【作者】 刘少华；

【导师】 苏珂；

【作者基本信息】 河北大学 ， 数学， 2023， 硕士

【摘要】 极大极小问题是一类特殊的非光滑问题,在工程设计、金融投资等实际领域都有着广泛的应用,求解该问题的算法也是多种多样的.本文在已有的基础上,针对极大极小问题进行如下研究:(1)提出一种基于信赖域序列二次规划方法的非单调无罚无滤子算法.对二次子问题进行修正以避免子问题可能不可行的情况出现.给出了带有可调算子的非单调的等价机制,代替滤子结构作为新的更新规则,从而能够一定程度上避免Maratos效应,使得算法更灵活.证明了该算法的全局收敛性和超线性收敛速度,并给出了数值结果.(2)提出一种带有可调算子的基于无序列二次规划的无罚无滤子算法来求解约束极大极小问题.首先,在每次迭代中,求解两个基于Karush-Kuhn-Tucker(KKT)条件的、由非线性互补函数组成的线性方程组来得到搜索方向.其次,通过工作集技术,进一步减小了计算量.同时,给出一个非单调平衡机制取代滤子结构,并根据每次迭代的结果更新可调算子,从而得到新的迭代点更新规则.最后给出算法的收敛性的证明,并且通过数值结果展示了算法的效率.(3)将已取得的研究成果扩展到半无限极大极小问题,给出一种求解半无限极大极小问题的信赖域序列二次规划无罚无滤子方法.更多还原

【Abstract】 The minimax problem is a special kind of nonsmooth problems,which is widely used in engineering design,financial investment strategy and other practical fields.The algorithms are also diverse.This thesis proposes a series of algorithms without a penalty function or a filter for minimax problems,and conducts the following research:(1)An adaptive nonmonotonic trust region sequential quadratic programming algorithm for solving constrained minimax problems is proposed.Corrected quadratic subproblems are used to ensure the feasibility of subproblems.An equivalent mechanism with adjustable operators is proposed to replace the filter structure as a new acceptable criterion.The adaptive parameters are adjusted according to the improvement of the current iteration to avoid the Maratos effect.At the same time,the algorithm combining non-monotonic techniques is proved to be more flexible than than before.(2)A penalty-free and filter-free quadratic programming algorithm with adjustable operators for minimax problems is proposed.Firstly,in each iteration,the search direction is obtained by solving two linear equations with the same coefficient matrixs,which are composed of two nonlinear complementarity functions based on Karush-Kuhn-Tucker(KKT)conditions.Secondly,combined with the active set technique,the computational scale is further reduced.Finally,a non-monotone equilibrium mechanism is used to replace the filter structure,and the adjustable operator is updated according to the results of each iteration,thus a new iteration point update rule is obtained.(3)Extending the results of research achieved to semi-infinite minimax problem,and gives a trust region sequential quadratic programming method without a penalty function or a filter for solving semi-infinite minimax problem.更多还原