We study the problem of minimizing a sum of p-norms where p is a fixed real number in the interval [1,∞]. Several practical algorithms have been proposed to solve this problem. However, none of them has a known polynomial time complexity. In this paper, we transform the problem into standard conic form. Unlike those in most convex optimization problems, the cone for the p-norm problem is not se...

We study two topologies $ \tau_{KR} and \tau_K on the space of measures a completely regular generated by Kantorovich–Rubinshtein Kantorovich seminorms analogous to their classical norms in case metric space. The topology coincides with weak nonnegative bounded uniformly tight sets measures. A sufficient condition is given for compactness topology. show that logarithmically concave measures, so...

Finite mixtures of parametric distributions are often used to model data of which it is known or suspected that there are subpopulations. Instead of a parametric model, a penalized likelihood smoothing algorithm is developed. The penalty is chosen to favor a log-concave result. The standard EM algorithm (“split and fit”) can be used. Theoretical results and applications are presented. © 2006 El...

In his study [7] on quartic integrals, Moll met a specialized family of Jacobi polynomials. Moll conjectured that the corresponding coefficient sequences are log-concave. In this paper we settle Moll’s conjecture by a non-trivial usage of computer algebra.

A graph with at most two vertices of the same degree is called antiregular [25], maximally nonregular [32] or quasiperfect [2]. If sk is the number of independent sets of cardinality k in a graph G, then I(G;x) = s0 + s1x+ ...+ sαx α is the independence polynomial of G [10], where α = α(G) is the size of a maximum independent set. In this paper we derive closed formulae for the independence pol...

In this paper, we generalize the well-known Nesterov’s accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using ...

In the paper, the authors first survey and review some logarithmically completely monotonic functions involving the gamma function and its various ratios, and then find sufficient conditions for a function involving the ratio of two gamma functions and originating from the coding gain to be logarithmically completely monotonic.

N/N !. In many cases, lnT ′ provides an asymptotically accurate estimate of lnT . The idea of the algorithm is to express T as the expectation of the permanent of an N ×N random matrix with exponentially distributed entries and approximate the expectation by the integral T ′ of an efficiently computable log-concave function on R. Applications to counting integer flows in graphs are also discussed.