求解非光滑复合约束优化问题的再分配束方法

Redistributed bundle method for solving nonsmooth composite constrained optimization problem

针对一类特殊的复合约束优化问题提出了再分配型束方法,其中目标函数和约束函数为具有lower- C 2性质的函数.利用改善函数,原约束问题可以被转化为无约束问题,并且新的目标函数也具有lower- C 2性质.再利用lower- C 2函数的性质,引入了凸化参数来改善子问题目标函数的凸性,并设计了相应的束方法.给出了原问题和新问题最优点的关系,并简单地给出了参数稳定性结论和算法的局部收敛性结论.通过对 H 2/H ∞ 综合问题的分析,利用提出的算法计算了最优的 H 2/H ∞ 动态控制器,表明了算法的有效性.

A redistributed-type bundle method is proposed for solving a specific kind of composite optimization problem whose objective function and constraint function are functions with the property of lower- C 2. By using improvement function, the original constrained problem is transformed into an unconstrained problem, in which the new objective function remains lower- C 2. By applying the properties of lower- C 2 function, a convexification parameter is introduced to improve the convexity of the objective function in the subproblem, and a bundle method is designed. The relationship between the optimum points of original problem and the new problem is given, together with the results about the stability of parameters and local convergence. Via analyzing H 2/H ∞ control problem, an optimal H 2/H ∞ dynamic controller is computed by the proposed method and the effectiveness of the method is demonstrated.