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This paper considers distributed optimization for minimizing the average of local nonconvex cost functions, by using local information exchange over undirected communication networks. To reduce the required communication capacity, we introduce an encoder--decoder scheme. By integrating them with distributed gradient tracking and proportional integral algorithms, respectively, we then propose two quantized distributed nonconvex optimization algorithms. Assuming the global cost function satisfies the Polyak--{\L}ojasiewicz condition, which does not require the global cost function to be convex and the global minimizer is not necessarily unique, we show that our proposed algorithms linearly converge to a global optimal point and that larger quantization level leads to faster convergence speed. Moreover, we show that a low data rate is sufficient to guarantee linear convergence when the algorithm parameters are properly chosen. The theoretical results are illustrated by numerical examples.

Graphene holds great potential for fabricating ultrathin selective membranes possessing high permeability without compromising selectivity and has attracted intensive interest in developing high-performance separation membranes for desalination, natural gas purification, hemodialysis, distillation, and other gas-liquid separation. However, the scalable and cost-effective synthesis of nanoporous graphene membranes, especially designing a method to produce an appropriate porous polymer substrate, remains very challenging. Here, we report a facile route to fabricate decimeter-scale (∼15 × 10 cm2) nanoporous atomically thin membranes (NATMs) via the direct casting of the porous polymer substrate onto graphene, which was produced by chemical vapor deposition (CVD). After the vapor-induced phase-inversion process under proper experimental conditions (60 °C and 60% humidity), the flexible nanoporous polymer substrate was formed. The resultant skin-free polymer substrate, which had the proper pore size and a uniform spongelike structure, provided enough mechanical support without reducing the permeance of the NATMs. It was demonstrated that after creating nanopores by the O2 plasma treatment, the NATMs were salt-resistant and simultaneously showed 3-5 times higher gas (CO2) permeance than the state-of-the-art commercial polymeric membranes. Therefore, our work provides guidance for the technological developments of graphene-based membranes and bridges the gap between the laboratory-scale "proof-of-concept" and the practical applications of NATMs in the industry.

Membrane acoustic metamaterial which is composed of a prestressed membrane and attached mass attracts considerable attention for its low-frequency (