Decentralized projected Riemannian gradient method for smooth optimization on compact submanifolds

We consider the problem of decentralized nonconvex optimization over a compact submanifold, where each local agent's objective function defined by the local dataset is smooth. Leveraging the powerful tool of proximal smoothness, we establish local linear convergence of the projected gradient descent method with unit step size for solving the consensus problem over the compact manifold. This serves as the basis for analyzing decentralized algorithms on manifolds. Then, we propose two decentralized methods, namely the decentralized projected Riemannian gradient descent (DPRGD) and the decentralized projected Riemannian gradient tracking (DPRGT) methods. We establish their convergence rates of $\mathcal{O}(1/\sqrt{K})$ and $\mathcal{O}(1/K)$, respectively, to reach a stationary point. To the best of our knowledge, DPRGT is the first decentralized algorithm to achieve exact convergence for solving decentralized optimization over a compact manifold. The key ingredients in the proof are the Lipschitz-type inequalities of the projection operator on the compact manifold and smooth functions on the manifold, which could be of independent interest. Finally, we demonstrate the effectiveness of our proposed methods compared to state-of-the-art ones through numerical experiments on eigenvalue problems and low-rank matrix completion.Comment: 32 page

Effect of Milling on the Properties of High Permeability Mn-Zn Ferrite Powders

AbstractHigh permeability Mn-Zn ferrites with broad frequency characteristic were prepared by the traditional ceramic method using the pre-sintering powders of Mn-Zn ferrite wastes. ICP Atomic Emission Spectrophotometer, scanning electron microscopy (SEM), Particle Sizer and LCR broadband digital bridge were used to investigate the influence of milling on the microstructure and properties of sintered Mn-Zn ferrite. During the milling, the impurities such as silicon and chromium were introduced into the powders as using steel balls. The particle size of powders decreased to a †minimum size with the increasing of milling time. As the particle size after milling was about 1 um, the initial permeability of Mn-Zn ferrites with small internal porosity, high density and uniform structure could reach 10000. Furthermore, the value were unaffected by the change in frequency 0 to 150 kHz

Riemannian Smoothing Gradient Type Algorithms]{Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space

In this paper, we introduce the notion of generalized $\epsilon$-stationarity for a class of nonconvex and nonsmooth composite minimization problems on compact Riemannian submanifold embedded in Euclidean space. To find a generalized $\epsilon$-stationarity point, we develop a family of Riemannian gradient-type methods based on the Moreau envelope technique with a decreasing sequence of smoothing parameters, namely Riemannian smoothing gradient and Riemannian smoothing stochastic gradient methods. We prove that the Riemannian smoothing gradient method has the iteration complexity of $\mathcal{O}(\epsilon^{-3})$ for driving a generalized $\epsilon$-stationary point. To our knowledge, this is the best-known iteration complexity result for the nonconvex and nonsmooth composite problem on manifolds. For the Riemannian smoothing stochastic gradient method, one can achieve the iteration complexity of $\mathcal{O}(\epsilon^{-5})$ for driving a generalized $\epsilon$-stationary point. Numerical experiments are conducted to validate the superiority of our algorithms

BmILF and I-motif Structure Are Involved in Transcriptional Regulation of \u3cem\u3eBmPOUM2\u3c/em\u3e in \u3cem\u3eBombyx mori\u3c/em\u3e

Guanine-rich and cytosine-rich DNA can form four-stranded DNA secondary structures called G-quadruplex (G4) and i-motif, respectively. These structures widely exist in genomes and play important roles in transcription, replication, translation and protection of telomeres. In this study, G4 and i-motif structures were identified in the promoter of the transcription factor gene BmPOUM2, which regulates the expression of the wing disc cuticle protein gene (BmWCP4) during metamorphosis. Disruption of the i-motif structure by base mutation, anti-sense oligonucleotides (ASOs) or inhibitory ligands resulted in significant decrease in the activity of the BmPOUM2 promoter. A novel i-motif binding protein (BmILF) was identified by pull-down experiment. BmILF specifically bound to the i-motif and activated the transcription of BmPOUM2. The promoter activity of BmPOUM2 was enhanced when BmILF was over-expressed and decreased when BmILF was knocked-down by RNA interference. This study for the first time demonstrated that BmILF and the i-motif structure participated in the regulation of gene transcription in insect metamorphosis and provides new insights into the molecular mechanism of the secondary structures in epigenetic regulation of gene transcription

Evaluation on Optimal Scale of Rural Fixed-asset Investment – Based on Microcosmic Perspective of Farmers’ Income Increase

The rural fundamental and productive fixed-asset investment not only makes active influence on the changes of farmers’ operational, wages and property income, but it also has an optimal scale range for farmers’ income increase. From the perspective of farmers’ income increase, this article evaluates the optimal scale of rural fixed-asset investment by setting up model with statistic data, and the results show that the optimal scale of per capita rural fixed-asset investment is 76.35% of per capita net income of rural residents, which has been reached in China in 2009. Therefore, compared with the adding of rural fixed-asset investment, a better income increase effect can be achieved through the adjustment of rural fixed-asset investment structure

Location and Expansion of Electric Bus Charging Stations Based on Gridded Affinity Propagation Clustering and a Sequential Expansion Rule

With the escalating contradiction between the growing demand for electric buses and limited supporting resources of cities to deploy electric charging infrastructure, it is a great challenge for decision-makers to synthetically plan the location and decide on the expansion sequence of electric charging stations. In light of the location decisions of electric charging stations having long-term impacts on the deployment of electric buses and the layout of city traffic networks, a comprehensive framework for planning the locations and deciding on the expansion of electric bus charging stations should be developed simultaneously. In practice, construction or renovation of a new charging station is limited by various factors, such as land resources, capital investment, and power grid load. Thus, it is necessary to develop an evaluation structure that combines these factors to provide integrated decision support for the location of bus charging stations. Under this background, this paper develops a gridded affinity propagation (AP) clustering algorithm that combines the superiorities of the AP clustering algorithm and the map gridding rule to find the optimal candidate locations for electric bus charging stations by considering multiple impacting factors such as land cost, traffic conditions, and so on. Based on the location results of the candidate stations, the expansion sequence of these candidate stations is proposed. In particular, a sequential expansion rule for planning the charging stations is proposed that considers the development trends of the charging demand. To verify the performance of the gridded AP clustering and the effectiveness of the proposed sequential expansion rule, an empirical investigation of Guiyang City, the capital of Guizhou province in China, is conducted. The results of the empirical investigation demonstrate that the proposed framework that helps find optimal locations for electric bus charging stations and the expansion sequence of these locations are decided with less capital investment pressure. This research shows that the combination of gridded AP clustering and the proposed sequential expansion rule can systematically solve the problem of finding the optimal locations and deciding on the best expansion sequence for electric bus charging stations, which denotes that the proposed structure is pretty pragmatic and would benefit the government for long-term investment in electric bus station deployment