一种非单调滤子信赖域算法解线性不等式约束优化

本文给出了一种新的多维滤子算法结合非单调信赖域策略解线性约束优化.目标函数及其投影梯度的分量组成了新的多维滤子，并且与信赖域半径有关.当信赖域半径充分小时，新的滤子能接受试探点，避免算法无限循环.非单调信赖域策略保证了新算法的整体收敛性.目前为止，多维滤子算法局部收敛性分析仍然没有解决，在合理假设下，我们分析了新算法的局部超线性收敛性.数值结果验证了算法的有效性.     

Non-monotone trust region methods for nonlinear equality constrained optimization without a penalty function

 We propose and analyze a class of penalty-function-free nonmonotone trust-region methods for nonlinear equality constrained optimization problems. The algorithmic framework yields global convergence without using a merit function and allows nonmonotonicity independently for both, the constraint violation and the value of the Lagrangian function. Similar to the Byrd–Omojokun class of algorithms, each step is composed of a quasi-normal and a tangential step. Both steps are required to satisfy a decrease condition for their respective trust-region subproblems. The proposed mechanism for accepting steps combines nonmonotone decrease conditions on the constraint violation and/or the Lagrangian function, which leads to a flexibility and acceptance behavior comparable to filter-based methods. We establish the global convergence of the method. Furthermore, transition to quadratic local convergence is proved. Numerical tests are presented that confirm the robustness and efficiency of the approach. Received: December 14, 2000 / Accepted: August 30, 2001 Published online: September 27, 2002 Key words. nonmonotone trust-region methods – sequential quadratic programming – penalty function – global convergence – equality constraints – local convergence – large-scale optimization Mathematics Subject Classification (2000): 65K05, 90C30     

e-filters of MS-algebras

The notion of e-filters is introduced in an MS-algebra and characterized. The concept of D-filters is introduced and a set of equivalent conditions under which every D-filter is an e-filter are given. The properties of the space of all prime e-filters of an MS-algebra are observed. The concept of D-prime filters is introduced and then a set of equivalent conditions are derived for a prime e-filter to become a D-prime filter in topological terms.     

Affine scaling inexact generalized Newton algorithm with interior backtracking technique for solving bound-constrained semismooth equations

We develop and analyze an affine scaling inexact generalized Newton algorithm in association with nonmonotone interior backtracking line technique for solving systems of semismooth equations subject to bounds on variables. By combining inexact affine scaling generalized Newton with interior backtracking line search technique, each iterate switches to inexact generalized Newton backtracking step to strict interior point feasibility. The global convergence results are developed in a very general setting of computing trial steps by the affine scaling generalized Newton-like method that is augmented by an interior backtracking line search technique projection onto the feasible set. Under some reasonable conditions we establish that close to a regular solution the inexact generalized Newton method is shown to converge locally p-order q-superlinearly. We characterize the order of local convergence based on convergence behavior of the quality of the approximate subdifferentials and indicate how to choose an inexact forcing sequence which preserves the rapid convergence of the proposed algorithm. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases.     

An inexact line search approach using modified nonmonotone strategy for unconstrained optimization

This paper concerns with a new nonmonotone strategy and its application to the line search approach for unconstrained optimization. It has been believed that nonmonotone techniques can improve the possibility of finding the global optimum and increase the convergence rate of the algorithms. We first introduce a new nonmonotone strategy which includes a convex combination of the maximum function value of some preceding successful iterates and the current function value. We then incorporate the proposed nonmonotone strategy into an inexact Armijo-type line search approach to construct a more relaxed line search procedure. The global convergence to first-order stationary points is subsequently proved and the R-linear convergence rate are established under suitable assumptions. Preliminary numerical results finally show the efficiency and the robustness of the proposed approach for solving unconstrained nonlinear optimization problems.     

Preconditioned Spectral Projected Gradient Method on Convex Sets

The spectral gradient method has proved to be effective for solving large-scale unconstrained optimization problems. It has been recently extended and combined with the projected gradient method for solving optimization problems on convex sets. This combination includes the use of nonmonotone line search techniques to preserve the fast local convergence. In this work we further extend the spectral choice of steplength to accept preconditioned directions when a good preconditioner is available. We present an algorithmthat combines the spectral projected gradient method with preconditioning strategies toincrease the local speed of convergence while keeping the global properties. We discuss implementation details for solving large-scale problems.     

Strong convergence properties of a modified nonmonotone smoothing algorithm for the SCCP

The smoothing algorithms have been successfully applied to solve the symmetric cone complementarity problem (denoted by SCCP), which in general have the global and local superlinear/quadratic convergence if the solution set of the SCCP is nonempty and bounded. Huang, Hu and Han [Science in China Series A: Mathematics, 52: 833–848, 2009] presented a nonmonotone smoothing algorithm for solving the SCCP, whose global convergence is established by just requiring that the solution set of the SCCP is nonempty. In this paper, we propose a new nonmonotone smoothing algorithm for solving the SCCP by modifying the version of Huang-Hu-Han’s algorithm. We prove that the modified nonmonotone smoothing algorithm not only is globally convergent but also has local superlinear/quadratical convergence if the solution set of the SCCP is nonempty. This convergence result is stronger than those obtained by most smoothing-type algorithms. Finally, some numerical results are reported.     

Nonmonotone filter DQMM method for the system of nonlinear equations

In this paper, we propose a nonmonotone filter Diagonalized Quasi-Newton Multiplier (DQMM) method for solving system of nonlinear equations. The system of nonlinear equations is transformed into a constrained nonlinear programming problem which is then solved by nonmonotone filter DQMM method. A nonmonotone criterion is used to speed up the convergence progress in some ill-conditioned cases. Under reasonable conditions, we give the global convergence properties. The numerical experiments are reported to show the effectiveness of the proposed algorithm.     

An inertial proximal scheme for nonmonotone mappings

We present an inertial proximal method for solving an inclusion involving a nonmonotone set-valued mapping enjoying some regularity properties. More precisely, we investigate the local convergence of an implicit scheme for solving inclusions of the type T(x)∋0 where T is a set-valued mapping acting from a Banach space into itself. We consider subsequently the case when T is strongly metrically subregular, metrically regular and strongly regular around a solution to the inclusion. Finally, we study the convergence of our algorithm under variational perturbations.     

Nonmonotone Levenberg–Marquardt Algorithms and Their Convergence Analysis

In this paper, two nonmonotone Levenberg–Marquardt algorithms for unconstrained nonlinear least-square problems with zero or small residual are presented. These algorithms allow the sequence of objective function values to be nonmonotone, which accelerates the iteration progress, especially in the case where the objective function is ill-conditioned. Some global convergence properties of the proposed algorithms are proved under mild conditions which exclude the requirement for the positive definiteness of the approximate Hessian T(x). Some stronger global convergence properties and the local superlinear convergence of the first algorithm are also proved. Finally, a set of numerical results is reported which shows that the proposed algorithms are promising and superior to the monotone Levenberg–Marquardt algorithm according to the numbers of gradient and function evaluations.     

An affine scaling interior trust-region method combining with line search filter technique for optimization subject to bounds on variables

This paper proposes and analyzes an affine scaling trust-region method with line search filter technique for solving nonlinear optimization problems subject to bounds on variables. At the current iteration, the trial step is generated by the general trust-region subproblem which is defined by minimizing a quadratic function subject only to an affine scaling ellipsoidal constraint. Both trust-region strategy and line search filter technique will switch to trail backtracking step which is strictly feasible. Meanwhile, the proposed method does not depend on any external restoration procedure used in line search filter technique. A new backtracking relevance condition is given which is weaker than the switching condition to obtain the global convergence of the algorithm. The global convergence and fast local convergence rate of this algorithm are established under reasonable assumptions. Preliminary numerical results are reported indicating the practical viability and show the effectiveness of the proposed algorithm.     

RECONDITIONED SPECTRAL PROJECTED GRADIENT METHOD ON CONVEX SETS

The spectral gradient method has proved to be effective for solving large-scale uncon-strained optimization problems.It has been recently extended and combined with theprojected gradient method for solving optimization problems on convex sets.This combi-nation includes the use of nonmonotone line search techniques to preserve the fast localconvergence.In this work we further extend the spectral choice of steplength to accept pre-conditioned directions when a good preconditioner is available.We present an algorithmthat combines the spectral projected gradient method with preconditioning strategies toincrease the local speed of convergence while keeping the global properties.We discussimplementation details for solving large-scale problems.     

Nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization

We propose a nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization. Unlike traditional trust region methods, the subproblem in our method is a simple conic model, where the Hessian of the objective function is approximated by a scalar matrix. The trust region radius is adjusted with a new self-adaptive adjustment strategy which makes use of the information of the previous iteration and current iteration. The new method needs less memory and computational efforts. The global convergence and Q-superlinear convergence of the algorithm are established under the mild conditions. Numerical results on a series of standard test problems are reported to show that the new method is effective and attractive for large scale unconstrained optimization problems.     

Riemann Rearrangement Theorem for some types of convergence

We reexamine the Riemann Rearrangement Theorem for different types of convergence and classify possible sum ranges for statistically convergent series and for series that converge along the 2n-filter.     

A nonmonotone adaptive trust region method for unconstrained optimization based on conic model

In this paper, we present a nonmonotone adaptive trust region method for unconstrained optimization based on conic model. The new method combines nonmonotone technique and a new way to determine trust region radius at each iteration. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments show that our algorithm is effective.     

Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming

Summary. This paper studies projected Barzilai-Borwein (PBB) methods for large-scale box-constrained quadratic programming. Recent work on this method has modified the PBB method by incorporating the Grippo-Lampariello-Lucidi (GLL) nonmonotone line search, so as to enable global convergence to be proved. We show by many numerical experiments that the performance of the PBB method deteriorates if the GLL line search is used. We have therefore considered the question of whether the unmodified method is globally convergent, which we show not to be the case, by exhibiting a counter example in which the method cycles. A new projected gradient method (PABB) is then considered that alternately uses the two Barzilai-Borwein steplengths. We also give an example in which this method may cycle, although its practical performance is seen to be superior to the PBB method. With the aim of both ensuring global convergence and preserving the good numerical performance of the unmodified methods, we examine other recent work on nonmonotone line searches, and propose a new adaptive variant with some attractive features. Further numerical experiments show that the PABB method with the adaptive line search is the best BB-like method in the positive definite case, and it compares reasonably well against the GPCG algorithm of Moré and Toraldo. In the indefinite case, the PBB method with the adaptive line search is shown on some examples to find local minima with better solution values, and hence may be preferred for this reason.This work was supported by the EPRSC in UK (no. GR/R87208/01) and the Chinese NSF grant (no. 10171104).Acknowledgement The authors would like to thank the two anonymous referees for their useful suggestions and comments.     

变步长非单调模式搜索法

A varied steplength nonmonotone pattern search method is proposed in this paper. The varied steplength search strategy is designed in this method such that the pattern direction is more approximated to efficient descent direction. The interpolation and nonmonotone technique are used for improving local search and global convergence. The theoretical and numerical results show that this method is an efficient direct search method.     

The Global Convergence of Self-Scaling BFGS Algorithm with Nonmonotone Line Search for Unconstrained Nonconvex Optimization Problems

The self-scaling quasi-Newton method solves an unconstrained optimization problem by scaling the Hessian approximation matrix before it is updated at each iteration to avoid the possible large eigenvalues in the Hessian approximation matrices of the objective function. It has been proved in the literature that this method has the global and superlinear convergence when the objective function is convex （or even uniformly convex）. We propose to solve unconstrained nonconvex optimization problems by a self-scaling BFGS algorithm with nonmonotone linear search. Nonmonotone line search has been recognized in numerical practices as a competitive approach for solving large-scale nonlinear problems. We consider two different nonmonotone line search forms and study the global convergence of these nonmonotone self-scale BFGS algorithms. We prove that, under some weaker condition than that in the literature, both forms of the self-scaling BFGS algorithm are globally convergent for unconstrained nonconvex optimization problems.     

基于混合非单调下降条件的直接搜索方法

基于混合非单调下降条件提出了一种网格步长的更新策略.这种策略要求发现强最小网格单元框时网格步长快速下降,而发现其他拟最小网格单元框时缓慢下降.这种混合非单调下降策略可以避免网格步长下降太快,又能够比单纯非单调下降条件更好地保证直接搜索算法的收敛性.基于这一策略,本文提出一个直接搜索算法,并证明了该算法的全局收敛性.数值试验表明,本文提出的算法是很有竞争力的直接搜索算法.     

A modified Broyden-like method for nonlinear complementarity problems

In this paper, we give a modified Broyden-like method for nonlinear complementarity problems, where we consider the smoothing parameter as an independent variable in the smoothing equation and adopt a modified nonmonotone step-length decision. We also analyze the global and local convergence of the modified method. Numerical experiments are also shown.