Finding global minima with a novel filled function for non-smooth unconstrained optimisation

This article presents a novel filled function approach for finding a better minimiser of both non-smooth and smooth unconstrained global optimisations. Unlike any traditional filled function, in theory, the filled function proposed in this article does not assume that the objective function is coercive. Moreover, the proposed function contains only one parameter, which can be easily adjusted. In the algorithm section, an algorithm based on the proposed filled function is presented for unconstrained global optimisation in which the objective function is assumed to be coercive. A numerical test is also provided. The preliminary computational results show that the proposed filled function approach is promising.     

求无约束优化问题的无参数填充函数法

填充函数作为求解优化问题的有效方法之一,以填充函数的基本思想为基础,构造了新的无参数填充函数,该函数形式简单,便于计算。分析了该函数的相关性质并设计了相应的算法,最后通过数值实验,结果表明提出的算法是可行的、有效的。     

A new class of filled functions with one parameter for global optimization

The filled function method is an efficient approach for finding global minimizers of multi-dimensional and nonlinear functions in the absence of any restrictions. In this paper, we give a new definition of filled function and the idea of constructing a new filled function, and then a new class of filled functions with one parameter on the basis of the new definition, which possesses better quality, is presented. Theoretical properties of the new class of filled functions are investigated. A new algorithm is developed from the new filled function method. The implementation of the algorithm on seven test problems with dimensions up to 30 is reported, and comparisons with other filled function methods demonstrate that the new algorithm is more efficient.     

A new spectral conjugate gradient method for large-scale unconstrained optimization

The spectral conjugate gradient methods, with simple construction and nice numerical performance, are a kind of effective methods for solving large-scale unconstrained optimization problems. In this paper, based on quasi-Newton direction and quasi-Newton condition, and motivated by the idea of spectral conjugate gradient method as well as Dai-Kou's selecting technique for conjugate parameter [SIAM J. Optim. 23 (2013), pp. 296–320], a new approach for generating spectral parameters is presented, where a new double-truncating technique, which can ensure both the sufficient descent property of the search directions and the bounded property of the sequence of spectral parameters, is introduced. Then a new associated spectral conjugate gradient method for large-scale unconstrained optimization is proposed. Under either the strong Wolfe line search or the generalized Wolfe line search, the proposed method is always globally convergent. Finally, a large number of comparison numerical experiments on large-scale instances from one thousand to two million variables are reported. The numerical results show that the proposed method is more promising.     

A nonmonotone ODE-based method for unconstrained optimization

This paper presents a new hybrid algorithm for unconstrained optimization problems, which combines the idea of the IMPBOT algorithm with the nonmonotone line search technique. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step, via a modified limited-memory BFGS two loop recursion that requires only matrix–vector products, thus reducing the computations and storage. Furthermore, when the trial step is not accepted, the proposed method performs a line search along it using a modified nonmonotone scheme, thus a larger stepsize can be yielded in each line search procedure. Under some reasonable assumptions, the convergence properties of the proposed algorithm are analysed. Numerical results are also reported to show the efficiency of this proposed method.     

A scaled conjugate gradient method for nonlinear unconstrained optimization

We propose a new optimization problem which combines the good features of the classical conjugate gradient method using some penalty parameter, and then, solve it to introduce a new scaled conjugate gradient method for solving unconstrained problems. The method reduces to the classical conjugate gradient algorithm under common assumptions, and inherits its good properties. We prove the global convergence of the method using suitable conditions. Numerical results show that the new method is efficient and robust.     

A second order method for unconstrained optimization

An algorithm using second derivatives for solving unconstrained optimization problems is presented. In this brief note the descent direction of the algorithm is based on a modification of the Newton direction, while the Armijo rule for choosing the stepsize is used. The rate of convergence of the algorithm is shown to be superlinear. Our computational experience shows that the method performs quite well and our numerical results are presented in Section 4.     

用填充函数改进的智能控制系统全局优化算法

为了提高全局优化算法的速度,提出了智能控制系统全局优化算法。该算法应用了闭环控制系统的反馈的思想,使得在寻优迭代过程中被优化函数的值不断接近设定值,直至达到其全局最优值。该算法的关键在于控制策略的设计和策略中的参数值的设定。为了降低参数初值设定的难度同时提高算法的寻优精度,利用填充函数法对智能控制系统全局优化算法进行改进。经12个标准的测试函数的验证,改进后的算法的速度较填充函数法快,算法的精度比智能控制系统全局优化算法高。     

基于Powell局部搜索策略的全局优化布谷鸟算法

为了解决布谷鸟搜索算法后期收敛速度慢、求解精度不高、易陷入局部最优等缺陷,提出了一种基于Powell局部搜索策略的全局优化布谷鸟搜索算法.算法将布谷鸟全局搜索能力与Powell方法的局部寻优性能有机地结合,并根据适应度值逐步构建精英种群候选解池在迭代后期牵引Powell搜索的局部优化,在保证求解速度、尽可能找到全局极值点的同时提高算法的求解精度.对52个典型测试函数实验结果表明,该算法相比于传统的布谷鸟搜索算法不仅寻优精度和寻优率有所提高,并且适应能力强、鲁棒性好,与最新提出的其他改进算法相比也具有一定的竞争优势.     

A New Subdivision Algorithm for the Bernstein Polynomial Approach to Global Optimization

In this paper,an improved algorithm is proposed for unconstrained global optimization to tackle non-convex nonlinear multivariate polynomial programming problems.The proposed algorithm is based on the Bernstein polynomial approach.Novel features of the proposed algorithm are that it uses a new rule for the selection of the subdivision point,modified rules for the selection of the subdivision direction,and a new acceleration device to avoid some unnecessary subdivisions.The performance of the proposed algorithm is numerically tested on a collection of 16 test problems.The results of the tests show the proposed algorithm to be superior to the existing Bernstein algorithm in terms of the chosen performance metrics.     

A new modified scaled conjugate gradient method for large-scale unconstrained optimization with non-convex objective function

In this paper, according to the fifth-order Taylor expansion of the objective function and the modified secant equation suggested by Li and Fukushima, a new modified secant equation is presented. Also, a new modification of the scaled memoryless BFGS preconditioned conjugate gradient algorithm is suggested which is the idea to compute the scaling parameter based on a two-point approximation of our new modified secant equation. A remarkable feature of the proposed method is that it possesses a globally convergent even without convexity assumption on the objective function. Numerical results show that the proposed new modification of scaled conjugate gradient is efficient.     

一种求解非线性约束优化全局最优的新方法

本文提出了一种求解非线性约束优化的全局最优的新方法—它是基于利用非线性互补函数和不断增加新的约束来重复解库恩-塔克条件的非线性方程组的新方法。因为库恩-塔克条件是非线性约束优化的必要条件,得到的解未必是非线性约束优化的全局最优解,为此,本文首次给出了通过利用该优化问题的先验知识,不断地增加约束来限制全局最优解范围的方法,一些仿真例子表明提出的方法和理论有效的,并且可行的。     

A cubic regularization algorithm for unconstrained optimization using line search and nonmonotone techniques

In recent years, cubic regularization algorithms for unconstrained optimization have been defined as alternatives to trust-region and line search schemes. These regularization techniques are based on the strategy of computing an (approximate) global minimizer of a cubic overestimator of the objective function. In this work we focus on the adaptive regularization algorithm using cubics (ARC) proposed in Cartis et al. [Adaptive cubic regularisation methods for unconstrained optimization. Part I: motivation, convergence and numerical results, Mathematical Programming A 127 (2011), pp. 245–295]. Our purpose is to design a modified version of ARC in order to improve the computational efficiency preserving global convergence properties. The basic idea is to suitably combine a Goldstein-type line search and a nonmonotone accepting criterion with the aim of advantageously exploiting the possible good descent properties of the trial step computed as (approximate) minimizer of the cubic model. Global convergence properties of the proposed nonmonotone ARC algorithm are proved. Numerical experiments are performed and the obtained results clearly show satisfactory performance of the new algorithm when compared to the basic ARC algorithm.     

A new subspace minimization conjugate gradient method based on tensor model for unconstrained optimization

ABSTRACT

A new subspace minimization conjugate gradient method based on tensor model is proposed and analysed. If the objective function is close to a quadratic, we construct a quadratic approximation model in a two-dimensional subspace to generate the search direction; otherwise, we construct a tensor model. It is remarkable that the search direction satisfies the sufficient descent property. We prove the global convergence of the proposed method under mild assumptions. Numerical comparisons are given with well-known CGOPT and CG_DESCENT and show that the proposed algorithm is very promising.     

解大规模无约束优化的自适应过滤信赖域法

提出一种解大规模无约束优化问题的自适应过滤信赖域法。用目标函数的梯度及迭代点的信息来构造目标函数海赛矩阵的近似数量矩阵,引进了过滤技术和自适应技术,大大提高了计算效率。从理论上证明了新算法的全局收敛性,数值试验结果也表明了新算法的有效性。     

Two modified spectral conjugate gradient methods and their global convergence for unconstrained optimization

In this paper, two modified spectral conjugate gradient methods which satisfy sufficient descent property are developed for unconstrained optimization problems. For uniformly convex problems, the first modified spectral type of conjugate gradient algorithm is proposed under the Wolfe line search rule. Moreover, the search direction of the modified spectral conjugate gradient method is sufficiently descent for uniformly convex functions. Furthermore, according to the Dai–Liao's conjugate condition, the second spectral type of conjugate gradient algorithm can generate some sufficient decent direction at each iteration for general functions. Therefore, the second method could be considered as a modification version of the Dai–Liao's algorithm. Under the suitable conditions, the proposed algorithms are globally convergent for uniformly convex functions and general functions. The numerical results show that the approaches presented in this paper are feasible and efficient.     

图的最大二等分问题的一种离散填充函数算法

图的最大二等分问题是一个经典的NP困难问题,有着广泛的应用背景。提出了一类求解最大二等分问题的离散填充函数算法。该算法采用快速的、基于迭代改进的算法作为局部搜索算法。构造了最大二等分问题的填充函数和辅助问题,并研究了该辅助问题的相关性质。利用局部搜索算法极大化辅助问题来寻找更好的解。用顶点数为800到10 000的大规模标准测试例子测试提出的算法。实验结果表明,该算法是有效的。     

非单调信赖域方法求解无约束非光滑优化问题

提出了非单调信赖域算法求解无约束非光滑优化问题,并和经典的信赖域方法作比较分析。同时,设定了一些条件,在这些假设条件下证明了该算法是整体收敛的。数值实验结果表明,非单调策略对无约束非光滑优化问题的求解是行之有效的,拓展了非单调信赖域算法的应用领域。     

A quasi-Newton method for unconstrained non-smooth problems

We present a method, based on a variational problem, for solving a non-smooth unconstrained optimization problem. We assume that the objective function is a Lipschitz continuous and a regular function. In this case the function of our variational problem is semismooth and a quasi-Newton method may be used to solve the variational problem. A convergence theorem for our algorithm and its discrete version is also proved. Preliminary computational results show that the method performs quite well and can compete with other methods.     

A heuristic iterated-subspace minimization method with pattern search for unconstrained optimization

Recently, an increasing attention was paid on different procedures for an unconstrained optimization problem when the information of the first derivatives is unavailable or unreliable. In this paper, we consider a heuristic iterated-subspace minimization method with pattern search for solving such unconstrained optimization problems. The proposed method is designed to reduce the total number of function evaluations for the implementation of high-dimensional problems. Meanwhile, it keeps the advantages of general pattern search algorithm, i.e., the information of the derivatives is not needed. At each major iteration of such a method, a low-dimensional manifold, the iterated subspace, is constructed. And an approximate minimizer of the objective function in this manifold is then determined by a pattern search method. Numerical results on some classic test examples are given to show the efficiency of the proposed method in comparison with a conventional pattern search method and a derivative-free method.