A trust region algorithm for nonsmooth optimization

A trust region algorithm is proposed for minimizing the nonsmooth composite functionF(x) = h(f(x)), wheref is smooth andh is convex. The algorithm employs a smoothing function, which is closely related to Fletcher's exact differentiable penalty functions. Global and local convergence results are given, considering convergence to a strongly unique minimizer and to a minimizer satisfying second order sufficiency conditions.     

Conditions for convergence of trust region algorithms for nonsmooth optimization

This paper discusses some properties of trust region algorithms for nonsmooth optimization. The problem is expressed as the minimization of a functionh(f(x), whereh(·) is convex andf is a continuously differentiable mapping from ℝ″ to ℝ‴. Bounds for the second order derivative approximation matrices are discussed. It is shown that Powel’s [7, 8] results hold for nonsmooth optimization.     

On the superlinear convergence of a trust region algorithm for nonsmooth optimization

It is proved that the second order correction trust region algorithm of Fletcher [5] ensures superlinear convergence if some mild conditions are satisfied.     

A trust region method for minimization of nonsmooth functions with linear constraints

We introduce a trust region algorithm for minimization of nonsmooth functions with linear constraints. At each iteration, the objective function is approximated by a model function that satisfies a set of assumptions stated recently by Qi and Sun in the context of unconstrained nonsmooth optimization. The trust region iteration begins with the resolution of an “easy problem”, as in recent works of Martínez and Santos and Friedlander, Martínez and Santos, for smooth constrained optimization. In practical implementations we use the infinity norm for defining the trust region, which fits well with the domain of the problem. We prove global convergence and report numerical experiments related to a parameter estimation problem. Supported by FAPESP (Grant 90/3724-6), FINEP and FAEP-UNICAMP. Supported by FAPESP (Grant 90/3724-6 and grant 93/1515-9).     

非线性优化的非单调信赖域算法及全局收敛性

Abstract. A trust region algorithm for equality constrained optimization is given in this paper.The algorithm does not enforce strict monotonicity of the merit function for every iteration.Global convergence of the algorithm is proved under the same conditions of usual trust regionmethod.     

A nonmonotonic trust region algorithm for a class of semi-infinite minimax programming

Based on the discretization methods for solving semi-infinite programming problems, this paper presents a new nonmonotonic trust region algorithm for a class of semi-infinite minimax programming problem. Under some mild assumptions, the global convergence of the proposed algorithm is given. Numerical tests are reported that show the efficiency of the proposed method.     

Global and local convergence of a nonmonotone trust region algorithm for equality constrained optimization

This paper presents a nonmonotone trust region algorithm for equality constrained optimization problems. Under certain conditions, we obtain not only the global convergence in the sense that every limit point is a stationary point but also the one step superlinear convergence rate. Numerical tests are also given to show the efficiency of the proposed algorithm.     

A hybrid trust region algorithm for unconstrained optimization

This paper presents a hybrid trust region algorithm for unconstrained optimization problems. It can be regarded as a combination of ODE-based methods, line search and trust region techniques. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step. Further, when the trial step is not accepted, the method performs an inexact line search along it instead of resolving a new linear system. Under reasonable assumptions, the algorithm is proven to be globally and superlinearly convergent. Numerical results are also reported that show the efficiency of this proposed method.     

On superlinear convergence of quasi-Newton methods for nonsmooth equations

We show that strong differentiability at solutions is not necessary for superlinear convergence of quasi-Newton methods for solving nonsmooth equations. We improve the superlinear convergence result of Ip and Kyparisis for general quasi-Newton methods as well as the Broyden method. For a special example, the Newton method is divergent but the Broyden method is superlinearly convergent.     

Nonmonotone trust region algorithm for unconstrained optimization problems

In this paper, a nonmonotone trust region algorithm for unconstrained optimization problems is presented. In the algorithm, a kind of nonmonotone technique, which is evidently different from Grippo, Lampariello and Lucidi’s approach, is used. Under mild conditions, global and local convergence results of the algorithm are established. Preliminary numerical results show that the new algorithm is efficient.     

An adaptive trust region method and its convergence

In this paper, a new trust region subproblem is proposed. The trust radius in the new subproblem adjusts itself adaptively. As a result, an adaptive trust region method is constructed based on the new trust region subproblem. The local and global convergence results of the adaptive trust region method are proved. Numerical results indicate that the new method is very efficient.     

A class of nonmonotone trust region algorithms for unconstrained optimization problems

A class of nonmonotone trust region algorithms is presented for unconstrained optimizations. Under suitable conditions, the global and Q-quadratic convergences of the algorithm are proved. Several rules of choosing trial steps and trust region radii are also discussed. Project supported by the National Natural Science Foundation of China (Grant No. 19136012).     

A globally convergent trust region algorithm for optimization with general constraints and simple bounds

1.IntroductionInthispaper,weconsiderthefollowingnonlinearprogr~ngproblemwherec(x)=(c,(x),c2(2),',We(.))',i(x)andci(x)(i=1,2,',m)arerealfunctions*ThisworkissupPOrtedbytheNationalNaturalScienceFOundationofChinaandtheManagement,DecisionandinformationSystemLab,theChineseAcademyofSciences.definedinD={xEReIISx5u}.Weassumethath相似文献   

On convergence rates of subgradient optimization methods

Rates of convergence of subgradient optimization are studied. If the step size is chosen to be a geometric progression with ratio the convergence, if it occurs, is geometric with rate. For convergence to occur, it is necessary that the initial step size be large enough, and that the ratio be greater than a sustainable ratez(), which depends upon a condition number, defined for both differentiable and nondifferentiable functions. The sustainable ratez() is closely related to the rate of convergence of the steepest ascent method for differentiable functions: in fact it is identical if the function is not too well conditioned.This research was supported in part by the D.G.E.S. (Quebec) and the N.R.C. of Canada under grants A8970 and A4152.     

A trust region algorithm for minimization of locally Lipschitzian functions

The classical trust region algorithm for smooth nonlinear programs is extended to the nonsmooth case where the objective function is only locally Lipschitzian. At each iteration, an objective function that carries both first and second order information is minimized over a trust region. The term that carries the first order information is an iteration function that may not explicitly depend on subgradients or directional derivatives. We prove that the algorithm is globally convergent. This convergence result extends the result of Powell for minimization of smooth functions, the result of Yuan for minimization of composite convex functions, and the result of Dennis, Li and Tapia for minimization of regular functions. In addition, compared with the recent model of Pang, Han and Rangaraj for minimization of locally Lipschitzian functions using a line search, this algorithm has the same convergence property without assuming positive definiteness and uniform boundedness of the second order term. Applications of the algorithm to various nonsmooth optimization problems are discussed.This author's work was supported in part by the Australian Research Council.This author's work was carried out while he was visiting the Department of Applied Mathematics at the University of New South Wales.     

A nonmonotone filter trust region method for nonlinear constrained optimization

In this paper, we present a nonmonotone filter trust region algorithm for solving nonlinear equality constrained optimization. Similar to Bryd–Omojokun class of algorithms, each step is composed of a quasi-normal step and a tangential step. This new method has more flexibility for the acceptance of the trial step compared to the filter methods, and requires less computational costs compared with the monotone methods. Under reasonable conditions, we give the globally convergence properties. Numerical tests are presented that confirm the efficiency of the approach.     

On convergence analysis of a derivative-free trust region algorithm for constrained optimization with separable structure

In this paper,we propose a derivative-free trust region algorithm for constrained minimization problems with separable structure,where derivatives of the objective function are not available and cannot be directly approximated.At each iteration,we construct a quadratic interpolation model of the objective function around the current iterate.The new iterates are generated by minimizing the augmented Lagrangian function of this model over the trust region.The filter technique is used to ensure the feasibility and optimality of the iterative sequence.Global convergence of the proposed algorithm is proved under some suitable assumptions.     

A modified integral global optimization method and its asymptotic convergence

In this paper, we propose a new integral global optimization algorithm for finding the solution of continuous minimization problem, and prove the asymptotic convergence of this algorithm. In our modified method we use variable measure integral, importance sampling and main idea of the cross-entropy method to ensure its convergence and efficiency. Numerical results show that the new method is very efficient in some challenging continuous global optimization problems.     

New filled functions for nonsmooth global optimization

The filled function method is an effective approach to find a global minimizer for a general class of nonsmooth programming problems with a closed bounded domain. This paper gives a new definition for the filled function, which overcomes some drawbacks of the previous definition. It proposes a two-parameter filled function and a one-parameter filled function to improve the efficiency of numerical computation. Based on these analyses, two corresponding filled function algorithms are presented. They are global optimization methods which modify the objective function as a filled function, and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found. Numerical results obtained indicate the efficiency and reliability of the proposed filled function methods.