基于降维的全局优化近似解法  

Approximate Solution of Global Optimization Problem Based on Dimensionality Reduction

作　　者：陈丹丹 王薇[1] 徐以汎[2] CHEN Dandan;WANG Wei;XU Yifan(Department of Mathematics,East China University of Science and Technology,Shanghai 200237,China;School of Management,Fudan University,Shanghai 200433,China)

机构地区：[1]华东理工大学数学系,上海200237 [2]复旦大学管理学院,上海200433

出　　处：《华东理工大学学报（自然科学版）》2019年第6期995-1000,共6页Journal of East China University of Science and Technology

基　　金：国家自然科学基金（71531005）

摘　　要：将降维应用到全局优化问题的求解中,提出了一个基于降维的全局优化近似算法,用以求解带箱约束的非线性全局优化问题。首先在区间[0,π]上构造一个新的降维公式,给出基于该降维变换曲线的α-致密度,再从降维曲线长度对该近似算法的计算量进行估计并给予证明,给出理论算法,最后给出了数值实验结果以说明算法的有效性。The idea of dimensionality reduction is to develop a n-variable objective function onto an α-dense curve which is subsequently transformed into a one-variable function. It can be proved that the global optima of the univariate function can approximate global optima of the original problem. The approximation degree of the approximate solution depends on the density of the n-dimensional space filled by the curve. When the point on the α-dense curve is in the feasible domain, an approximate solution with sufficient accuracy can be obtained. In some cases, the reducing dimension technology is combined with other algorithms with the aim to explore a new way towards the global optimization. When α-dense curve h is constructed in the feasible set X, a n-dimensional function can be approximated by a one-dimensional function. With increase in the numbers of independent variables, solving the global optima of multivariate function turns to be more complicated and needs more calculations. In order to minimize computational complexity, length of the α-dense curve was computed to obtain general classes of reducing transformations having minimal properties, and the properties of dimensionality reduction need to be investigated. A global optimization approximation algorithm based on dimensionality reduction method was proposed to solve the nonlinear global optimization problem with box constraints. Firstly, a new reduction transformation was constructed on the interval[0, π], and the density of the α-dense curve was given. Secondly, the amount of calculation according to the approximate algorithm was estimated from the length of the α-dense curve and the proof was given. Thirdly, a theoretical algorithm was proposed. Finally, the results of numerical experiments were listed to show the effectiveness of the algorithm.

关 键 词：全局最优化 降维变换 α-致密 近似 

分 类 号：O221.2[理学—运筹学与控制论]