Algorithm for Locating All Local Minima of Multidimensional Function

DOI：10.3724/SP.J.1001.2013.04337

 作者 单位 E-mail 刘杰 西安电子科技大学 理学院, 陕西 西安 710071西安科技大学 理学院, 陕西 西安 710054 tears191@foxmail.com 王宇平 西安电子科技大学 计算机学院, 陕西 西安 710071

为求出具有箱式约束的非线性全局优化问题所有的局部极小点,提出了一种基于Multistart 方法的新算法.结合目标函数在可行域内的总变差、下降率和凹凸性等信息,构造了一个刻划局部极小点分布的G-度量.将可行域剖分为若干个小区域,把初始点按G-度量值的比例分配在每块区域上,使得局部极小点密集的区域能够被分配较多的初始点进行搜索;给出了有效初始点的判断条件为了进一步减少局部优化算法的运行次数.针对G-度量计算量较大的问题,设计了相应的近似计算方法,降低了计算量.选择了4 个2 维~10 维具有大量局部极小点的测试函数进行求解,与Multisatart 和Minfinder 算法的实验结果进行对比,表明了该方法在收敛速度和搜索全部局部极小点上都有了较大的改进和提高.

This paper focuses on locating all local minima of box-constrained, non-linear optimization problems. A new algorithm based on Multistart method is proposed. A quality measure called G-measure is constructed to measure the local minima of a multidimensional continuous and differentiable function distribution inside bounded domain. This paper measures the distribution of local minima in three facets: Gradient, convexity and concavity, and rate of decline. Feasible region is divided into several small regions, and each is assigned a set of initial points in proportion to its G-measures. More initial points can be allocated in the region which includes more local minima. A condition judging whether an initial point is effective is aimed to decrease the run times of local optimal technique. The approximate computing method is constructed to reduce computational complexity of G-measure. Several benchmarks with large quantities of local minima are chosen. The performance of this new method is compared with that of Multistart and Minfinder on benchmark problems. Experimental results show that the proposed method performs better in search efficiency.
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