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.     

A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC^1 CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, a new trust region algorithm for nonlinear equality constrained LC^1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subproblem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.     

AN INTERIOR TRUST REGION ALGORITHM FOR NONLINEAR MINIMIZATION WITH LINEAR CONSTRAINTS

AbstractAn interior trust-region-based algorithm for linearly constrained minimization problems is proposed and analyzed. This algorithm is similar to trust region algorithms for unconstrained minimization: a trust region subproblem on a subspace is solved in each iteration. We establish that the proposed algorithm has convergence properties analogous to those of the trust region algorithms for unconstrained minimization. Namely, every limit point of the generated sequence satisfies the Krush-Kuhn-Tucker (KKT) conditions and at least one limit point satisfies second order necessary optimality conditions. In addition, if one limit point is a strong local minimizer and the Hessian is Lipschitz continuous in a neighborhood of that point, then the generated sequence converges globally to that point in the rate of at least 2-step quadratic. We are mainly concerned with the theoretical properties of the algorithm in this paper. Implementation issues and adaptation to large-scale problems will be addressed in a     

一类非线性互补问题的信赖域算法

In this paper,an ODE-type trust region algorithm for solving a class of nonlinear complementarity problems is proposed.A feature of this algorithm is that only the solution of linear systems of equations is required at each iteration,thus avoiding the need for solving a quadratic subproblem with a trust region bound.Under some conditions,it is proven that this algorithm is globally and locally superlinear convergent.The limited numerical examples show its efficiency.     

A CLASS OF TRUST REGION METHODS FOR LINEAR INEQUALITY CONSTRAINED OPTIMIZATION AND ITS THEORY ANALYSIS：Ⅰ. ALGORITHM AND GLOBAL CONVERGENCE

A class of trust region methods tor solving linear inequality constrained problems is propo6ed in this paper. It is shown that the algorithm is of global convergence. The algorithm uses a version of the two-slded projection and the strategy of the unconstrained trust region methods. It keeps the good convergence properties of the unconstrained case and has the merits of the projection method. In some sense, our algorithm can be regarded as an extension and improvement of the projected type algorithm.     

采用不定Dogleg路径的非单调自适应信赖域算法解无约束极小化(英文)

In this paper, we combine the nonmonotone and adaptive techniques with trust region method for unconstrained minimization problems. We set a new ratio of the actual descent and predicted descent. Then, instead of the monotone sequence, the nonmonotone sequence of function values are employed. With the adaptive technique, the radius of trust region △k can be adjusted automatically to improve the efficiency of trust region methods. By means of the Bunch-Parlett factorization, we construct a method with indefinite dogleg path for solving the trust region subproblem which can handle the indefinite approximate Hessian Bk. The convergence properties of the algorithm are established. Finally, detailed numerical results are reported to show that our algorithm is efficient.     

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 TRUST REGION METHOD WITH A CONIC MODEL FOR NONLINEARLY CONSTRAINED OPTIMIZATION

Trust region methods are powerful and effective optimization methods.The conic model method is a new type of method with more information available at each iteration than standard quadratic-based methods.The advantages of the above two methods can be combined to form a more powerful method for constrained optimization.The trust region subproblem of our method is to minimize a conic function subject to the linearized constraints and trust region bound.At the same time,the new algorithm still possesses robust global properties.The global convergence of the new algorithm under standard conditions is established.     

CONIC TRUST REGION METHOD FOR LINEARLY CONSTRAINED OPTIMIZATION

In this paper we present a trust region method of conic model for linearly constrained optimization problems.We discuss trust region approaches with conic model subproblems.Some equivalent variation properties and optimality conditions are given.A trust region algorithm based on conic model is constructed.Global convergence of the method is established.     

信赖域策略的内点投影既约Hessian方法解非线性约束优化

A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on variables is proposed. In order to deal with large problems,a pair of trust region subproblems in horizontal and vertical subspaces is used to replace the general full trust region subproblem. The horizontal trust region subproblem in the algorithm is only a general trust region subproblem while the vertical trust region subproblem is defined by a parameter size of the vertical direction subject only to an ellipsoidal constraint. Both trust region strategy and line search technique at each iteration switch to obtaining a backtracking step generated by the two trust region subproblems. By adopting the l1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion and the second order correction step are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases.     

一个锥模型的自适应信赖域算法及其收敛性

本文研究了无约束最优化问题的基于锥模型的自适应信赖域算法.利用理论分析得到一个新的自适应信赖域半径.算法在每步迭代中以变化的速率、当前迭代点的信息以及水平向量信息调节信赖域半径的大小.从理论上证明了新算法的全局收敛性和Q-二阶收敛性.用数值试验验证了新算法的有效性.推广了已有的自适应信赖域算法的可行性和有效性.     

A new trust region method with adaptive radius

In this paper we develop a new trust region method with adaptive radius for unconstrained optimization problems. The new method can adjust the trust region radius automatically at each iteration and possibly reduces the number of solving subproblems. We investigate the global convergence and convergence rate of this new method under some mild conditions. Theoretical analysis and numerical results show that the new adaptive trust region radius is available and reasonable and the resultant trust region method is efficient in solving practical optimization problems. The work was supported in part by NSF grant CNS-0521142, USA.     

Nonmonotone adaptive trust region method

In this paper, we propose a nonmonotone adaptive trust region method for unconstrained optimization problems. This method can produce an adaptive trust region radius automatically at each iteration and allow the functional value of iterates to increase within finite iterations and finally decrease after such finite iterations. This nonmonotone approach and adaptive trust region radius can reduce the number of solving trust region subproblems when reaching the same precision. The global convergence and convergence rate of this method are analyzed under some mild conditions. Numerical results show that the proposed method is effective in practical computation.     

带有固定步长的非单调自适应信赖域算法

提出了求解无约束优化问题带有固定步长的非单调自适应信赖域算法.信赖域半径的修正采用自适应技术,算法在试探步不被接受时,采用固定步长寻找下一迭代点.并在适当的条件下,证明算法具有全局收敛性和超线性收敛性.初步的数值试验表明算法对高维问题具有较好的效果.     

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.     

AN ADAPTIVE NONMONOTONIC TRUST REGION METHOD WITH CURVILINEAR SEARCHES

In this paper,an algorithm for unconstrained optimization that employs both trustregion techniques and curvilinear searches is proposed.At every iteration,we solve thetrust region subproblem whose radius is generated adaptively only once.Nonmonotonicbacktracking curvilinear searches are performed when the solution of the subproblem isunacceptable.The global convergence and fast local convergence rate of the proposedalgorithms are established under some reasonable conditions.The results of numericalexperiments are reported to show the effectiveness of the proposed algorithms.     

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.     

一类新拟牛顿非单调信赖域算法

提出了一类新的求解无约束最优化问题的新拟牛顿非单调信赖域算法.采用加权的r_k用以调整信赖域半径,在适当的条件下,证明了算法的全局收敛性.数值结果表明算法的有效性.     

无约束优化的自适应信赖域方法

本文对无约束优化问题提出一个自适应信赖域方法,每次迭代都充分利用前迭代点的信息自动产生一个恰当的信赖域半径,在此区域内,二次模型与原目标函数尽可能一致,避免盲目的尝试,提高了计算效率。文中在通常条件下证明了全局收敛性及局部超线性收敛结果,给出了新算法与传统信赖域方法的数值结果,证实了新方法的有效性。