关于退化双重非负投影问题
On Degenerate Doubly Nonnegative Projection Problems
摘要：
双非负（DNN）锥是所有元素非负的正半定矩阵的集合，是计算上难以处理的完全正锥的一个流行的近似。用Newton型方法计算大规模矩阵在DNN锥上的投影的主要困难在于约束非退化性的可能失效，这是非线性规划线性无关约束规格的推广.这种失效导致代表Karush-Kuhn-Tucker最优性条件的非光滑方程的雅可比矩阵的奇异性，从而阻止半光滑Newton-CG方法以期望的收敛速度求解该方程。在本文中，我们克服了上述困难，通过求解一系列更好的条件非光滑方程产生的增广拉格朗日方法（ALM），而不是解决上述奇异方程。利用半正定锥的法锥的度量次正则性，得到了保证问题的对偶二次增长条件的充分条件，从而进一步证明了算法的渐近超线性收敛.困难的随机生成的情况下，从半定规划库的数值结果，以证明计算DNN投影到一个非常高的精度的算法的效率。
英文摘要：
The doubly nonnegative (DNN) cone, being the set of all positive semidefinite matrices whose elements are nonnegative, is a popular approximation of the computationally intractable completely positive cone. The major difficulty for implementing a Newton-type method to compute the projection of a given large scale matrix onto the DNN cone lies in the possible failure of the constraint nondegeneracy, a generalization of the linear independence constraint qualification for nonlinear programming. Such a failure results in the singularity of the Jacobian of the nonsmooth equation representing the Karush-Kuhn-Tucker optimality condition that prevents the semismooth Newton-CG method from solving it with a desirable convergence rate. In this paper, we overcome the aforementioned difficulty by solving a sequence of better conditioned nonsmooth equations generated by the augmented Lagrangian method (ALM) instead of solving one above mentioned singular equation. By leveraging on the metric subregularity of the normal cone associated with the positive semidefinite cone, we derive sufficient conditions to ensure the dual quadratic growth condition of the underlying problem, which further leads to the asymptotically superlinear convergence of the proposed ALM. Numerical results on difficult randomly generated instances and from the semidefinite programming library are presented to demonstrate the efficiency of the algorithm for computing the DNN projection to a very high accuracy.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
