Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

Purification of hyaluronidase as an anticancer agent inhibiting CD44

Hyaluronidase (Hyal) can be employed to accomplish a diversity of complications related to hyaluronic acid (HA). Hyal contains some classes of catalysts that cleave HA. This enzyme is detected in several human tissues as well as in animal venoms, pathogenic organisms and cancers. Destructive cancer cells regularly increase the CD44 receptor existing in a cell membrane. This receptor acts as an exact receptor for HA, and HA is recognized to motivate the migration, spread, attack and metastasis of cancer cells. Nearly all of the methods used to purify Hyal are highly costly and not proper for industrial applications. This survey aims to review different methods of Hyal purification, which acts as an anticancer agent by degrading HA in tissues and thus inhibiting the CD44–HA interaction. Hyal can be successfully employed in the management of cancer, which is associated with HA–CD44. This review has described different methods for Hyal purification to prepare an origin to develop a novel purification technique for this highly appreciated protein. Using multiple columns is not applicable for the purification of Hyal and thus cannot be used at the industrial level. It is better to use affinity chromatography of anti-Hyal for Hyal with one-step purification.     

Synthesis,structural characterization and DNA-binding studies of a new cobalt (II) complex: (HAcr)<Subscript>4</Subscript>[Co(Sal)<Subscript>3</Subscript>]

A new cobalt (II) coordination compound was synthesized using proton transfer mechanism. The reaction between CoCl₂·2H₂O, Salicylic acid (H₂Sal) and acridine (Acr) gave a new coordination compound formulated as (HAcr)₄[Co(Sal)₃], which was characterized by elemental analysis, NMR, IR and UV/Vis spectroscopies. The interaction of this complex with DNA has been investigated in vitro using UV absorption, fluorescence spectroscopy, viscosity measurements and gel electrophoresis methods. The intrinsic binding constant has been estimated to be 5.8 × 10⁵ M^?1 using UV absorption. The interaction of DNA–Co (II) complex caused quenching in fluorescence. The binding constant, the number of binding site and Stern–Volmer quenching constant have been calculated to be 7.7 × 10⁴ M^?1, 1.143 and 1.5 × 10⁴ Lmol^?1, respectively. The increase in the viscosity of DNA with increasing the concentration of the Co (II) complex and the observations of other experiments suggest that the cobalt (II) complex binds to DNA by partial intercalation binding mode. Furthermore, the interaction of DNA–Co (II) complex was confirmed using gel electrophoresis studies. Moreover, molecular docking technique predicted partial intercalation binding mode for the complex.     

Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers

The Douglas–Rachford and alternating direction method of multipliers are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of Douglas–Rachford (resp. alternating direction method of multipliers) when the involved functions (resp. their Legendre–Fenchel conjugates) are moreover partly smooth. More precisely, when the two functions (resp. their conjugates) are partly smooth relative to their respective smooth submanifolds, we show that Douglas–Rachford (resp. alternating direction method of multipliers) (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide conditions sufficient for finite convergence. The obtained results are illustrated by several concrete examples and supported by numerical experiments.     

Catalytic study of the copper-based magnetic nanocatalyst on the aerobic oxidation of alcohols in water

A copper-based magnetic nanocatalyst has been prepared by co-precipitation method and characterized by FESEM, EDS, TEM, XRD, XRF, ICP–OES, FTIR, and BET analysis. This new nanocatalyst displays a good activity toward the aerobic oxidation of a wide range of alcohols in water. Moreover, it is recyclable up to five following runs by simple filtration without any significant loss of its catalytic activity.

     

Poly(4-vinylpyridine)：As a green, efficient and commercial available basic catalyst for the synthesis of chromene derivatives

Poly(4-vinylpyridine) is reported as a green,commercial available and efficient basic recyclable catalyst for the synthesis of chromene derivatives.This catalyst can be easily recovered by simple filtration and recycled up to 5 consecutive runs without any loss of its efficiency.     

Study of kinetic,thermodynamic, and isotherm of Sr adsorption from aqueous solutions on graphene oxide (GO) and (aminomethyl)phosphonic acid–graphene oxide (AMPA–GO)

Journal of Radioanalytical and Nuclear Chemistry - In this work, graphene oxide (GO) and (aminomethyl)phosphonic acid–Graphene oxide (AMPA–GO) adsorbents were prepared by the modified...