Convergence of a projected gradient method with trust region for nonlinear constrained optimization†

In this paper we describe a projected gradient algorithm with trust region, introducing a nondifferentiable merit function for solving nonlinear constrained optimization problems. We show that this method is globally convergent even if conditions are weak. It is also proved that, when the strict complementarity condition holds, the proposed algorithm can be solved by an equality constrained problem, allowing locally rate of superlinear convergence.     

A nonmonotone trust region method with new inexact line search for unconstrained optimization

In this paper, a new nonmonotone inexact line search rule is proposed and applied to the trust region method for unconstrained optimization problems. In our line search rule, the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value, instead of the current objective function value . We can obtain a larger stepsize in each line search procedure and possess nonmonotonicity when incorporating the nonmonotone term into the trust region method. Unlike the traditional trust region method, the algorithm avoids resolving the subproblem if a trial step is not accepted. Under suitable conditions, global convergence is established. Numerical results show that the new method is effective for solving unconstrained optimization problems.     

无约束最优化锥模型拟牛顿信赖域方法的收敛性(英)

本文研究无约束最优化雄模型拟牛顿信赖域方法的全局收敛性．文章给出了确保这类方法全局收敛的条件．文章还证明了，当用拆线法来求这类算法中锥模型信赖域子问题的近似解时，确保全局收敛的条件得到满足     

无约束最优化的信赖域BB法

梯度法是求解无约束最优化的一类重要方法.步长选取的好坏与梯度法的数值表现息息相关.注意到BB步长隐含了目标函数的二阶信息,本文将BB法与信赖域方法相结合,利用BB步长的倒数去近似目标函数的Hesse矩阵,同时利用信赖域子问题更加灵活地选取梯度法的步长,给出求解无约束最优化问题的单调和非单调信赖域BB法.在适当的假设条件下,证明了算法的全局收敛性.数值试验表明,与已有的求解无约束优化问题的BB类型的方法相比,非单调信赖域BB法中e_k=‖x_k-x~*‖的下降呈现更明显的阶梯状和单调性,因此收敛速度更快.     

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.     

On memory gradient method with trust region for unconstrained optimization

In this paper we present a new memory gradient method with trust region for unconstrained optimization problems. The method combines line search method and trust region method to generate new iterative points at each iteration and therefore has both advantages of line search method and trust region method. It sufficiently uses the previous multi-step iterative information at each iteration and avoids the storage and computation of matrices associated with the Hessian of objective functions, so that it is suitable to solve large scale optimization problems. We also design an implementable version of this method and analyze its global convergence under weak conditions. This idea enables us to design some quick convergent, effective, and robust algorithms since it uses more information from previous iterative steps. Numerical experiments show that the new method is effective, stable and robust in practical computation, compared with other similar methods.     

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.     

On the global convergence of trust region algorithms for unconstrained minimization

Many trust region algorithms for unconstrained minimization have excellent global convergence properties if their second derivative approximations are not too large [2]. We consider how large these approximations have to be, if they prevent convergence when the objective function is bounded below and continuously differentiable. Thus we obtain a useful convergence result in the case when there is a bound on the second derivative approximations that depends linearly on the iteration number.     

A New Nonmonotonic Trust Region Algorithm for A Class of Unconstrained Nonsmooth Optimization

This paper presents a new trust region algorithm for solving a class of composite nonsmooth optimizations. It is distinguished by the fact that this method does not enforce strict monotonicity of the objective function values at successive iterates and that this method extends the existing results for this type of nonlinear optimization with smooth, or piecewise smooth, or convex objective functions or their composition. It is proved that this algorithm is globally convergent under certain conditions. Finally, some numerical results for several optimization problems are reported which show that the nonmonotonic trust region method is competitive with the usual trust region method.     

A New Nonmonotonic Trust Region Algorithm for A Class of Unconstraied Nonsmooth Optimization

This paper preasents a new trust region algorithm for solving a class of composite nonsmooth optimizations.It is distinguished by the fact that this method does not enforce strict monotonicity of the objective function values at successive itereates and that this method extends the existing results for this type of nonlinear optimization with smooth ,or piecewis somooth,or convex objective functions or their composition It is pyoved that this algorithm is globally convergent under certain conditions.Finally,some numerical results for several optimization problems are reported which show that the nonmonotonic trust region method is competitive with the usual trust region method.     

A Line Search and Trust Region Algorithm with Trust Region Radius Converging to Zero

In this paper, we present a new line search and trust region algorithm for unconstrained optimization problem with the trust region radius converging to zero. The new trust region algorithm performs a backtracking line search from the failed, point instead of resolving the subproblem when the trial step results in an increase in the objective function. We show that the algorithm preserves the convergence properties of the traditional trust region algorithms. Numerical results are also given.     

一类变分不等式问题的信赖域算法

基于J.M.Peng研究一类变分不等式问题（简记为VIP）时所提出的价值函数，本文提出了求解强单调的VIP的一个新的信赖域算法。和已有的处理VIP的信赖域方法不同的是：它在每步迭代时，不必求解带信赖域界的子问题，仅解一线性方程组而求得试验步。这样，计算的复杂性一般来说可降低。在通常的假设条件下，文中还证明了算法的整体收敛性。最后，在梯度是半光滑和约束是矩形域的假设下，该算法还是超线性收敛的。     

一类拟牛顿非单调信赖域算法及其收敛性

本文提出了一类求解无约束最优化问题的非单调信赖域算法.将非单调Wolfe线搜索技术与信赖域算法相结合,使得新算-法不仅不需重解子问题,而且在每步迭代都满足拟牛顿方程同时保证目标函数的近似Hasse阵Bk的正定性.在适当的条件下,证明了此算法的全局收敛性.数值结果表明该算法的有效性.     

A trust region and affine scaling interior point method for nonconvex minimization with linear inequality constraints

A trust region and affine scaling interior point method (TRAM) is proposed for a general nonlinear minimization with linear inequality constraints [8]. In the proposed approach, a Newton step is derived from the complementarity conditions. Based on this Newton step, a trust region subproblem is formed, and the original objective function is monotonically decreased. Explicit sufficient decrease conditions are proposed for satisfying the first order and second order necessary conditions.?The objective of this paper is to establish global and local convergence properties of the proposed trust region and affine scaling interior point method. It is shown that the proposed explicit decrease conditions are sufficient for satisfy complementarity, dual feasibility and second order necessary conditions respectively. It is also established that a trust region solution is asymptotically in the interior of the proposed trust region subproblem and a properly damped trust region step can achieve quadratic convergence. Received: January 29, 1999 / Accepted: November 22, 1999?Published online February 23, 2000     

Truncated Trust Region Methods Based on Preconditioned Iterative Subalgorithms for Large Sparse Systems of Nonlinear Equations

This paper is devoted to globally convergent methods for solving large sparse systems of nonlinear equations with an inexact approximation of the Jacobian matrix. These methods include difference versions of the Newton method and various quasi-Newton methods. We propose a class of trust region methods together with a proof of their global convergence and describe an implementable globally convergent algorithm which can be used as a realization of these methods. Considerable attention is concentrated on the application of conjugate gradient-type iterative methods to the solution of linear subproblems. We prove that both the GMRES and the smoothed COS well-preconditioned methods can be used for the construction of globally convergent trust region methods. The efficiency of our algorithm is demonstrated computationally by using a large collection of sparse test problems.     

A non-monotone trust region algorithm for unconstrained optimization with dynamic reference iteration updates using filter

We present a non-monotone trust region algorithm for unconstrained optimization. Using the filter technique of Fletcher and Leyffer, we introduce a new filter acceptance criterion and use it to define reference iterations dynamically. In contrast with the early filter criteria, the new criterion ensures that the size of the filter is finite. We also show a correlation between problem dimension and the filter size. We prove the global convergence of the proposed algorithm to first- and second-order critical points under suitable assumptions. It is significant that the global convergence analysis does not require the common assumption of monotonicity of the sequence of objective function values in reference iterations, as assumed by the standard non-monotone trust region algorithms. Numerical experiments on the CUTEr problems indicate that the new algorithm is competitive compared to some representative non-monotone trust region algorithms.     

求解非光滑凸规划的一种混合束方法

提出一种求解非光滑凸规划问题的混合束方法. 该方法通过对目标函数增加迫近项, 且对可行域增加信赖域约束进行迭代, 做为迫近束方法与信赖域束方法的有机结合, 混合束方法自动在二者之间切换, 收敛性分析表明该方法具有全局收敛性. 最后的数值算例验证了算法的有效性．     

一类新的非单调信赖域算法

提出了一类带线性搜索的非单调信赖域算法.算法将非单调Armijo线性搜索技术与信赖域方法相结合,使算法不需重解子问题.而且由于采用了MBFGS校正公式,使矩阵Bk能较好地逼近目标函数的Hesse矩阵并保持正定传递.在较弱的条件下,证明了算法的全局收敛性.数值结果表明算法是有效的.     

A Trust Region Method for Optimization Problem with Singular Solutions

In this paper, we propose a trust region method for minimizing a function whose Hessian matrix at the solutions may be singular. The global convergence of the method is obtained under mild conditions. Moreover, we show that if the objective function is LC ² function, the method possesses local superlinear convergence under the local error bound condition without the requirement of isolated nonsingular solution. This is the first regularized Newton method with trust region technique which possesses local superlinear (quadratic) convergence without the assumption that the Hessian of the objective function at the solution is nonsingular. Preliminary numerical experiments show the efficiency of the method. This work is partly supported by the National Natural Science Foundation of China (Grant Nos. 70302003, 10571106, 60503004, 70671100) and Science Foundation of Beijing Jiaotong University (2007RC014).     

一种解无约束优化问题的新的非单调自适应信赖域方法

本文提出了一种解无约束优化问题的新的非单调自适应信赖域方法.这种方法借助于目标函数的海赛矩阵的近似数量矩阵来确定信赖域半径.在通常的条件下,给出了新算法的全局收敛性以及局部超线性收敛的结果,数值试验验证了新的非单调方法的有效性.