Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points
Augmented Lagrangians with constrained subproblems and convergence to second-order stationary points
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DOI：
10.1007/s10589-017-9937-2
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发表时间：
2017-08
影响因子：
2.2
通讯作者：
E. Birgin;G. Haeser;A. Ramos
中科院分区：
文献类型：
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作者：
E. Birgin;G. Haeser;A. Ramos
这项工作考虑了收敛到二阶驻点的增广拉格朗日方法，其中任何约束都可以被惩罚或对子问题执行。每个子问题的解决可以通过任何能够返回近似二阶驻点的数值算法来完成。所开发的全局收敛理论比当前已知的收敛到二阶点的算法更强，因为除了一般较低级方法引入的灵活性之外，它还包括对子问题解决的宽松要求。所提出的方法依赖于弱约束条件，允许拉格朗日乘子在解处无界。它还表明，子问题的二阶解析增加了找到可行点的机会，从某种意义上说，对于最小化平方不可行性的问题，极限点是二阶平稳的。该方法的适用性通过球约束子问题的数值例子进行了说明。
Augmented Lagrangian methods with convergence to second-order stationary points in which any constraint can be penalized or carried out to the subproblems are considered in this work. The resolution of each subproblem can be done by any numerical algorithm able to return approximate second-order stationary points. The developed global convergence theory is stronger than the ones known for current algorithms with convergence to second-order points in the sense that, besides the flexibility introduced by the general lower-level approach, it includes a loose requirement for the resolution of subproblems. The proposed approach relies on a weak constraint qualification, that allows Lagrange multipliers to be unbounded at the solution. It is also shown that second-order resolution of subproblems increases the chances of finding a feasible point, in the sense that limit points are second-order stationary for the problem of minimizing the squared infeasibility. The applicability of the proposed method is illustrated in numerical examples with ball-constrained subproblems.