The proximal point mapping is the basis of many optimization techniques for convex functions. By means of variational analysis, the concept of proximal mapping was recently extended to nonconvex functions that are prox-regular and prox-bounded. In such a setting, the proximal point mapping is locally Lipschitz continuous and its set of fixed points coincide with the critical points of the origi...

The main aim of this paper is twofold. Our first objective to study a new system generalized multivalued variational-like inequalities in Banach spaces and establish its equivalence with fixed point problems utilizing the concept P-?-proximal mapping. obtained alternative equivalent formulation used iterative algorithm for finding approximate solution suggested. Under some appropriate assumptio...

This paper studies a class of problems consisting of minimizing a continuously differentiable function penalizedwith the so-called `0-norm over a symmetric set. These problems are hard to solve, yet prominent in many fields and applications.We first study the proximal mapping with respect to the `0-norm over symmetric sets, and provide an efficient method to attainit. The method is ...

In this paper, we present a novel yet simple homotopy proximal mapping algorithm for compressive sensing. The algorithm adopts a simple proximal mapping for l1 norm regularization at each iteration and gradually reduces the regularization parameter of the l1 norm. We prove a global linear convergence for the proposed homotopy proximal mapping (HPM) algorithm for solving compressive sensing unde...

In this paper, we present a novel yet simple homotopy proximal mapping algorithm for compressive sensing. The algorithm adopts a simple proximal mapping of the l1 norm at each iteration and gradually reduces the regularization parameter for the l1 norm. We prove a global linear convergence of the proposed homotopy proximal mapping (HPM) algorithm for solving compressive sensing under three diff...

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...

Low-rank inducing unitarily invariant norms have been introduced to convexify problems with low-rank/sparsity constraint. They are the convex envelope of a unitary norm and indicator function an upper bounding rank The most well-known member this family is so-called nuclear norm. To solve optimization involving such proximal splitting methods, efficient ways evaluating mapping low-rank needed. ...

BACKGROUND Quantitative magnetic resonance imaging (MRI) techniques, such as T2 and T2 star (T2*) mapping, have been used to evaluate ligamentous tissue in vitro and to identify significant changes in structural integrity of a healing ligament. These studies lay the foundation for a clinical study that uses quantitative mapping to evaluate ligaments in vivo, particularly the posterior cruciate ...