An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone - Khanh Hien, Le Thi (2015)
In the first part of this thesis, using barrier based smoothing approximation, we extend the non-interior continuation method proposed in [B. Chen and N. Xiu, SIAM J. Optim. 9(1999), 605--623] for complementarity problem over non-negative orthant to an inexact non-interior continuation method for variational inequalities over general closed convex sets. The Newton equations involved in the method are solved inexactly to handle large scale problems. The method is proved to have global linear and local quadratic convergence under suitable assumptions. We give application of the algorithm to variational inequalities over non-negative orthant, positive semidefinite cone, epigraph of $l_∞$ norm cone, epigraph of $l_1$ norm cone, epigraph of matrix operator norm cone and epigraph of matrix nuclear norm cone. We also report numerical performance of the algorithm to prove its efficiency. The second part of this thesis studies differential properties of Euclidean projection onto the high dimensional power cones $K^{α}_{m,n}=\{(x,z)\in \mathbb{R}^m_+ ×\mathbb{R}^n, \norm{z} ≤ \prod\limits_{i=1}^m x_i^{α_i}\}$, where $0<α_i$ and $\sum\limits_{i=1}^m α_i=1$. We find projector's formulas, its directional derivative formulas, its first order Fr\'echet derivative formulas for $K^{α}_{m,n}$. We also consider strongly semismoothness of Euclidean projection onto the cone when $m=2$. Euclidean projector onto certain power cones is the first example of semismooth but non-strongly semismooth projector onto a convex cone.