A new supermemory gradient method for unconstrained optimization problems

This paper presents a new supermemory gradient method for unconstrained optimization problems. It can be regarded as a combination of ODE-based methods, line search and subspace techniques. The main characteristic of this method is that, at each iteration, a lower dimensional system of linear equations is solved only once to obtain a trial step, thus avoiding solving a quadratic trust region subproblem. Another is that when a trial step is not accepted, this proposed method generates an iterative point whose step-length satisfies Armijo line search rule, thus avoiding resolving linear system of equations. Under some reasonable assumptions, the method is proven to be globally convergent. Numerical results show the efficiency of this proposed method in practical computation.     

Combining trust region and linesearch algorithm for equality constrained optimization

In this paper, a combining trust region and line search algorithm for equality constrained optimization is proposed. At each iteration, we only need to solve the trust region subproblem once, when the trust region trial step can not be accepted, we switch to line search to obtain the next iteration. Hence, the difficulty of repeated solving trust region subproblem in an iterate is avoided. In order to allow the direction of negative curvature, we add second correction step in trust region step and employ nommonotone technique in line search. The global convergence and local superlinearly rate are established under certain assumptions. Some numerical examples are given to illustrate the efficiency of the proposed algorithm.     

线性不等式约束优化的弧线路径信赖域算法

提供了弧线路径结合仿射内点信赖域策略的非单调回代算法解线性不等式约束的优化问题.基于仿射投影的信赖域子问题获得新的搜索方向,采用弧线路径的近似信赖域和线搜索结合技术得到回代步,获得新的步长.通过证明所提供的弧线路径具有一系列良好性质,从而在合理的条件下,证明所提供的算法不仅具有整体收敛性,而且保持算法的局部超线性收敛速率.数值测试表明了算法的有效性与可靠性.     

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.     

A nonmonotone conic trust region method based on line search for solving unconstrained optimization

In this paper, we present a nonmonotone conic trust region method based on line search technique for unconstrained optimization. The new algorithm can be regarded as a combination of nonmonotone technique, line search technique and conic trust region method. When a trial step is not accepted, the method does not resolve the trust region subproblem but generates an iterative point whose steplength satisfies some line search condition. The function value can only be allowed to increase when trial steps are not accepted in close succession of iterations. The local and global convergence properties are proved under reasonable assumptions. Numerical experiments are conducted to compare this method with the existing methods.     

An ODE-based trust region method for unconstrained optimization problems

In this paper, a new trust region algorithm is proposed for solving unconstrained optimization problems. This method can be regarded as a combination of trust region technique, fixed step-length and ODE-based methods. A feature of this proposed method is that at each iteration, only a system of linear equations is solved to obtain a trial step. Another is that when a trial step is not accepted, the method generates an iterative point whose step-length is defined by a formula. Under some standard assumptions, it is proven that the algorithm is globally convergent and locally superlinear convergent. Preliminary numerical results are reported.     

Two accelerated nonmonotone adaptive trust region line search methods

Hybridizing monotone and nonmonotone approaches, we employ a modified trust region ratio in which more information is provided about the agreement between the exact and the approximate models. Also, we use an adaptive trust region radius as well as two accelerated Armijo-type line search strategies to avoid resolving the trust region subproblem whenever a trial step is rejected. We show that the proposed algorithm is globally and locally superlinearly convergent. Comparative numerical experiments show practical efficiency of the proposed accelerated adaptive trust region algorithm.     

A new hybrid method for nonlinear complementarity problems

In this paper, a new hybrid method is proposed for solving nonlinear complementarity problems (NCP) with P ₀ function. In the new method, we combine a smoothing nonmonotone trust region method based on a conic model and line search techniques. We reformulate the NCP as a system of semismooth equations using the Fischer-Burmeister function. Using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing nonmonotone trust region algorithm of a conic model for solving the NCP with P ₀ functions. This is different from the classical trust region methods, in that when a trial step is not accepted, the method does not resolve the trust region subproblem but generates an iterative point whose steplength is defined by a line search. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, the superlinear convergence of the algorithm is established without a strict complementarity condition.     

Combining nonmonotone conic trust region and line search techniques for unconstrained optimization

In this paper, we propose a trust region method for unconstrained optimization that can be regarded as a combination of conic model, nonmonotone and line search techniques. Unlike in traditional trust region methods, the subproblem of our algorithm is the conic minimization subproblem; moreover, our algorithm performs a nonmonotone line search to find the next iteration point when a trial step is not accepted, instead of resolving the subproblem. The global and superlinear convergence results for the algorithm are established under reasonable assumptions. Numerical results show that the new method is efficient for unconstrained optimization problems.     

基于非单调技术的ODE型混合方法

基于非单调线搜索技术和IMPBOT算法,提出了一个求解无约束优化问题的ODE型混合方法.该方法的主要特点是：为了求得试验步,该方法在每次迭代时不必求解带信赖域界的子问题,仅需要求解一线性方程组系统;当试验步不被接受时,该方法就执行改进的Wolfe-型非单调线搜索来获得下一个新的迭代点,从而避免了反复求解线性方程组系统. 在一定条件下,所提算法还是整体收敛和超线性收敛的. 数值试验结果表明该方法是有效的.     

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.     

An ODE-Based Trust Region Filter Algorithm for Unconstrained Optimization

In this article, an ODE-based trust region filter algorithm for unconstrained optimization is proposed. It can be regarded as a combination of trust region and filter techniques with ODE-based methods. Unlike the existing trust-region-filter methods and ODE-based methods, a distinct feature of this method is that at each iteration, a reduced linear system is solved to obtain a trial step, thus avoiding solving a trust region subproblem. Under some standard assumptions, it is proven that the algorithm is globally convergent. Preliminary numerical results show that the new algorithm is efficient for large scale problems.     

一个带固定步长的ODE型算法

提出了一种新的求解无约束优化问题的ODE型方法,其特点是:它在每次迭代时仅求解一个线性方程组系统来获得试探步;若该试探步不被接受,算法就沿着该试探步的方向求得下一个迭代点,其中步长通过固定公式计算得到.这样既避免了传统的ODE型算法中为获得可接受的试探步而重复求解线性方程组系统,又不必执行线搜索,从而减少了计算量.在适当的条件下,还证明了新算法的整体收敛性和局部超线性收敛性.数值试验结果表明:提出的算法是有效的.     

An iterative method for solving semismooth equations

In this paper, we combine trust region technique with line search technique to develop an iterative method for solving semismooth equations. At each iteration, a trust region subproblem is solved. The solution of the trust region subproblem provides a descent direction for the norm of a smoothing function. By using a backtracking line search, a steplength is determined. The proposed method shares advantages of trust region methods and line search methods. Under appropriate conditions, the proposed method is proved to be globally and superlinearly convergent. In particular, we show that after finitely many iterations, the unit step is always accepted and the method reduces to a smoothing Newton method.     

一类带线搜索的非单调信赖域新算法

结合非单调信赖域方法,和非单调线搜索技术,提出了一种新的无约束优化算法.信赖域方法的每一步采用线搜索,使得迭代每一步都充分下降加快了迭代速度.在一定条件下,证明了算法具有全局收敛性和局部超线性.收敛速度.数值试验表明算法是十分有效的.     

基于信赖域技术的处理带线性约束优化的内点算法

基于信赖域技术,本文提出了一个求解带线性等式和非负约束优化问题的内点算法,其特点是:为了求得搜索方向,算法在每一步迭代时仅需要求解一线性方程组系统,从而避免了求解带信赖域界的子问题,然后利用非精确的Armijo线搜索法来得到下一个迭代内点. 从数值计算的观点来看,这种技巧可减少计算量.在适当的条件下,文中还证明了该算法所产生的迭代序列的每一个聚点都是原问题的KKT点.     

INTERIOR POINT PROJECTED REDUCED HESSIAN METHOD WITH TRUST REGION STRATEGY FOR NONLINEAR CONSTRAINED OPTIMIZATION

§1　IntroductionIn this paper we analyze an interior point scaling projected reduced Hessian methodwith trust region strategy for solving the nonlinear equality constrained optimizationproblem with nonnegative constraints on variables:min　f(x)s.t.　c(x) =0 (1.1)x≥0where f∶Rn→R is the smooth nonlinear function,notnecessarily convex and c(x)∶Rn→Rm(m≤n) is the vector nonlinear function.There are quite a few articles proposing localsequential quadratic programming reduced Hessian methods…     

带非线性不等式约束优化问题的信赖域算法

借助于KKT条件和NCP函数,提出了求解带非线性不等式约束优化问题的信赖域算法.该算法在每一步迭代时,不必求解带信赖域界的二次规划子问题,仅需求一线性方程组系统.在适当的假设条件下,它还是整体收敛的和局部超线性收敛的.数值实验结果表明该方法是有效的.     

非单调技术预条件弧线路径信赖域解无约束优化

本文提供修正近似信赖域类型路经三类预条件弧线路径方法解无约束最优化问题.使用对称矩阵的稳定Bunch-Parlett易于形成信赖域子问题的弧线路径,使用单位下三角矩阵作为最优路径和修正梯度路径的预条件因子.运用预条件因子改进Hessian矩阵特征值分布加速预条件共轭梯度路径收敛速度.基于沿着三类路径信赖域子问题产生试探步,将信赖域策略与非单调线搜索技术相结合作为新的回代步.理论分析证明在合理条件下所提供的算法是整体收敛性,并且具有局部超线性收敛速率,数值结果表明算法的有效性.     

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.