等式约束的一类新的光滑精确罚函数  

A New Type of Smooth Exact Penalty Function for Equality Constraints

作　　者：房明磊 盛雨婷 丁德凤 FANG Minglei;SHENG Yuting;DING Defeng(School of Mathematics and Big data,Anhui University of Science and Technology,Huainan Anhui 232001,China)

机构地区：[1]安徽理工大学数学与大数据学院,安徽淮南232001

出　　处：《安徽理工大学学报（自然科学版）》2024年第1期89-95,共7页Journal of Anhui University of Science and Technology:Natural Science

基　　金：安徽省高校自然科学基金项目(KJ2021A0451);安徽省自然科学基金项目(2008085MA01)。

摘　　要：目的为了有效地处理含等式、不等式的约束优化问题,追求具有更优性质、更简单形式的罚函数以及高效快速的算法。方法提出了一类新的罚函数,用于解决等式约束优化问题。结果通过证明,得到了新罚函数的性质:光滑性和精确性。同时,可以通过适当选择罚参数的值来实现最优解的求解。特别在Mangasarian-Fromovitz(M-F)约束条件下,可以证明当罚参数取得足够大时,无约束优化问题的局部极小点也是原等式约束优化问题的局部极小点。结合提出的罚函数形式,给出了相应的罚函数算法。通过数值实验的结果,可以验证罚函数算法在求解等式约束优化问题上是可行的。结论新提出的罚函数形式具备了更优的性质,可以有效地将等式约束优化问题转化为无约束优化问题,并利用罚函数算法来求解,为约束优化问题提供了一种新的参考方法。Objective To effectively handle constrained optimization problems containing equations and inequalities,and to strive for the penalty functions with better properties and simpler forms,and the efficient and fast algorithms.Methods A new type of penalty function was proposed to solve the equality constrained optimization problems.Results The properties of the new penalty function smoothness and accuracy were obtained through the proof.Meanwhile,the optimal solution was obtained by appropriately selecting the value of penalty parameters.Specifically,under the constraints of Mangasarian Fromovitz(M-F),it was proven that when the penalty parameter was sufficiently large,the local minimum of the unconstrained optimization problem was also the local minimum of the original equality constrained optimization problem.Based on the proposed penalty function form,the corresponding penalty function algorithm was provided.The results of numerical experiments verified the feasibility of the penalty function algorithm in solving equality constrained optimization problems.Conclusion The newly proposed penalty function form has better properties,able to effectively transform equality constrained optimization problems into unconstrained optimization problems easy to be solved by using penalty function algorithms,providing a new reference method for constrained optimization problems.

关 键 词：等式约束 罚函数 KKT点 Mangasarian-Fromovitz约束条件 

分 类 号：O221.2[理学—运筹学与控制论]