Strongly convex functions, Moreau envelopes and the generic nature of convex functions with strong minimizers

In this work, using Moreau envelopes, we define a complete metric for the set of proper lower semicontinuous convex functions. Under this metric, the convergence of each sequence of convex functions is epi-convergence. We show that the set of strongly convex functions is dense but it is only of the first category. On the other hand, it is shown that the set of convex functions with strong minima is of the second category

Compositions and Averages of Two Resolvents: Relative Geometry of Fixed Points Sets and a Partial Answer to a Question by C. Byrne

We show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators. Recently, the proximal average has attracted considerable attention in convex analysis. Our results imply that the minimizers of proximal-average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions. When both convex functions in the proximal average are indicator functions of convex sets, least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings. This provides a partial answer to a question posed by C. Byrne. A novelty of our approach is to use the notion of resolvent average and proximal average

Energy-Efficient Optimization for Wireless Information and Power Transfer in Large-Scale MIMO Systems Employing Energy Beamforming

In this letter, we consider a large-scale multiple-input multiple-output (MIMO) system where the receiver should harvest energy from the transmitter by wireless power transfer to support its wireless information transmission. The energy beamforming in the large-scale MIMO system is utilized to address the challenging problem of long-distance wireless power transfer. Furthermore, considering the limitation of the power in such a system, this letter focuses on the maximization of the energy efficiency of information transmission (bit per Joule) while satisfying the quality-of-service (QoS) requirement, i.e. delay constraint, by jointly optimizing transfer duration and transmit power. By solving the optimization problem, we derive an energy-efficient resource allocation scheme. Numerical results validate the effectiveness of the proposed scheme.Comment: 4 pages, 3 figures. IEEE Wireless Communications Letters 201