Global Convergence Rate Research Articles

Coordinated planning and operation of PV- hydrogen integrated distribution network incorporating daily-seasonal green hydrogen storage and EV charging station

The current study introduces an optimal planning and operational framework for a Distribution Network (DN) that integrates Photovoltaic (PV)-green Hydrogen (H2)-based energy system with Electric Vehicle Charging Station (EVCS). An efficient operational strategy is proposed considering both short-term H2 storage (STHS) and long-term H2 storage (LTHS), ensuring uninterrupted power supply. Besides, a novel Improved Harris Hawk Optimization method is developed to enhance the global search capabilities and convergence rate in optimizing the capacity of Solar module, Electrolyzer, and Fuel cells. The objective of the proposed model is to minimize the device costs, energy loss, and carbon emissions, adhering to all operational constraints. The proposed work is tested on 33-bus radial DN and 51-bus real Indian DN considering the time frame of 10 years. The analysis reveals that the PV-STHS-LTHS configuration, reduces total planning costs by 12% and 15% as compared to the base case in the 33-bus and 51-bus systems, respectively. Additionally, the integration of both daily and seasonal storage facilities with PV systems leads to a reduction in carbon emissions by 76% and 81% in the 33-bus and 51-bus systems, respectively. Besides the carbon reduction, the incorporation of daily and cross-seasonal H2 storage increases oxygen release into the environment by 13% for the 33-bus system and 15% for the 51-bus system.

A diffusive predator-prey system with hunting cooperation in predators and prey-taxis: II stationary pattern formation

This paper is a continuation of the study conducted by Ko and Ryu (2024) [5], which introduces and analyzes a generalized predator-prey reaction-diffusion system incorporating (repulsive) prey-taxis and a hunting cooperation effect in predators, under homogeneous Neumann boundary conditions. In the study, the existence and uniqueness of global and classical solutions for the time- and space-dependent system are analytically examined. Furthermore, the study examines the local and global stability and convergence rate at the constant predator-extinction and coexistence states. In our paper, we analyze the stationary system corresponding to the system in [5], with a specific focus on examining the existence and nonexistence of positive and nonconstant solutions. The nonexistence occurs when the diffusion rate of prey is sufficiently high. On the other hand, the existence occurs when the prey-tactic rate is sufficiently high, indicating a strong repulsive prey-taxis, and the diffusion rate of prey is sufficiently low. For this investigation, we separately employ the energy method and the Leray-Schauder degree theory.

Globalization of convergence of the constrained piecewise Levenberg–Marquardt method

We develop linesearch algorithms intended for globalization of convergence of the piecewise Levenberg–Marquardt method for constrained piecewise smooth equation. Conditions ensuring global convergence properties and asymptotic superlinear convergence rate are proposed. The peculiarities of the global convergence results in the piecewise smooth case are discussed and illustrated by examples. We also provide numerical results for unconstrained and constrained reformulations of nonlinear complementarity problems, comparing the performance of the globalized piecewise Levenberg–Marquardt algorithm with some relevant alternatives.

Global convergence rates from relaxed Euler equations to Navier–Stokes equations with Oldroyd-type constitutive laws

In a previous work (Peng and Zhao 2022 J. Math. Fluid Mech. 24 29), it is proved that the 1D full compressible Navier–Stokes equations for a Newtonian fluid can be approximated globally-in-time by a relaxed Euler-type system with Oldroyd’s derivatives and a revised Cattaneo’s constitutive law. These two relaxations turn the whole system into a first-order quasilinear hyperbolic one with partial dissipation. In this paper, we establish the global convergence rates between the smooth solutions to the relaxed Euler-type system and the Navier–Stokes equations over periodic domains. For this purpose, we use stream function techniques together with energy estimates for error systems. These techniques may be applicable to more complicated systems.

An improved moth flame optimization for optimal DG and battery energy storage allocation in distribution systems

Deploying distributed generators (DGs) powered by renewable energy poses a significant challenge for effective power system operation. Optimally scheduling DGs, especially photovoltaic (PV) systems and wind turbines (WTs), is critical because of the unpredictable nature of wind speed and solar radiation. These intermittencies have posed considerable challenges to power grids, including power oscillation, increased losses, and voltage instability. To overcome these challenges, the battery energy storage (BES) system supports the PV unit, while the biomass aids the WT unit, mitigating power fluctuations and boosting supply continuity. Therefore, the main innovation of this study is presenting an improved moth flame optimization algorithm (IMFO) to capture the optimal scheduling of multiple dispatchable and non-dispatchable DGs for mitigating energy loss in power grids, considering different dynamic load characteristics. The IMFO algorithm comprises a new update position expression based on a roulette wheel selection strategy as well as Gaussian barebones (GB) and quasi-opposite-based learning (QOBL) mechanisms to enhance exploitation capability, global convergence rate, and solution precision. The IMFO algorithm's success rate and effectiveness are evaluated using 23rd benchmark functions and compared with the basic MFO algorithm and other seven competitors using rigorous statistical analysis. The developed optimizer is then adopted to study the performance of the 69-bus and 118-bus distribution grids, considering deterministic and stochastic DG's optimal planning. The findings reflect the superiority of the developed algorithm against its rivals, emphasizing the influence of load types and varying generations in DG planning. Numerically, the optimal deployment of BES + PV and biomass + WT significantly maximizes the energy loss reduction percent to 68.3471 and 98.0449 for the 69-bus's commercial load type and to 54.833 and 52.0623 for the 118-bus's commercial load type, respectively, confirming the efficacy of the developed algorithm for maximizing the performance of distribution systems in diverse situations.

New global exponential stability conditions for nonlinear delayed differential systems with three kinds of time-varying delays

For a class of nonlinear differential systems with heterogeneous time-varying delays, including distributed, leakage and transmission time-varying delays, a novel global exponential stability (GES) analysis method was developed. Based on the GES definition, some sufficient conditions and rigorous convergence analysis of nonlinear delayed differential systems are presented directly, which ensure all states to be globally exponentially convergent. The proposed analysis method not only avoids the construction of the Lyapunov–Krasovskii functional, but also uses some simple integral reduction techniques to determine the global exponential convergence rate. Furthermore, the main advantages and low calculation complexity are demonstrated through a theoretical comparison. Finally, three numerical examples are provided to verify the effectiveness of the theoretical results.

Joint sparse optimization: lower-order regularization method and application in cell fate conversion

Multiple measurement signals are commonly collected in practical applications, and joint sparse optimization adopts the synchronous effect within multiple measurement signals to improve model analysis and sparse recovery capability. In this paper, we investigate the joint sparse optimization problem via ℓp,q regularization ( 0⩽q⩽1⩽p ) in three aspects: theory, algorithm and application. In the theoretical aspect, we introduce a weak notion of joint restricted Frobenius norm condition associated with the ℓp,q regularization, and apply it to establish an oracle property and a recovery bound for the ℓp,q regularization of joint sparse optimization problem. In the algorithmic aspect, we apply the well-known proximal gradient algorithm to solve the ℓp,q regularization problems, provide analytical formulas for proximal subproblems of certain specific ℓp,q regularizations, and establish the global convergence and linear convergence rate of the proximal gradient algorithm under some mild conditions. More importantly, we propose two types of proximal gradient algorithms with the truncation technique and the continuation technique, respectively, and establish their convergence to the ground true joint sparse solution within a tolerance relevant to the noise level and the recovery bound under the assumption of restricted isometry property. In the aspect of application, we develop a novel method, based on joint sparse optimization with lower-order regularization and proximal gradient algorithm, to infer the master transcription factors for cell fate conversion, which is a powerful tool in developmental biology and regenerative medicine. Numerical results indicate that the novel method facilitates fast identification of master transcription factors, give raise to the possibility of higher successful conversion rate and in the hope of reducing biological experimental cost.

Application of a Dirichlet Distribution-Based Ensemble Surrogate Model in Aerodynamic Optimization

Surrogate models have been widely applied in the aerodynamic optimization of aircrafts, whereas the traditional individual surrogate models have the defects of low robustness and applicability. In this study, a novel ensemble surrogate model is proposed and applied in the multi-objective optimization of the airfoil. The backpropagation neural network, deep belief network, and kriging surrogate models are selected as the member surrogate models, and the Dirichlet distribution strategy is introduced to adaptively generate the weights of the member surrogate models in constructing the ensemble surrogate model. An improved multi-objective particle swarm optimization (MOPSO) framework is established by employing the α-stable distribution function to enhance the global convergence rate of the algorithm. Based on the improved MOPSO framework in which the ensemble surrogate model is embedded, the multi-objective optimization of the airfoil is conducted. The results indicate that the proposed ensemble-surrogate-model-based optimization obtains better aerodynamic performance of the airfoil under multiple operating conditions, compared to the individual-surrogate-model-based optimization.

Estimating Latent-Variable Panel Data Models Using Parameter-Expanded SEM Methods

This article presents new estimation algorithms for three types of dynamic panel data models with latent variables: factor models, discrete choice models, and persistent-transitory quantile processes. The new methods combine the parameter expansion (PX) ideas of Liu, Rubin, and Wu with the stochastic expectation-maximization (SEM) algorithm in likelihood and moment-based contexts. The goal is to facilitate convergence in models with a large space of latent variables by improving algorithmic efficiency. This is achieved by specifying expanded models within the M step. Effectively, we are proposing new estimators for the pseudo-data within iterations that take into account the fact that the model of interest is misspecified for draws based on parameter values far from the truth. We establish the asymptotic equivalence of the likelihood-based PX-SEM to an alternative SEM algorithm with a smaller expected fraction of missing information compared to the standard SEM based on the original model, implying a faster global convergence rate. Finally, in simulations we show that the new algorithms significantly improve the convergence speed relative to standard SEM algorithms, sometimes dramatically so, by reducing the total computing time from hours to a few minutes.

Convergence rate analysis of the gradient descent–ascent method for convex–concave saddle-point problems

In this paper, we study the gradient descent–ascent method for convex–concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming performance estimation method. The given convergence rate incorporates most parameters of the problem and it is exact for a large class of strongly convex-strongly concave saddle-point problems for one iteration. We also investigate the algorithm without strong convexity and we provide some necessary and sufficient conditions under which the gradient descent–ascent enjoys linear convergence.

Anderson acceleration for PWR whole-core pin-by-pin Nu-TH coupling calculation in NECP-Bamboo2.0

For PWR whole-core pin-by-pin neutronics/thermal-hydraulics (Nu-TH) coupling analysis, the Picard iteration strategy is widely employed benefitting from the convenience of implementation. Nevertheless, it is limited to the global multi-physics convergence rate regarding its oscillatory convergence behaviors in the local solution of each single-physics field. In this paper, the Anderson Acceleration (AA) scheme is employed and investigated to enhance the convergence performance of the pin-by-pin Nu and subchannel TH coupling analysis code NECP-Bamboo2.0. Different from the classical relaxation scheme where the relaxation factor is predetermined and fixed, the AA scheme utilizes the linear combination of several previous solutions to correct the current solution. The coefficients of this linear combination can be obtained by solving the residual equation, making the approach more robust. In addition, the selection of the accelerated solution vector is investigated based on the tightly coupling strategy of Nu-TH in NECP-Bamboo2.0. It is illustrated that the neutron-transport solution vector could have superior theoretical acceleration performance compared to the commonly used TH solution vector in the practical PWR analysis scenario. Finally, the AA scheme in NECP-Bamboo2.0 is validated using the VERA#6 3D single-assembly transport/TH coupling benchmark problem and the VERA#7 3D whole-core multi-physics coupling benchmark. It is demonstrated by the numerical results that the AA scheme possesses good robustness and computational efficiency. For the 3D whole-core VERA#7 benchmark, compared with the classic Picard iteration with a relaxation factor of 0.5, the AA scheme based on the neutron-transport solution vector can reduce the number of global iterations and calculation time by around 80%.

A stochastic gradient method with variance control and variable learning rate for Deep Learning

In this paper we study a stochastic gradient algorithm which rules the increase of the mini-batch size in a predefined fashion and automatically adjusts the learning rate by means of a monotone or non-monotone line search procedure. The mini-batch size is incremented at a suitable a priori rate throughout the iterative process in order that the variance of the stochastic gradients is progressively reduced. The a priori rate is not subject to restrictive assumptions, allowing for the possibility of a slow increase in the mini-batch size. On the other hand, the learning rate can vary non-monotonically throughout the iterations, as long as it is appropriately bounded. Convergence results for the proposed method are provided for both convex and non-convex objective functions. Moreover it can be proved that the algorithm enjoys a global linear rate of convergence on strongly convex functions. The low per-iteration cost, the limited memory requirements and the robustness against the hyperparameters setting make the suggested approach well-suited for implementation within the deep learning framework, also for GPGPU-equipped architectures. Numerical results on training deep neural networks for multi-class image classification show a promising behaviour of the proposed scheme with respect to similar state of the art competitors.

Asynchronous Parallel Large-Scale Gaussian Process Regression.

Gaussian process regression (GPR) is an important nonparametric learning method in machine learning research with many real-world applications. It is well known that training large-scale GPR is a challenging task due to the required heavy computational cost and large volume memory. To address this challenging problem, in this article, we propose an asynchronous doubly stochastic gradient algorithm to handle the large-scale training of GPR. We formulate the GPR to a convex optimization problem, i.e., kernel ridge regression. After that, in order to efficiently solve this convex kernel problem, we first use the random feature mapping method to approximate the kernel model and then utilize two unbiased stochastic approximations, i.e., stochastic variance reduced gradient and stochastic coordinate descent, to update the solution asynchronously and in parallel. In this way, our algorithm scales well in both sample size and dimensionality, and speeds up the training computation. More importantly, we prove that our algorithm has a global linear convergence rate. Our experimental results on eight large-scale benchmark datasets with both regression and classification tasks show that the proposed algorithm outperforms the existing state-of-the-art GPR methods.

A linearized L2-1[formula omitted] Galerkin FEM for Kirchhoff type quasilinear subdiffusion equation with memory

This article aims to construct a novel linearized Galerkin FEM based on Alikhanov’s L2-1σ scheme to solve a Kirchhoff type quasilinear subdiffusion equation with memory Dα. Near time t=0, the weak singularity of the solution to the equation Dα is taken into account by using the graded mesh in the time direction. The nonlocal behavior of Kirchhoff term causes huge computer storage thereby increasing computational cost. We reduce these costs by linearizing the Kirchhoff term. We derive a priori bounds and global rates of convergence for the developed numerical scheme. We prove that the convergence rate of the proposed numerical scheme is first-order in space and second-order in time in L∞(0,T;L2(Ω)). In addition, we prove the same convergence rate in L∞(0,T;H01(Ω)) by making use of a discrete Laplacian operator. A numerical experiment is presented to verify the theoretical results.

Stabilized BB projection algorithm for large-scale convex constrained nonlinear monotone equations to signal and image processing problems

The Barzilai–Borwein (BB) method is a popular and efficient gradient method for solving large-scale unconstrained optimization problems. In general, it converges much faster than the steepest descent (Cauchy) method. However, it may not converge, even when the objective function is strongly convex. To overcome this, by virtue of bounding the distance between sequential iterates, [Burdakov et al. (2019)] proposed a stabilized BB method. In this paper, by combining the stabilized BB method with a projection approach, we develop a stabilized BB projection (SBBP) method to solve nonlinear monotone equations with convex constraints. SBBP does not involve computing matrices proving the usefulness for solving large-scale problems. Its global convergence and Q-linear convergence rate are also established. We report numerical experiments on recovering sparse signals and restoring blurred images arising from compressive sensing, demonstrating the usefulness of SBBP.

Distributed adaptive greedy quasi-Newton methods with explicit non-asymptotic convergence bounds

Though quasi-Newton methods have been extensively studied in the literature, they either suffer from local convergence or use a series of line searches for global convergence which is not easy to implement in the distributed setting. In this work, we first propose a line search free greedy quasi-Newton (GQN) method with adaptive steps and establish explicit non-asymptotic bounds for both the global convergence rate and local superlinear rate. Our novel idea lies in the design of multiple GQN updates, which only involves computing Hessian-vector products, to control the Hessian approximation error, and a simple mechanism to adjust stepsizes to ensure the objective function improvement per iterate. Then, we extend it to the master–worker framework and propose a distributed adaptive GQN method whose communication cost is comparable with that of first-order methods, yet it retains the superb convergence property of its centralized counterpart. Finally, we demonstrate the advantages of our methods via numerical experiments.

基于EMSDBO算法农业植保无人机三维航迹多目标优化研究

Both cruising ability and safety should be considered in the 3D inspection path planning of agricultural unmanned aerial vehicles (UAVs). Specific to a complex working environment, the 3D inspection environment of agricultural UAVs was simulated through terrain modeling and threat modeling. First, the dynamic constraints of flight approaching rate and response time were added to the threat cost, and the 3D mission space model and flight path cost function were constructed considering the influence of UAVs’ turning performance. Second, the offset estimation strategy, variable spiral search strategy, quasi-reverse learning strategy and dimension-by-dimension mutation strategy were introduced into the dung beetle optimizer (DBO) algorithm to improve the global optimization ability and convergence rate of the algorithm. By establishing a three-dimensional trajectory planning model for unmanned aerial vehicles, the trajectory planning is transformed into a multi-objective function optimization problem, and an improved algorithm is used to solve the three-dimensional trajectory planning of unmanned aerial vehicles. The fitness is evaluated by considering the objective function of trajectory cost, terrain cost, and danger level, and the trajectory planning is iteratively optimized. The results indicate that the proposed improved dung beetle algorithm for trajectory planning has lower overall cost and stability in adapting to different complex terrain environments.

An efficient inertial subspace minimization CG algorithm with convergence rate analysis for constrained nonlinear monotone equations

Conjugate gradient (CG) methods are efficient algorithms for unconstrained optimization. An inertial step strategy is usually used to accelerate iterative methods. In 1995, Yuan and Stoer (1995) introduced a subspace study on CG algorithms and presented some subspace minimization CG (SMCG) methods. Motivated by these approaches, we incorporate a new inertial step strategy in SMCG methods and present an efficient inertial SMCG method for solving constrained nonlinear monotone equations by combining with the projection technique. The proposed inertial step strategy makes use of information from recent iterations. It can be considered as an extension of the known inertial step strategy. The resulting direction always satisfies the sufficient descent property and that of trust region. Indeed, these properties do not depend on line search rules. The global convergence and Q-linear convergence rate of the proposed method are established under mild assumptions, without any Lipschitz continuity. Numerical results indicate that the proposed method is very promising.

On constrained Kaczmarz algorithm with momentum for image reconstruction

SummaryThe Kaczmarz algorithm is widely used in image reconstruction. In this paper, we propose an enhanced version of the Kaczmarz algorithm by integrating Nesterov momentum into the Kaczmarz algorithm, denoted as the Kaczmarz algorithm with momentum. Our innovation offers a novel approach to accelerate the convergence rate of the Kaczmarz algorithm significantly. Based on greedy rules for selecting work row, we prove that the Kaczmarz method with momentum can achieve global linear convergence rates. Furthermore, we extend a constrained version of the Kaczmarz algorithm with Nesterov momentum for improved image reconstruction and prove its convergence. Numerical experiments show that the proposed algorithm outperforms the original Kaczmarz algorithm in terms of efficiency for image reconstruction.

The exact worst-case convergence rate of the alternating direction method of multipliers

AbstractRecently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We also study the linear and R-linear convergence of ADMM in terms of dual objective. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak–Łojasiewicz (PŁ) inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two scenarios.