A descent algorithm without line search for unconstrained optimization

In this paper, a new descent algorithm for solving unconstrained optimization problem is presented. Its search direction is descent and line search procedure can be avoided except for the first iteration. It is globally convergent under mild conditions. The search direction of the new algorithm is generalized and convergence of corresponding algorithm is also proved. Numerical results show that the algorithm is efficient for given test problems.     

Multiple reduced gradient method for multiobjective optimization problems

Multiobjective optimization has a large number of real-life applications. Under this motivation, in this paper, we present a new method for solving multiobjective optimization problems with both linear constraints and bound constraints on the variables. This method extends, to the multiobjective setting, the classical reduced gradient method for scalar-valued optimization. The proposed algorithm generates a feasible descent direction by solving an appropriate quadratic subproblem, without the use of any scalarization approaches. We prove that the sequence generated by the algorithm converges to Pareto-critical points of the problem. We also present some numerical results to show the efficiency of the proposed method.     

一种改进的GDF多目标优化算法

本文提出一种交互式非线性多目标优化算法，该算法是ＧＤＦ多目标优化算法的改进，具有这样的特点：算法采用了既约设计空间策略，具有良好的收敛性；算法生成的迭代点是有效解；算法具有多种一维搜索准则；对于线性多目标问题，算法只需一次交互迭代即可示出多目标问题的最优解。     

不等式约束优化一个新型可行QP-free算法

本文对非线性不等式约束优化问题提出了一个新的可行 QP-free 算法. 新算法保存了现有算法的优点, 并具有以下特性: (1) 算法每次迭代只需求解三个具有相同系数矩阵的线性方程组, 计算量小; (2) 可行下降方向只需通过求解一个线性方程组即可获得, 克服了以往分别求解两个线性方程组获得下降方向和可行方向, 然后再做凸组合的困难;(3) 迭代点均为可行点, 并不要求是严格内点; (4) 算法中采用了试探性线搜索,可以进一步减少计算量; (5) 算法中参数很少,数值试验表明算法具有较好的数值效果和较强的稳定性.     

A New Hybrid Evolutionary Multiobjective Algorithm Guided by Descent Directions

Hybridization of local search based algorithms with evolutionary algorithms is still an under-explored research area in multiobjective optimization. In this paper, we propose a new multiobjective algorithm based on a local search method. The main idea is to generate new non-dominated solutions by adding a linear combination of descent directions of the objective functions to a parent solution. Additionally, a strategy based on subpopulations is implemented to avoid the direct computation of descent directions for the entire population. The evaluation of the proposed algorithm is performed on a set of benchmark test problems allowing a comparison with the most representative state-of-the-art multiobjective algorithms. The results show that the proposed approach is highly competitive in terms of the quality of non-dominated solutions and robustness.     

Convergence of descent method with new line search

An efficient descent method for unconstrained optimization problems is line search method in which the step size is required to choose at each iteration after a descent direction is determined. There are many ways to choose the step sizes, such as the exact line search, Armijo line search, Goldstein line search, and Wolfe line search, etc. In this paper we propose a new inexact line search for a general descent method and establish some global convergence properties. This new line search has many advantages comparing with other similar inexact line searches. Moreover, we analyze the global convergence and local convergence rate of some special descent methods with the new line search. Preliminary numerical results show that the new line search is available and efficient in practical computation.     

Strongly sub-feasible direction method for constrained optimization problems with nonsmooth objective functions

In this paper, we propose a strongly sub-feasible direction method for the solution of inequality constrained optimization problems whose objective functions are not necessarily differentiable. The algorithm combines the subgradient aggregation technique with the ideas of generalized cutting plane method and of strongly sub-feasible direction method, and as results a new search direction finding subproblem and a new line search strategy are presented. The algorithm can not only accept infeasible starting points but also preserve the “strong sub-feasibility” of the current iteration without unduly increasing the objective value. Moreover, once a feasible iterate occurs, it becomes automatically a feasible descent algorithm. Global convergence is proved, and some preliminary numerical results show that the proposed algorithm is efficient.     

New spectral PRP conjugate gradient method for unconstrained optimization

In this paper, a new spectral PRP conjugate gradient algorithm has been developed for solving unconstrained optimization problems, where the search direction was a kind of combination of the gradient and the obtained direction, and the steplength was obtained by the Wolfe-type inexact line search. It was proved that the search direction at each iteration is a descent direction of objective function. Under mild conditions, we have established the global convergence theorem of the proposed method. Numerical results showed that the algorithm is promising, particularly, compared with the existing several main methods.     

修正PRP共轭梯度方法求解无约束最优化问题

基于著名的PRP共轭梯度方法,利用CG_DESCENT共轭梯度方法的结构,本文提出了一种求解大规模无约束最优化问题的修正PRP共轭梯度方法。该方法在每一步迭代中均能够产生一个充分下降的搜索方向,且独立于任何线搜索条件。在标准Wolfe线搜索条件下,证明了修正PRP共轭梯度方法的全局收敛性和线性收敛速度。数值结果展示了修正PRP方法对给定的测试问题是非常有效的。     

修正PRP共轭梯度方法求解无约束最优化问题

基于著名的PRP共轭梯度方法,利用CG_DESCENT共轭梯度方法的结构,本文提出了一种求解大规模无约束最优化问题的修正PRP共轭梯度方法。该方法在每一步迭代中均能够产生一个充分下降的搜索方向,且独立于任何线搜索条件。在标准Wolfe线搜索条件下,证明了修正PRP共轭梯度方法的全局收敛性和线性收敛速度。数值结果展示了修正PRP方法对给定的测试问题是非常有效的。     

A nonmonotone trust-region line search method for large-scale unconstrained optimization

We consider an efficient trust-region framework which employs a new nonmonotone line search technique for unconstrained optimization problems. Unlike the traditional nonmonotone trust-region method, our proposed algorithm avoids resolving the subproblem whenever a trial step is rejected. Instead, it performs a nonmonotone Armijo-type line search in direction of the rejected trial step to construct a new point. Theoretical analysis indicates that the new approach preserves the global convergence to the first-order critical points under classical assumptions. Moreover, superlinear and quadratic convergence are established under suitable conditions. Numerical experiments show the efficiency and effectiveness of the proposed approach for solving unconstrained optimization problems.     

一类非单调修正PRP算法的全局收敛性

本文给出一类非单调线性搜索下的修正PRP算法,该方法保证每次迭代中的搜索方向是充分下降的.在较弱的条件下,我们证明了此类非单调修正PRP算法具有全局收敛性.     

改进鲸鱼优化算法在电力负荷调度中的应用

为了提高鲸鱼优化算法(WOA)的全局优化性能, 提出了一种基于黄金分割搜索的改进鲸鱼优化算法(GWOA)。首先利用黄金分割搜索对WOA的初始种群进行初始化, 使得初始种群能够尽可能的靠近全局最优解, 然后利用黄金分割搜索所形成的变区间, 进行变区间黄金分割非均匀变异操作, 以增加WOA的粒子多样性和提高粒子跳出局部最优陷阱的能力, 从而改善WOA的寻优性能。选取了15个大规模测试函数进行数值仿真测试, 仿真结果和统计分析表明GWOA的寻优性能要优于对比文献的改进鲸鱼优化算法(IWOA)。此外, 将GWOA用于对工程实际应用领域中的电力负荷优化调度问题进行实例分析, 实例应用结果表明, GWOA能有效对电力负荷优化调度问题进行寻优求解。     

Steepest descent preconditioning for nonlinear GMRES optimization

Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N‐GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, whereas the second employs a predefined small step. A simple global convergence proof is provided for the N‐GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioning for N‐GMRES optimization is also motivated by relating it to standard non‐preconditioned GMRES for linear systems in the case of a standard quadratic optimization problem with symmetric positive definite operator. Numerical tests on a variety of model problems show that the N‐GMRES optimization algorithm is able to very significantly accelerate convergence of stand‐alone steepest descent optimization. Moreover, performance of steepest‐descent preconditioned N‐GMRES is shown to be competitive with standard nonlinear conjugate gradient and limited‐memory Broyden–Fletcher–Goldfarb–Shanno methods for the model problems considered. These results serve to theoretically and numerically establish steepest‐descent preconditioned N‐GMRES as a general optimization method for unconstrained nonlinear optimization, with performance that appears promising compared with established techniques. In addition, it is argued that the real potential of the N‐GMRES optimization framework lies in the fact that it can make use of problem‐dependent nonlinear preconditioners that are more powerful than steepest descent (or, equivalently, N‐GMRES can be used as a simple wrapper around any other iterative optimization process to seek acceleration of that process), and this potential is illustrated with a further application example. Copyright © 2012 John Wiley & Sons, Ltd.     

A Two-Term PRP-Based Descent Method

In this paper, by the use of the project of the PRP (Polak–Ribiére–Polyak) conjugate gradient direction, we develop a PRP-based descent method for solving unconstrained optimization problem. The method provides a sufficient descent direction for the objective function. Moreover, if exact line search is used, the method reduces to the standard PRP method. Under suitable conditions, we show that the method with some backtracking line search or the generalized Wolfe-type line search is globally convergent. We also report some numerical results and compare the performance of the method with some existing conjugate gradient methods. The results show that the proposed method is efficient.     

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.     

A Projected Gradient Method for Vector Optimization Problems

Vector optimization problems are a significant extension of multiobjective optimization, which has a large number of real life applications. In vector optimization the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative orthant. We consider extensions of the projected gradient gradient method to vector optimization, which work directly with vector-valued functions, without using scalar-valued objectives. We provide a direction which adequately substitutes for the projected gradient, and establish results which mirror those available for the scalar-valued case, namely stationarity of the cluster points (if any) without convexity assumptions, and convergence of the full sequence generated by the algorithm to a weakly efficient optimum in the convex case, under mild assumptions. We also prove that our results still hold when the search direction is only approximately computed.     

解新锥模型信赖域子问题的折线法

本文以新锥模型信赖域子问题的最优性条件为理论基础,认真讨论了新子问题的锥函数性质,分析了此函数在梯度方向及与牛顿方向连线上的单调性.在此基础上本文提出了一个求解新锥模型信赖域子问题折线法,并证明了这一子算法保证解无约束优化问题信赖域法全局收敛性要满足的下降条件.本文获得的数值实验表明该算法是有效的.     

A Modified Hestenes-Stiefel Conjugate Gradient Algorithm for Large-Scale Optimization

Mathematical programming is a rich and well-developed area in operations research. Nevertheless, there remain many challenging problems in this area, one of which is the large-scale optimization problem. In this article, a modified Hestenes and Stiefel (HS) conjugate gradient (CG) algorithm with a nonmonotone line search technique is presented. This algorithm possesses information about not only the gradient value but also the function value. Moreover, the sufficient descent condition holds without any line search. The global convergence is established for nonconvex functions under suitable conditions. Numerical results show that the proposed algorithm is advantageous to existing CG methods for large-scale optimization problems.     

Global convergence of a modified Fletcher–Reeves conjugate gradient method with Armijo-type line search

In this paper, we are concerned with the conjugate gradient methods for solving unconstrained optimization problems. It is well-known that the direction generated by a conjugate gradient method may not be a descent direction of the objective function. In this paper, we take a little modification to the Fletcher–Reeves (FR) method such that the direction generated by the modified method provides a descent direction for the objective function. This property depends neither on the line search used, nor on the convexity of the objective function. Moreover, the modified method reduces to the standard FR method if line search is exact. Under mild conditions, we prove that the modified method with Armijo-type line search is globally convergent even if the objective function is nonconvex. We also present some numerical results to show the efficiency of the proposed method.Supported by the 973 project (2004CB719402) and the NSF foundation (10471036) of China.