Two optimal Dai–Liao conjugate gradient methods

Two adaptive choices for the parameter of Dai–Liao conjugate gradient (CG) method are suggested. One of which is obtained by minimizing the distance between search directions of Dai–Liao method and a three-term CG method proposed by Zhang et al. and the other one is obtained by minimizing Frobenius condition number of the search direction matrix. Global convergence analyses are made briefly. Numerical results are reported; they demonstrate effectiveness of the suggested adaptive choices.     

On optimality of two adaptive choices for the parameter of Dai–Liao method

Orthonormal matrices are a class of well-conditioned matrices with the least spectral condition number. Here, at first it is shown that a recently proposed choice for parameter of the Dai–Liao nonlinear conjugate gradient method makes the search direction matrix as close as possible to an orthonormal matrix in the Frobenius norm. Then, conducting a brief singular value analysis, it is shown that another recently proposed choice for the Dai–Liao parameter improves spectral condition number of the search direction matrix. Thus, theoretical justifications of the two choices for the Dai–Liao parameter are enhanced. Finally, some comparative numerical results are reported.     

Exponential Convergence of Time-Delay Systems in the Presence of Bounded Disturbances

In this paper, the problem of control design for exponential convergence of state/input delay systems with bounded disturbances is considered. Based on the Lyapunov–Krasovskii method combining with the delay-decomposition technique, a new sufficient condition is proposed for the existence of a state feedback controller, which guarantees that all solutions of the closed-loop system converge exponentially (with a pre-specified convergence rate) within a ball whose radius is minimized. The obtained condition is given in terms of matrix inequalities with one parameter needing to be tuned, which can be solved by using a one-dimensional search method with Matlab’s LMI Toolbox to minimize the radius of the ball. Two numerical examples are given to illustrate the superiority of the proposed method.     

结合Armijo步长搜索的一类新的记忆梯度算法及其收敛特征

对求解无约束规划的超记忆梯度算法中线搜索方向中的参数,给了一个假设条件,从而确定了它的一个新的取值范围,保证了搜索方向是目标函数的充分下降方向,由此提出了一类新的记忆梯度算法.在去掉迭代点列有界和Armijo步长搜索下,讨论了算法的全局收敛性,且给出了结合形如共轭梯度法FR,PR,HS的记忆梯度法的修正形式.数值实验表明,新算法比Armijo线搜索下的FR、PR、HS共轭梯度法和超记忆梯度法更稳定、更有效.     

Some new three-term Hestenes–Stiefel conjugate gradient methods with affine combination

In this paper, a modified Hestenes–Stiefel conjugate gradient method for unconstrained problems is developed, which can achieves the twin goals of generating sufficient descent direction at each iteration as well as being close to the Newton direction. In our methods, the hybridization parameter can also be obtained based on other kinds of conjugacy conditions. Under mild condition, we establish their global convergence for general objective functions. Numerical experimentation with the new method indicates that it efficiently solves the test problems and therefore is promising.     

An Accelerated Three-Term Conjugate Gradient Method with Sufficient Descent Condition and Conjugacy Condition

An accelerated three-term conjugate gradient method is proposed, in which the search direction can satisfy the sufficient descent condition as well as extended Dai–Liao conjugacy condition. Different from the existent methods, a dynamical compensation strategy in our proposed method is considered, that is Li–Fushikuma-type quasi-Newton equation is satisfied as much as possible, otherwise, to some extent, the singular values of iteration matrix of search directions will adaptively clustered, which substantially benefits acceleration the convergence or reduction in the condition number of iteration matrix. Global convergence is established under mild conditions for general objective functions. We also report some numerical results to show its efficiency.     

A NEW ADAPTIVE SUBSPACE MINIMIZATION THREE-TERM CONJUGATE GRADIENT ALGORITHM FOR UNCONSTRAINED OPTIMIZATION

A new adaptive subspace minimization three-term conjugate gradient algorithm with nonmonotone line search is introduced and analyzed in this paper.The search directions are computed by minimizing a quadratic approximation of the objective function on special subspaces,and we also proposed an adaptive rule for choosing different searching directions at each iteration.We obtain a significant conclusion that the each choice of the search directions satisfies the sufficient descent condition.With the used nonmonotone line search,we prove that the new algorithm is globally convergent for general nonlinear functions under some mild assumptions.Numerical experiments show that the proposed algorithm is promising for the given test problem set.     

A new subspace minimization conjugate gradient method with nonmonotone line search for unconstrained optimization

A new subspace minimization conjugate gradient algorithm with a nonmonotone Wolfe line search is proposed and analyzed. In the scheme, we propose two choices of the search direction by minimizing a quadratic approximation of the objective function in special subspaces, and state criterions on how to choose the direction. Under given conditions, we obtain the significant conclusion that each choice of the direction satisfies the sufficient descent property. Based on the idea on how the function is close to a quadratic function, a new strategy for choosing the initial stepsize is presented for the line search. With the used nonmonotone Wolfe line search, we prove the global convergence of the proposed method for general nonlinear functions under mild assumptions. Numerical comparisons are given with well-known CGOPT and CG_DESCENT and show that the proposed algorithm is very promising.     

A Dai-Liao conjugate gradient algorithm with clustering of eigenvalues

A new value for the parameter in Dai and Liao conjugate gradient algorithm is presented. This is based on the clustering of eigenvalues of the matrix which determine the search direction of this algorithm. This value of the parameter lead us to a variant of the Dai and Liao algorithm which is more efficient and more robust than the variants of the same algorithm based on minimizing the condition number of the matrix associated to the search direction. Global convergence of this variant of the algorithm is briefly discussed.     

Global Convergence of a Memory Gradient Method for Unconstrained Optimization

Memory gradient methods are used for unconstrained optimization, especially large scale problems. The first idea of memory gradient methods was proposed by Miele and Cantrell (1969) and Cragg and Levy (1969). In this paper, we present a new memory gradient method which generates a descent search direction for the objective function at every iteration. We show that our method converges globally to the solution if the Wolfe conditions are satisfied within the framework of the line search strategy. Our numerical results show that the proposed method is efficient for given standard test problems if we choose a good parameter included in the method.     

NEW HYBRID CONJUGATE GRADIENT METHOD AS A CONVEX COMBINATION OF LS AND FR METHODS

In this paper, we present a new hybrid conjugate gradient algorithm for unconstrained optimization. This method is a convex combination of Liu-Storey conjugate gradient method and Fletcher-Reeves conjugate gradient method. We also prove that the search direction of any hybrid conjugate gradient method, which is a convex combination of two conjugate gradient methods, satisfies the famous D-L conjugacy condition and in the same time accords with the Newton direction with the suitable condition. Furthermore, this property doesn't depend on any line search. Next, we also prove that, moduling the value of the parameter t,the Newton direction condition is equivalent to Dai-Liao conjugacy condition.The strong Wolfe line search conditions are used.The global convergence of this new method is proved.Numerical comparisons show that the present hybrid conjugate gradient algorithm is the efficient one.     

无约束优化问题的对角稀疏拟牛顿法

对无约束优化问题提出了对角稀疏拟牛顿法,该算法采用了Armijo非精确线性搜索,并在每次迭代中利用对角矩阵近似拟牛顿法中的校正矩阵,使计算搜索方向的存贮量和工作量明显减少,为大型无约束优化问题的求解提供了新的思路．在通常的假设条件下,证明了算法的全局收敛性,线性收敛速度并分析了超线性收敛特征。数值实验表明算法比共轭梯度法有效,适于求解大型无约束优化问题．     

A Globally Convergent Method of Constrained Minimization by Solving Subproblems of the Conic Model

A new method for nonlinearly constrained optimization problems is proposed. The method consists of two steps. In the first step, we get a search direction by the linearly constrained subproblems based on conic functions. In the second step, we use a differentiable penalty function, and regard it as the metric function of the problem. From this, a new approximate solution is obtained. The global convergence of the given method is also proved.     

A new regularized quasi-Newton method for unconstrained optimization

In this paper, we propose a new regularized quasi-Newton method for unconstrained optimization. At each iteration, a regularized quasi-Newton equation is solved to obtain the search direction. The step size is determined by a non-monotone Armijo backtracking line search. An adaptive regularized parameter, which is updated according to the step size of the line search, is employed to compute the next search direction. The presented method is proved to be globally convergent. Numerical experiments show that the proposed method is effective for unconstrained optimizations and outperforms the existing regularized Newton method.     

A new family of hybrid three-term conjugate gradient methods with applications in image restoration

In this paper, based on the hybrid conjugate gradient method and the convex combination technique, a new family of hybrid three-term conjugate gradient methods are proposed for solving unconstrained optimization. The conjugate parameter in the search direction is a hybrid of Dai-Yuan conjugate parameter and any one. The search direction then is the sum of the negative gradient direction and a convex combination in relation to the last search direction and the gradient at the previous iteration. Without choosing any specific conjugate parameters, we show that the search direction generated by the family always possesses the descent property independent of line search technique, and that it is globally convergent under usual assumptions and the weak Wolfe line search. To verify the effectiveness of the presented family, we further design a specific conjugate parameter, and perform medium-large-scale numerical experiments for smooth unconstrained optimization and image restoration problems. The numerical results show the encouraging efficiency and applicability of the proposed methods even compared with the state-of-the-art methods.

     

凸约束伪单调方程组的无导数投影算法

基于HS共轭梯度法的结构,本文在弱假设条件下建立了一种求解凸约束伪单调方程组问题的迭代投影算法.该算法不需要利用方程组的任何梯度或Jacobian矩阵信息,因此它适合求解大规模问题.算法在每一次迭代中都能产生充分下降方向,且不依赖于任何线搜索条件.特别是,我们在不需要假设方程组满足Lipschitz条件下建立了算法的全...     

A one-parametric class of smoothing functions and an improved regularization Newton method for the NCP

In this paper, we introduce a one-parametric class of smoothing functions, which enjoys some favourable properties and includes two famous smoothing functions as special cases. Based on this class of smoothing functions, we propose a regularization Newton method for solving the non-linear complementarity problem. The main feature of the proposed method is that it uses a perturbed Newton equation to obtain the direction. This not only allows our method to have global and local quadratic convergences without strict complementarity conditions, but also makes the regularization parameter converge to zero globally Q-linearly. In addition, we use a new non-monotone line search scheme to obtain the step size. Some numerical results are reported which confirm the good theoretical properties of the proposed method.     

An Optimal Extension of the Polak–Ribière–Polyak Conjugate Gradient Method

Based on a singular value analysis on an extension of the Polak–Ribière–Polyak method, a nonlinear conjugate gradient method with the following two optimal features is proposed: the condition number of its search direction matrix is minimum and also, the distance of its search direction from the search direction of a descent nonlinear conjugate gradient method proposed by Zhang et al. is minimum. Under proper conditions, global convergence of the method can be achieved. To enhance e?ciency of the proposed method, Powell’s truncation of the conjugate gradient parameters is used. The method is computationally compared with the nonlinear conjugate gradient method proposed by Zhang et al. and a modified Polak–Ribière–Polyak method proposed by Yuan. Results of numerical comparisons show e?ciency of the proposed method in the sense of the Dolan–Moré performance profile.     

一类改进的非凸二次规划有效集方法

一类改进的非凸二次规划有效集方法修乃华（河北师范学院数学系）ＡＣＬＡＳＳＯＦＩＭＰＲＯＶＥＤＡＣＴＩＶＥＳＥＴＭＥＴＨＯＤＳＦＯＲＮＯＮＣＯＮＶＥＸＱＵＡＤＲＡＴＩＣＰＲＯＧＲＡＭＭＩＮＧＰＲＯＢＬＥＭ￥ＸｉｕＮａｉ－ｈｕａ（Ｄｅｐｔ．ｏｆＭａｔｈ．...