论文标题
非线性半决赛编程及其含义的强大差异足够
Strong Variational Sufficiency for Nonlinear Semidefinite Programming and its Implications
论文作者
论文摘要
强大的差异足以是一种新提出的属性，事实证明，在乘数方法的收敛分析中，它非常有用。但是，这种特性所暗示的非多层问题仍然是一个难题。在本文中，我们证明了强大的变分充足与非线性半限制编程（NLSDP）的强二阶足够条件（SOSC）之间的等效性，而无需乘数或任何其他约束资格的唯一性。基于此特征，在没有约束资格的情况下，可以在强大的SOSC下建立增强Lagrangian方法（ALM）的局部收敛性。此外，在强大的SOSC下，我们可以应用半平滑的牛顿方法来解决NLSDP的ALM子问题，作为增强拉格朗日函数的广义Hessian的正面确定性。
Strong variational sufficiency is a newly proposed property, which turns out to be of great use in the convergence analysis of multiplier methods. However, what this property implies for non-polyhedral problems remains a puzzle. In this paper, we prove the equivalence between the strong variational sufficiency and the strong second order sufficient condition (SOSC) for nonlinear semidefinite programming (NLSDP), without requiring the uniqueness of multiplier or any other constraint qualifications. Based on this characterization, the local convergence property of the augmented Lagrangian method (ALM) for NLSDP can be established under strong SOSC in the absence of constraint qualifications. Moreover, under the strong SOSC, we can apply the semi-smooth Newton method to solve the ALM subproblems of NLSDP as the positive definiteness of the generalized Hessian of augmented Lagrangian function is satisfied.
