonuniform Fbs with any given set of decimators that are not all equal, there is no PCFB if the input spectrum is strictly monotone.

A Univariate Extension of Jensen's Inequality

Abstract: The res ult in this paper exp lains some of the qualitative nature of Jensen's inequality It is shown that the more disperse the distribution of a random variable is, the small er is the expectation of any concave function of it This result can be used to show the inadequacy of some current methods of reporting environmental data by using geometric means, and it extends the result of I Billick, D Shier, and C H Spiegelman, where symmetry of th e error in environmental mellsurements is assumed