Local pointwise convergence of the 3D finite element

For an elliptic problem with variable coefficients in three dimensions,this article discusses local pointwise convergence of the three-dimensional(3D)finite element.First,the Green's function and the derivative Green's function are introduced.Secondly,some relationship of norms such as L^(2)-norms,W^(1,∞)-norms,and negative-norms in locally smooth subsets of the domainΩis derived.Finally,local pointwise convergence properties of the finite element approximation are obtained.

ON THE LOCALIZATION AND CONVERGENCE OF MULTIPLE FOURIER INTEGRAL BY BOCHNER-RIESZ MEANS

In this paper we consider lim _(R-) B_R^(f,x_0), in one case that f_x_0 (t) is a ABMV function on [0, ∞], and in another case that f∈L_(m-1)~1(R~) and x^k/~kf∈BV(R) when |k| = m-1 and f(x) = 0 when |x -x_0|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]).

A GLOBAL LINEAR AND LOCAL QUADRATIC SINGLE-STEP NONINTERIOR CONTINUATION METHOD FOR MONOTONE SEMIDEFINITE COMPLEMENTARITY PROBLEMS

A noninterior continuation method is proposed for semidefinite complementarity problem （SDCP）. This method improves the noninterior continuation methods recently developed for SDCP by Chen and Tseng. The main properties of our method are： （i） it is well d.efined for the monotones SDCP; （ii） it has to solve just one linear system of equations at each step; （iii） it is shown to be both globally linearly convergent and locally quadratically convergent under suitable assumptions.

Convergence Analysis of General Version of Gauss-Type Proximal Point Method for Metrically Regular Mappings

We introduce and study in the present paper the general version of Gauss-type proximal point algorithm (in short GG-PPA) for solving the inclusion , where T is a set-valued mapping which is not necessarily monotone acting from a Banach space X to a subset of a Banach space Y with locally closed graph. The convergence of the GG-PPA is present here by choosing a sequence of functions with , which is Lipschitz continuous in a neighbourhood O of the origin and when T is metrically regular. More precisely, semi-local and local convergence of GG-PPA are analyzed. Moreover, we present a numerical example to validate the convergence result of GG-PPA.

HOMOCENTRIC CONVERGENCE BALL OF THE SECANT METHOD

A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper. For every convergence theorem, a convergence ball is respectively introduced, where the hypothesis conditions of the corresponding theorem can be satisfied. Since all of these convergence balls have the same center x^＊, they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.

Coevolutionary Framework for Generalized Multimodal Multi-Objective Optimization

Most multimodal multi-objective evolutionary algorithms(MMEAs)aim to find all global Pareto optimal sets(PSs)for a multimodal multi-objective optimization problem(MMOP).However,in real-world problems,decision makers(DMs)may be also interested in local PSs.Also,searching for both global and local PSs is more general in view of dealing with MMOPs,which can be seen as generalized MMOPs.Moreover,most state-of-theart MMEAs exhibit poor convergence on high-dimension MMOPs and are unable to deal with constrained MMOPs.To address the above issues,we present a novel multimodal multiobjective coevolutionary algorithm(Co MMEA)to better produce both global and local PSs,and simultaneously,to improve the convergence performance in dealing with high-dimension MMOPs.Specifically,the Co MMEA introduces two archives to the search process,and coevolves them simultaneously through effective knowledge transfer.The convergence archive assists the Co MMEA to quickly approach the Pareto optimal front.The knowledge of the converged solutions is then transferred to the diversity archive which utilizes the local convergence indicator and the-dominance-based method to obtain global and local PSs effectively.Experimental results show that Co MMEA is competitive compared to seven state-of-the-art MMEAs on fifty-four complex MMOPs.

SMOOTHING NEWTON ALGORITHM FOR THE CIRCULAR CONE PROGRAMMING WITH A NONMONOTONE LINE SEARCH

In this paper, we present a nonmonotone smoothing Newton algorithm for solving the circular cone programming(CCP) problem in which a linear function is minimized or maximized over the intersection of an affine space with the circular cone. Based on the relationship between the circular cone and the second-order cone(SOC), we reformulate the CCP problem as the second-order cone problem(SOCP). By extending the nonmonotone line search for unconstrained optimization to the CCP, a nonmonotone smoothing Newton method is proposed for solving the CCP. Under suitable assumptions, the proposed algorithm is shown to be globally and locally quadratically convergent. Some preliminary numerical results indicate the effectiveness of the proposed algorithm for solving the CCP.

A Quadratically Approximate Framework for Constrained Optimization,Global and Local Convergence

This paper presents a quadratically approximate algorithm framework （QAAF） for solving general constrained optimization problems, which solves, at each iteration, a subproblem with quadratic objective function and quadratic equality together with inequality constraints. The global convergence of the algorithm framework is presented under the Mangasarian-Fromovitz constraint qualification （MFCQ）, and the conditions for superlinear and quadratic convergence of the algorithm framework are given under the MFCQ, the constant rank constraint qualification （CRCQ） as well as the strong second-order sufficiency conditions （SSOSC）. As an incidental result, the definition of an approximate KKT point is brought forward, and the global convergence of a sequence of approximate KKT points is analysed.

SOME IMPROVED PROJECTED QUASI-NEWTON ALGORITHMS AND THEIR CONVERGENCE Ⅱ.LOCAL CONVERGENCE RATE AND NUMERICAL TESTS

For the improved two-sided projected quasi-Newton algorithms, which were presented in PartI, we prove in this paper that they are locally one-step or two-step superlinearly convergent. Numerical tests are reported thereafter. Results by solving a set of typical problems selectedfrom literature have demonstrated the extreme importance of these modifications in making Nocedal& Overton's original methon practical. Furthermore, these results show that the improved algoritnmsare very competitive in comparison with some highly praised sequential quadratic programmingmethods.

NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities （BVIs） was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it. Global and local superlinear/ quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.

Improved Dual Algorithm for Constrained Optimization Problems

One class of effective methods for the optimization problem with inequality constraints are to transform the problem to a unconstrained optimization problem by constructing a smooth potential function. In this paper, we modifies a dual algorithm for constrained optimization problems and establishes a corresponding improved dual algorithm; It is proved that the improved dual algorithm has the local Q-superlinear convergence; Finally, we performed numerical experimentation using the improved dual algorithm for many constrained optimization problems, the numerical results are reported to show that it is valid in practical computation.

On the local convergence of a stochastic semismooth Newton method for nonsmooth nonconvex optimization

In this work,we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth convex terms in the objective function.We assume that the gradient and Hessian information of the smooth part of the objective function can only be approximated and accessed via calling stochastic firstand second-order oracles.The approach combines stochastic semismooth Newton steps,stochastic proximal gradient steps and a globalization strategy based on growth conditions.We present tail bounds and matrix concentration inequalities for the stochastic oracles that can be utilized to control the approximation errors via appropriately adjusting or increasing the sampling rates.Under standard local assumptions,we prove that the proposed algorithm locally turns into a pure stochastic semismooth Newton method and converges r-linearly or r-superlinearly with high probability.

Local Convergence for a Fifth Order Traub-Steffensen-Chebyshev-Like Composition Free of Derivatives in Banach Space

We present the local convergence analysis of a fifth order Traub-Steffensen-Chebyshev-like composition for solving nonlinear equations in Banach spaces.In earlier studies,hypotheses on the Fréchet derivative up to the fifth order of the operator un-der consideration is used to prove the convergence order of the method although only divided differences of order one appear in the method.That restricts the applicability of the method.In this paper,we extended the applicability of the fifth order Traub-Steffensen-Chebyshev-like composition without using hypotheses on the derivatives of the operator involved.Our convergence conditions are weaker than the conditions used in earlier studies.Numerical examples where earlier results cannot apply to solve equa-tions but our results can apply are also given in this study.

A NEW ADAPTIVE TRUST REGION ALGORITHM FOR OPTIMIZATION PROBLEMS

It is well known that trust region methods are very effective for optimization problems. In this article, a new adaptive trust region method is presented for solving uncon- strained optimization problems. The proposed method combines a modified secant equation with the BFGS updated formula and an adaptive trust region radius, where the new trust region radius makes use of not only the function information but also the gradient information. Under suitable conditions, global convergence is proved, and we demonstrate the local superlinear convergence of the proposed method. The numerical results indicate that the proposed method is very efficient.

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.

PARALLEL NONLINEAR MULTISPLITTING RELAXATION METHODS

By further generalizing Frommer's results in the sense of nonlinear multisplitting, we build a class of nonlinear multisplitting AOR-type methods, which covers many rather practical nonlinear multisplitting relaxation methods such as multisplitting AOR-Newton method, multisplitting AOR-chord method and multisplitting AOR-Steffensen method, etc.. Furthermore,a general convergence theorem for the nonlinear multisplitting AOR-type methods and the local convergence for the multisplitting AOR-Newton method are discussed in detail.A lot of numerical tests show that our new methods are feasible and satisfactory.

Convergence of a Non-interior Continuation Algorithm for the Monotone SCCP

It is well known that the symmetric cone complementarity problem（SCCP） is a broad class of optimization problems which contains many optimization problems as special cases.Based on a general smoothing function,we propose in this paper a non-interior continuation algorithm for solving the monotone SCCP.The proposed algorithm solves at most one system of linear equations at each iteration.By using the theory of Euclidean Jordan algebras,we show that the algorithm is globally linearly and locally quadratically convergent under suitable assumptions.

SECOND-ORDER CONVERGENCE PROPERTIES OF TRUST-REGION METHODS USING INCOMPLETE CURVATURE INFORMATION, WITH AN APPLICATION TO MULTIGRID OPTIMIZATION

Convergence properties of trust-region methods for unconstrained nonconvex optimization is considered in the case where information on the objective function＇s local curvature is incomplete, in the sense that it may be restricted to a fixed set of ＂test directions＂ and may not be available at every iteration. It is shown that convergence to local ＂weak＂ minimizers can still be obtained under some additional but algorithmically realistic conditions. These theoretical results are then applied to recursive multigrid trust-region methods, which suggests a new class of algorithms with guaranteed second-order convergence properties.

On the Convergence of Broyden-Like Methods

The author provides a finer local as well as semilocM convergence analysis of a certain class of Broyden-like methods for solving equations containing a nondifferentiable term on the m-dimensional Euclidean space （m ≥ 1 a natural number）.

A STOCHASTIC NEWTON METHOD FOR NONLINEAR EQUATIONS

In this paper,we study a stochastic Newton method for nonlinear equations,whose exact function information is difficult to obtain while only stochastic approximations are available.At each iteration of the proposed algorithm,an inexact Newton step is first computed based on stochastic zeroth-and first-order oracles.To encourage the possible reduction of the optimality error,we then take the unit step size if it is acceptable by an inexact Armijo line search condition.Otherwise,a small step size will be taken to help induce desired good properties.Then we investigate convergence properties of the proposed algorithm and obtain the almost sure global convergence under certain conditions.We also explore the computational complexities to find an approximate solution in terms of calls to stochastic zeroth-and first-order oracles,when the proposed algorithm returns a randomly chosen output.Furthermore,we analyze the local convergence properties of the algorithm and establish the local convergence rate in high probability.At last we present preliminary numerical tests and the results demonstrate the promising performances of the proposed algorithm.