标题： 基于有限多个向量值函数的集值映射的集合优化问题的拟牛顿方法 显示英文标题

标题： Quasi-Newton Method for Set Optimization Problems with Set-Valued Mapping Given by Finitely Many Vector-Valued Functions

摘要： 在本文中，我们提出了一种拟牛顿方法，用于求解无约束集优化问题的弱极小解，相对于下集无序关系而言。 所考虑的集值目标映射由有限个二次连续可微的向量值函数给出。 为了借助所提出的拟牛顿方法找到弱极小点的必要最优性条件，我们使用了划分的概念，并构建了一族向量优化问题。 为了找到弱极小点的必要最优性条件的评估，涉及计算每个目标函数的近似海森矩阵，这是通过向量优化问题的拟牛顿方案完成的。 在所提出的拟牛顿方法中，我们推导出一个迭代点序列，该序列收敛到满足弱极小点所推导出的必要最优性条件的点。 之后，我们从这一族向量优化问题中找到一个适当选择的向量优化问题的下降方向，并从当前迭代更新到下一个迭代。 所提出的集优化问题的拟牛顿方法并不是向量优化问题拟牛顿方法的直接扩展，因为所选择的向量优化问题在迭代过程中会变化。 分析了所提出方法的良定义性和收敛性。 在静止点的一些正则性条件、非静止点的条件、拟牛顿方向范数的有界性以及存在满足Armijo条件的步长的条件下，推导出了所提出算法的收敛性。 在海森矩阵近似函数的一致连续性条件下，我们获得了所提出方法的局部超线性收敛性。 显示英文摘要

摘要： In this article, we propose a quasi-Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The set-valued objective mapping under consideration is given by a finite number of vector-valued functions that are twice continuously differentiable. To find the necessary optimality condition for weak minimal points with the help of the proposed quasi-Newton method, we use the concept of partition and formulate a family of vector optimization problems. The evaluation of necessary optimality condition for finding the weakly minimal points involves the computation of the approximate Hessian of every objective function, which is done by a quasi-Newton scheme for vector optimization problems. In the proposed quasi-Newton method, we derive a sequence of iterative points that exhibits convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. After that, we find a descent direction for a suitably chosen vector optimization problem from this family of vector optimization problems and update from the current iterate to the next iterate. The proposed quasi-Newton method for set optimization problems is not a direct extension of that for vector optimization problems, as the selected vector optimization problem varies across the iterates. The well-definedness and convergence of the proposed method are analyzed. The convergence of the proposed algorithm under some regularity condition of the stationary points, a condition on nonstationary points, the boundedness of the norm of quasi-Newton direction, and the existence of step length that satisfies the Armijo condition are derived. We obtain a local superlinear convergence of the proposed method under uniform continuity of the Hessian approximation function.