A SUPERLINEARLY CONVERGENT SPLITTING FEASIBLE SEQUENTIAL QUADRATIC OPTIMIZATION METHOD FOR TWO-BLOCK LARGE-SCALE SMOOTH OPTIMIZATION

This paper discusses the two-block large-scale nonconvex optimization problem with general linear constraints.Based on the ideas of splitting and sequential quadratic optimization(SQO),a new feasible descent method for the discussed problem is proposed.First,we consider the problem of quadratic optimal(QO)approximation associated with the current feasible iteration point,and we split the QO into two small-scale QOs which can be solved in parallel.Second,a feasible descent direction for the problem is obtained and a new SQO-type method is proposed,namely,splitting feasible SQO(SF-SQO)method.Moreover,under suitable conditions,we analyse the global convergence,strong convergence and rate of superlinear convergence of the SF-SQO method.Finally,preliminary numerical experiments regarding the economic dispatch of a power system are carried out,and these show that the SF-SQO method is promising.

一类非线性代数方程组的Newton-Triangle Splitting迭代法

Triangle Splitting迭代方法是求解大型稀疏非Hermitian正定线性代数方程组的一种有效迭代算法.为了有效求解大型稀疏且Jacobi矩阵为非Hermitian正定的非线性代数方程组,本文将Triangle Splitting迭代方法作为不精确Newton方法的内迭代求解器,构造了不精确Newton-Triangle Splitting迭代方法.在适当的约束条件下,给出了该方法的两类局部收敛性定理.通过数值实验结果验证了该方法的可行性和有效性,并说明了该方法在计算时间和迭代次数方面比Newton-BTSS迭代方法更有优势.

UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS

WT5,5”BX] A new class of numerical schemes is proposed to solve convection diffusion equations by combining the upwind technique and the method of operator splitting. For every time step, the multi dimensional approximation is performed in several independent directions alternatively, while the upwind technique is applied to treat the convection term in every individual direction. This scheme possesses maximum principle. Stability and convergence are analysed by energy method.[WT5,5”HZ]

THE UPWIND OPERATOR SPLITTING FINITE DIFFERENCE METHOD FOR COMPRESSIBLE TWO-PHASE DISPLACEMENT PROBLEM AND ANALYSIS

For compressible two-phase displacement problem, a kind of upwind operator splitting finite difference schemes is put forward and make use of operator splitting, of calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates are adopted. Optimal order estimates in L 2 norm are derived to determine the error, in the approximate solution.

Configurable topological beam splitting via antichiral gyromagnetic photonic crystal

Antichiral gyromagnetic photonic crystal(GPC)in a honeycomb lattice with the two interpenetrating triangular sublattices A and B magnetically biased in opposite directions can realize antichiral one-way edge states propagating along the same direction at its two parallel edges.Here,we report the construction and observation of topological beam splitting with the easily adjustable right-to-left ratio in an antichiral GPC.The splitter is compact and configurable,has high trans-mission efficiency,and allows for multi-channel utilization,crosstalk-proof,and robust against defects and obstacles.This magnificent performance is attributed to the peculiar property that antichiral one-way edge states exist only at zigzag edge but not at armchair edge of antichiral GPC.When we combine two rectangular antichiral GPCs holding left-and right-propagating antichiral one-way edge states respectively,bidirectionally radiating one-way edge states at two paral-lel zigzag edges can be achieved.Our observations can enrich the understanding of fundamental physics and expand to-pological photonic applications.

New hybrid inertial CQ projection algorithms with line-search process for the split feasibility problem

In this paper, we propose two hybrid inertial CQ projection algorithms with linesearch process for the split feasibility problem. Based on the hybrid CQ projection algorithm, we firstly add the inertial term into the iteration to accelerate the convergence of the algorithm, and adopt flexible rules for selecting the stepsize and the shrinking projection region, which makes an optimal stepsize available at each iteration. The shrinking projection region is the intersection of three sets, which are the set C and two hyperplanes. Furthermore, we modify the Armijo-type line-search step in the presented algorithm to get a new algorithm.The algorithms are shown to be convergent under certain mild assumptions. Besides, numerical examples are given to show that the proposed algorithms have better performance than the general CQ algorithm.

Convergence of Bregman Peaceman–Rachford Splitting Method for Nonconvex Nonseparable Optimization

This work is about a splitting method for solving a nonconvex nonseparable optimization problem with linear constraints,where the objective function consists of two separable functions and a coupled term.First,based on the ideas from Bregman distance and Peaceman–Rachford splitting method,the Bregman Peaceman–Rachford splitting method with different relaxation factors for the multiplier is proposed.Second,the global and strong convergence of the proposed algorithm are proved under general conditions including the region of the two relaxation factors as well as the crucial Kurdyka–Łojasiewicz property.Third,when the associated Kurdyka–Łojasiewicz property function has a special structure,the sublinear and linear convergence rates of the proposed algorithm are guaranteed.Furthermore,some preliminary numerical results are shown to indicate the effectiveness of the proposed algorithm.

An Iterative Method for Split Variational Inclusion Problem and Split Fixed Point Problem for Averaged Mappings

In this paper, we use resolvent operator technology to construct a viscosity approximate algorithm to approximate a common solution of split variational inclusion problem and split fixed point problem for an averaged mapping in real Hilbert spaces. Further, we prove that the sequences generated by the proposed iterative method converge strongly to a common solution of split variational inclusion problem and split fixed point problem for averaged mappings which is also the unique solution of the variational inequality problem. The results presented here improve and extend the corresponding results in this area.

求解分裂可行问题的次梯度投影松弛算法

在无限维Hilbert空间中,区别于现有许多算法中的正交投影,采用次梯度投影法,提出求解分裂可行问题的次梯度投影松弛算法,并利用次梯度算子的cutter性质以及分类讨论的思想,证明了次梯度投影松弛算法生成的序列弱收敛于分裂可行问题的解.

混合Bregman投影算法在Banach空间中分裂不动点问题的强收敛性

在p-一致凸且一致光滑的Banach空间中,利用Bregman投影,构造一新的混合投影迭代算法,逼近Bregman拟严格伪压缩映射不动点集和分裂可行性问题的公共解.目的是将2017年Chen J Z,Hu H Y和Ceng L C的研究结果中的迭代系数α_(n)须满足0<c≤a_(n)≤d<1证明对α_(n)≡1或α_(n)≡0时亦成立.所得的结果是对2017年Chen J Z,Hu H Y和Ceng L C相应结果的拓展和补充.

求解随机微分方程split-step欧拉方法的收敛性

给出随机微分方程的split-step欧拉格式的算法,并证明了当方程的偏移系数和扩散系数均满足线性增长条件和李普希兹条件的情况下,此方法用以求解随机微分方程的收敛性,并且求出强收敛的阶是1/2.同时证明了split-step近似解的均方收敛理论.

Bi-extrapolated subgradient projection algorithm for solving multiple-sets split feasibility problem

This paper deals with a bi-extrapolated subgradient projection algorithm by intro- ducing two extrapolated factors in the iterative step to solve the multiple-sets split feasibility problem. The strategy is intend to improve the convergence. And its convergence is proved un- der some suitable conditions. Numerical results illustrate that the bi-extrapolated subgradient projection algorithm converges more quickly than the existing algorithms.

GENERAL SPLIT FEASIBILITY PROBLEMS FOR TWO FAMILIES OF NONEXPANSIVE MAPPINGS IN HILBERT SPACES

The purpose of this article is to introduce a general split feasibility problems for two families of nonexpansive mappings in Hilbert spaces. We prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split feasibility problem. Our results extend and improve some recent known results.

VISCOSITY APPROXIMATION METHODS FOR THE SPLIT EQUALITY COMMON FIXED POINT PROBLEM OF QUASI-NONEXPANSIVE OPERATORS

Let H;, H;, H;be real Hilbert spaces, let A : H;→ H;, B : H;→ H;be two bounded linear operators. The split equality common fixed point problem（SECFP） in the infinite-dimensional Hilbert spaces introduced by Moudafi（Alternating CQ-algorithm for convex feasibility and split fixed-point problems. Journal of Nonlinear and Convex Analysis）is to find x ∈ F（U）, y ∈ F（T） such that Ax = By,（1）where U : H;→ H;and T : H;→ H;are two nonlinear operators with nonempty fixed point sets F（U） = {x ∈ H;: Ux = x} and F（T） = {x ∈ H;: Tx = x}. Note that,by taking B = I and H;= H;in（1）, we recover the split fixed point problem originally introduced in Censor and Segal. Recently, Moudafi introduced alternating CQ-algorithms and simultaneous iterative algorithms with weak convergence for the SECFP（1） of firmly quasi-nonexpansive operators. In this paper, we introduce two viscosity iterative algorithms for the SECFP（1） governed by the general class of quasi-nonexpansive operators. We prove the strong convergence of algorithms. Our results improve and extend previously discussed related problems and algorithms.

FURTHER INVESTIGATION INTO APPROXIMATION OF A COMMON SOLUTION OF FIXED POINT PROBLEMS AND SPLIT FEASIBILITY PROBLEMS

The purpose of this paper is to study and analyze an iterative method for finding a common element of the solution set ~ of the split feasibility problem and the set F（T） of fixed points of a right Bregman strongly nonexpansive mapping T in the setting of p- uniformly convex Banach spaces which are also uniformly smooth. By combining Mann＇s iterative method and the Halpern＇s approximation method, we propose an iterative algorithm for finding an element of the set F（T）∩Ω moreover, we derive the strong convergence of the proposed algorithm under appropriate conditions and give numerical results to verify the efficiency and implementation of our method. Our results extend and complement many known related results in the literature.

带时滞随机泛函微分方程的Split-step算法

针对一类带有泊松跳的时变时滞随机泛函微分方程,基于Euler-Maruyama算法,给出了Split-step算法。在带跳时滞随机泛函微分方程的系数满足全局Lipschitz条件、线性增长条件和初值函数具有Hlder连续性的条件下,证明了文中的Split-step算法在均方意义下以0.5阶矩收敛。最后通过几个实例进行了数值模拟,验证了算法的有效性。

A Split Least-squares Characteristic Procedure for Convection-dominated Parabolic Integro-differential Equations

In this paper, we combine a split least-squares procedure with the method of characteristics to treat convection-dominated parabolic integro-differential equations. By selecting the least-squares functional properly, the procedure can be split into two independent sub-procedures, one of which is for the primitive unknown and the other is for the flux. Choosing projections carefully, we get optimal order H^1 （Ω） and L^2（Ω） norm error estimates for u and sub-optimal （L^2（Ω））^d norm error estimate for σ. Numerical results are presented to substantiate the validity of the theoretical results.

General Modified Split-Step Balanced Methods for Stiff Stochastic Differential Equations

A class of general modified split-step balanced methods proposed in the paper can be applied to solve stiff stochastic differential systems with m-dimensional multiplicative noise.Compared to some other already reported split-step balanced methods,the drift increment function of the methods can be taken from any chosen one-step ordinary differential equations(ODEs)solver.The schemes is proved to be strong convergent with order one.For the mean-square stability analysis,the investigation is confined to two cases.Some numerical experiments are reported to testify the performance and the effectiveness of the methods.

On Globally Q-Linear Convergence of a Splitting Method for Group Lasso

In this paper,we discuss a splitting method for group Lasso.By assuming that the sequence of the step lengths has positive lower bound and positive upper bound(unrelated to the given problem data),we prove its Q-linear rate of convergence of the distance sequence of the iterates to the solution set.Moreover,we make comparisons with convergence of the proximal gradient method analyzed very recently.

Local Linear Convergence of an ADMM-Type Splitting Framework for Equality Constrained Optimization

We establish local convergence results for a generic algorithmic framework for solving a wide class of equality constrained optimization problems.The framework is based on applying a splitting scheme to the augmented Lagrangian function that includes as a special case the well-known alternating direction method of multipliers(ADMM).Our local convergence analysis is free of the usual restrictions on ADMM-like methods,such as convexity,block separability or linearity of constraints.It offers a much-needed theoretical justification to the widespread practice of applying ADMM-like methods to nonconvex optimization problems.