We provide a proximal average with repect to $1$-coercive Legendre function. In the sense of Bregman distance, envelope is convex combination envelopes individual functions. The mapping convexified mappings Techniques from variational analysis keys for average.

The recently introduced proximal average of two convex functions is a convex function with many useful properties. In this paper, we introduce and systematically study the proximal average for finitely many convex functions. The basic properties of the proximal average with respect to the standard convex-analytical notions (domain, Fenchel conjugate, subdifferential, proximal mapping, epi-conti...

Bauschke, Lucet, and Trienis [SIAM Rev., 50 (2008), pp. 115–132] developed the concept of the proximal average of two convex functions. In this work we show the relationship between the proximal average and the Moreau envelope and exploit this relationship to develop stability theory for a generalized proximal average function. This approach allows us to extend the concept of the proximal avera...

The NC-proximal average is a parametrized function used to continuously transform one proper, lsc, prox-bounded function into another. Until now it has been defined for two functions. The purpose of this article is to redefine it so that any finite number of functions may be used. The layout generally follows that of [11], extending those results to the more general case and in some instances g...

Objective- To measure statistically bone cortex to diameter  (C/D) ratio in metocarpo/metatarsal and proximal phalanges in camel. This ratio may be used in diagnosing possible metabolic and nutritional diseases in camel. Design- Experimental study. Animals- Twelve camel limbs (6 fore limbs & 6 hind limbs)      Procedures- This study was conducted on dorsopalmar/dorsoplantar radiographs of t...

We propose a fast first-order method to solve multi-term nonsmooth composite convex minimization problems by employing a recent proximal average approximation technique and a novel adaptive parameter tuning technique. Thanks to this powerful parameter tuning technique, the proximal gradient step can be performed with a much larger stepsize in the algorithm implementation compared with the prior...

Sparse modeling has been highly successful in many realworld applications. While a lot of interests have been on convex regularization, recent studies show that nonconvex regularizers can outperform their convex counterparts in many situations. However, the resulting nonconvex optimization problems are often challenging, especially for composite regularizers such as the nonconvex overlapping gr...