An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone

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 va...

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Main Author: Khanh Hien, Le Thi
Other Authors: Chua Chek Beng
Format: Theses and Dissertations
Language:English
Published: 2015
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Online Access:http://hdl.handle.net/10356/63797
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Institution: Nanyang Technological University
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spelling sg-ntu-dr.10356-637972023-02-28T23:56:09Z An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone Khanh Hien, Le Thi Chua Chek Beng School of Physical and Mathematical Sciences DRNTU::Science::Mathematics 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 $l_\infty$ 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 $K^{\alpha}_{m,n}=\{(x,z)\in \mathbb{R}^m_+ \times\mathbb{R}^n, \norm{z} \leq \prod\limits_{i=1}^m x_i^{\alpha_i}\}$, where $0<\alpha_i$ and $\sum\limits_{i=1}^m \alpha_i=1$. We find projector's formulas, its directional derivative formulas, its first order Fr\'echet derivative formulas for $K^{\alpha}_{m,n}$. 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. ​Doctor of Philosophy (SPMS) 2015-05-19T03:40:08Z 2015-05-19T03:40:08Z 2015 2015 Thesis Le Thi, K. H. (2015). An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone. Doctoral thesis, Nanyang Technological University, Singapore. http://hdl.handle.net/10356/63797 en 122 p. application/pdf
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic DRNTU::Science::Mathematics
spellingShingle DRNTU::Science::Mathematics
Khanh Hien, Le Thi
An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone
description 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 $l_\infty$ 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 $K^{\alpha}_{m,n}=\{(x,z)\in \mathbb{R}^m_+ \times\mathbb{R}^n, \norm{z} \leq \prod\limits_{i=1}^m x_i^{\alpha_i}\}$, where $0<\alpha_i$ and $\sum\limits_{i=1}^m \alpha_i=1$. We find projector's formulas, its directional derivative formulas, its first order Fr\'echet derivative formulas for $K^{\alpha}_{m,n}$. 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.
author2 Chua Chek Beng
author_facet Chua Chek Beng
Khanh Hien, Le Thi
format Theses and Dissertations
author Khanh Hien, Le Thi
author_sort Khanh Hien, Le Thi
title An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone
title_short An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone
title_full An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone
title_fullStr An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone
title_full_unstemmed An inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone
title_sort inexact non-interior continuation method for variational inequalities and differential properties of euclidean projection onto power cone
publishDate 2015
url http://hdl.handle.net/10356/63797
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