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基于正弦型光滑打磨函数对0-1规划问题的连续化求解方法

隋允康, 李臻臻, 李宏, 陈国庆   

  1. 1. 北京工业大学机械工程与电子技术学院, 北京 100124 2. 内蒙古师范大学数学科学学院, 呼和浩特 010022
    3. 内蒙古大学数学科学学院, 呼和浩特 010021
  • 收稿日期:2016-09-30 出版日期:2017-09-15 发布日期:2017-09-15
  • 通讯作者: 隋允康 ysui@bjut.edu.cn
  • 基金资助:

    国家自然科学基金(No.11672103), 内蒙古自治区自然科学基金(No.2014MS0119), 内蒙古自治区高等学校科学研究项目(No. NJSY16030), 内蒙古师范大学科研基金(Nos. 2016ZRYB001, 2016ZRYB002)

Continuous solution method for 0-1 programming based on the sinusoidal smooth polish function

SUI Yunkang, LI Zhenzhen, LI Hong, CHEN Guoqing   

  1. 1. College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China 2. School of Mathematical Sciences, Inner Mongolia Normal University, Hohhot 010022, China 3. School of Mathematical Sciences, Inner Mongolia University,  Hohhot 010021, China
  • Received:2016-09-30 Online:2017-09-15 Published:2017-09-15

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摘要：

传统的求解0-1规划问题方法大多属于直接离散的解法. 现提出一个包含严格转换和近似逼近三个步骤的连续化解法: (1) 借助阶跃函数把0-1离散变量转化为[0,1]区间上的连续变量; (2) 对目标函数采用逼近折中阶跃函数近光滑打磨函数, 约束条件采用线性打磨函数逼近折中阶跃函数, 把0-1规划问题由离散问题转化为连续优化模型; (3) 利用高阶光滑的解法求解优化模型. 该方法打破了特定求解方法仅适用于特定类型0-1规划问题惯例, 使求解0-1规划问题的方法更加一般化. 在具体求解时, 采用正弦型光滑打磨函数来逼近折中阶跃函数, 计算效果很好.

关键词: 阶跃函数, 0-1规划问题, 离散, 连续, 折中阶跃函数, 光滑打磨函数, 正弦型打磨函数

Abstract:

Traditional solutions of 0-1 programming problems belong mostly to direct discrete solving methods. A strict conversion for the problem and approximate continuous solution are proposed in this paper which have three steps: (1) 0-1 discrete variables were expressed as continuous variables on the interval [0, 1] by means of a step function; (2) Objective function is approximated to take more near smooth polish function to approach tradeoff step function, and every constraint function is approximated to take linear polish function to approach tradeoff step function, then 0-1 programming problem is transformed to continuous optimization model from discrete problem; (3) The model is solved by using the method with high smoothness solution. This method breaks certain solving method applies only to certain types of 0-1 programming, so it can solve more general problems. During the solving process, a sinusoidal polish function is taken to obtain very good computational results.

Key words: step function, 0-1 programming, discrete, continuous, tradeoff step function, smooth polish function, sinusoidal polish function