A Newton-type active set algorithm for large-scale minimization subject to polyhedral constraints is proposed. The algorithm consists of a gradient projection step, a second-order Newton-type step in the null space of the constraint matrix, and a set of rules for branching between the two steps. We show that the proposed method asymptotically takes the Newton step when the active constraints are linearly independent and a strong second-order sufficient optimality condition holds. We also show that the method has a quadratic rate of convergence under standard conditions. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEst and on a specific class of problems for which finding second-order stationary points is critical.
Semismooth Newton methods for degenerate polyhedral projection
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）；Fudan University（复旦大学）
A Hybrid Quasi-Newton Projected-Gradient Method with Application to Lasso and Basis-Pursuit Denoise
本文提出了一种新的算法，用于Rn中多面体集上凸函数的优化。该算法扩展了谱投影梯度法，尽可能将有限内存BFGS迭代限制在当前人脸上。我们在适当的条件下证明了算法的收敛性，并将该算法应用于解决Lasso问题，从而通过van den Berg和Friedlander [SIAM Journal on Scientific Computing，31（2），2008]提出的寻根框架解决了基追踪去噪问题。该算法特别适用于简单区域，也可用于解决有界约束问题以及限制在单纯形的问题。
