不等式约束极大极小问题的可行下降束方法

A feasible descent bundle method for inequality constrained Minimax problems

本文提出一个求解不等式约束极大极小问题的可行下降束方法.该方法的主要特点有(1)借助于函数的次梯度及束方法思想,不需要假设原问题的分量函数具备光滑性;(2)利用部分割平面模型技术,每次无效步迭代仅需利用一个分量函数的函数值和次梯度产生新的割平面,从而有效减少了计算量;(3)能够保证有效迭代点的可行性及目标函数的下降性;(4)引入次梯度聚集技术,对束集中的次梯度进行聚集,克服了数值计算和存储的困难;(5)算法具备全局收敛性,且初步的数值试验表明算法是有效的.

In this paper, a feasible descent bundle method for solving inequality constrained Minimax problems is proposed. The main features of the method are：（1） By using the subgradients of functions and the idea of bundle method, it does not assume that the component functions of the original problem are smooth;（2） by adopting the technique of partial cutting-planes model, at each null step, only the function value and subgradient of one component function are needed to generate the new cutting plane, and therefore the computational cost is reduced effectively;（3） it can ensure the feasibility of the serious iterates and the decsent of the objective function;（4） by introducing the technique of subgradient aggregation to aggregate the subgradients in the bundle, the difficulty of numerical calculation and storage is overcome;（5） the algorithm is proved to be globally convergent, and the preliminary numerical results show that the proposed algorithm is effective.