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.     

Air-Stable Hexagonal Bipyramidal Dysprosium(III) Single-Ion Magnets with Nearly Perfect D6h Local Symmetry

Eight-coordinated Dy^III centres with D_6h symmetry are expected to act as high-performance single-molecule magnets (SMMs) due to the simultaneous fulfilment of magnetic axiality and a high coordination number (a requisite for air stability). But the experimental realization is challenging due to the requirement of six coordinating atoms in the equatorial plane of the hexagonal bipyramid; this is usually too crowded for the central Dy^III ion. Here a hexaaza macrocyclic Schiff base ligand and finetuned axial alkoxide/phenol-type ligands are used to show that a family of hexagonal bipyramidal Dy^III complexes can be isolated. Among them, three complexes possess nearly perfect D_6h local symmetry. The highest effective magnetic reversal barrier is found at 1338(3) K and an open hysteresis temperature of 6 K at the field sweeping rate of 1.2 mT s⁻¹; this represents a new record for D_6h SMMs.     

A New Approach for the Synthesis of Miktoarm Star Polymers Through a Combination of Thiol–Epoxy “Click” Chemistry and ATRP/Ring‐Opening Polymerization Techniques

A new approach was developed for synthesis of certain A₃B₃‐type of double hydrophilic or amphiphilic miktoarm star polymers using a combination of “grafting onto” and “grafting from” methods. To achieve the synthesis of desired miktoarm star polymers, acetyl protected poly(ethylene glycol) (PEG) thiols (M_n = 550 and 2000 g mol^?1) were utilized to generate A₃‐type of homoarm star polymers through an in situ protective group removal and a subsequent thiol–epoxy “click” reaction with a tris‐epoxide core viz. 1,1,1‐tris(4‐hydroxyphenyl)ethane triglycidyl ether. The secondary hydroxyl groups generated adjacent to the core upon the thiol–epoxy reaction were esterified with α‐bromoisobutyryl bromide to install atom transfer radical polymerization (ATRP) initiating sites. ATRP of N‐isopropylacrylamide (NIPAM) using the three‐arm star PEG polymer fitted with ATRP initiating sites adjacent to the core afforded A₃B₃‐type of double hydrophilic (PEG)₃[poly(N‐isopropylacrylamide)] (PNIPAM)₃ miktoarm star polymers. Furthermore, the generated hydroxyl groups were directly used as initiator for ring‐opening polymerization of ε‐caprolactone to prepare A₃B₃‐type of amphiphilic (PEG)₃[poly(ε‐caprolactone)]₃ miktoarm star polymers. The double hydrophilic (PEG)₃(PNIPAM)₃ miktoarm star polymers showed lower critical solution temperature around 34 °C. The preliminary transmission electron microscopy analysis indicated formation of self‐assembly of (PEG)₃(PNIPAM)₃ miktoarm star polymer in aqueous solution. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 146–156     

Switching from linear to cyclic δ‐Polyvalerolactone synthesized via zeolitic imidazolate framework as a catalyst: A promising approach

A series of spray dried zeolitic imidazolate frameworks (ZIFs = ZIF‐8, ZIF‐67, and Zn/Co‐ZIF) are used as a catalyst for the bulk ring‐opening polymerization of δ‐valerolactone without any co‐catalyst to generate polyvalerolactone. Interestingly, using the same catalyst under the same reaction conditions could manipulate the structure of the product polymer, and thus its physical properties. Thus, using a dried substrate leads to the formation of the cyclic polymer while a linear polymer was formed on using the commercially available substrate. An activated monomer mechanism has been suggested where the propagating zinc alkoxide undergoes an intramolecular transesterification to release cyclic or linear polyvalerolactone. The ROP of δ‐VL without drying shows that the polymeric zwitterions have little tendency to cyclize in the presence of moisture. At 140 °C, ZIF‐8 shows a superior catalytic activity resulting in the production of cyclic polyvalerolactone having a high molecular weight as compared to ZIF‐67 or Zn/Co‐ZIF due to the presence of highly active sites. The catalyst could be recycled and reused without any significant loss of catalytic activity.     

Nanoscale C‐Rich SixC1−x Bus/Ring Waveguide Based Cross‐Wavelength Data Converter

The carbon‐rich silicon carbide (C‐rich Si_xC_1?x) micro‐ring channel waveguide with asymmetric core aspect is demonstrated for all‐optical cross‐wavelength pulsed return‐to‐zero on‐off keying (PRZ‐OOK) data conversion. Enhanced nonlinear optical Kerr switching enables 12‐Gbit per second data processing with optimized modulation depth. The inverse tapered waveguide at end‐face further enlarges the edge‐coupling efficiency, and the asymmetric channel waveguide distinguishes the polarization modes. To prevent data shape distortion, the bus/ring gap spacing is adjusted to control the quality factor (Q‐factor) of the micro‐ring. Designing the waveguide cross section at 500 × 350 nm² provides the C‐rich Si_xC_1?x channel waveguide to induce strong transverse electric mode (TE‐mode) confinement with a large Kerr nonlinearity of 2.44 × 10^?12 cm² W^?1. Owing to the trade‐off between the Q‐factor and the on/off extinction ratio, the optimized bus/ring gap spacing of 1400 nm is selected to provide a coupling ratio at 5–6% for compromising the modulation depth and the switching throughput. Such a C‐rich Si_xC_1?x micro‐ring with asymmetric channel waveguide greatly enhances the cross‐wavelength data conversion efficiency to favor its on‐chip all‐optical data processing applications for future optoelectronic interconnect circuits.     

Pentapnictogen heterocyclic monoanions: Structure,stability, and aromaticity

Inorganic planar ring-shape molecules with 4n + 2 π electrons are always the focus of experimental synthesis and theoretical research due to their potential aromaticity and stability. In this work, the whole series of five-membered heterocycle monoanions X _nY_5-n⁻ (X, Y = group 15 elements; n = 1-4) were thoroughly investigated by means of density functional theory calculations. They all have large formation energies and HOMO-LUMO gap energies, suggesting the potential thermodynamic and kinetic stability. Their aromaticities are comparable to that of typical aromatic hydrocarbons. Their thermal stabilities were firmly established by the ab initio molecular dynamics simulations. As most of them are predicted for the first time, their various spectra were simulated for experimental characterization. Furthermore, we demonstrate that these five-membered cyclic anions can be employed as η⁵-ligand to construct novel all-inorganic metallocenes, which may serve as the building blocks of low-dimensional nanomaterials.