Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: S. Joe Qin. Vita. Includes bibliographical references.

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2006. / Includes bibliographical references (leaves 155-163). Also available in electronic version.

Thesis (Ph. D.)--Civil and Environmental Engineering, Georgia Institute of Technology, 2008. / Jae-Hong Kim, Committee Chair ; Philip J.W. Roberts, Committee Member ; E. Michael Perdue, Committee Member ; Sotira Yiacoumi, Committee Member ; Thorsten Stoesser, Committee Member.

Thesis (Ph. D.)--University of Washington, 1986. / Vita. Bibliography: leaves [81]-83.

Thesis (Ph.D.)--Hong Kong University of Science and Technology, 2007. / Includes bibliographical references (leaves 120-125). Also available in electronic version.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Thesis (M.S.)-- Carnegie-Mellon University, 1983. / Bibliography : p. 72-73.

Thesis (M.S.) -- Worcester Polytechnic Institute. / Keywords: Robust estimation; portfolio optimization. Includes bibliographical references (leaf 24).

This thesis focuses on optimization problems with generalized orthogonality constraints, which may also contains linear equality constraints. These problems appear in many areas, such as machine learning, signal processing, computer vision and so on.;Many problems in this form are NP hard. One challenge posed by generalized orthogonality constraints is local minimizers loaded by nonconvex constraints. More-over, the generalized orthogonality constraints are numerically expensive to preserve during iterations.;This thesis is mainly divided into two parts. The first part is focused on solving generalized orthogonality constrained optimization problems with differentiable objective functions. For this class of optimization problems, a generalized gradient.;This thesis is mainly divided into two parts. The first part is focused on solving generalized orthogonality constrained optimization problems with differentiable objective functions. For this class of optimization problems, a generalized gradient flow is proposed, which is contained on the constraints set if the initial condition satisfies generalized orthogonality constrains. The week convergence of the generalizedgradient flow is given. A discrete iterative scheme is also proposed to make the gradient flow method computable. In addition, we analyze the relationship between our discrete iteration scheme and some existing constraint preserving methods, and the relationship between our discrete iteration scheme and the inexact forward-backward method, respectively. Several problems which also can be solved by the generalized gradient flow are given. Furthermore, we also propose an optimal gradient flow by an alyzing the first order optimality condition.;The second part of this thesis is devoted to study of the generalized orthogonality constrained optimization problems with nondifferentiable objective functions. An approximate augmented Lagrangian method is used to deal with this class of problems. The global convergence is presented. We also extend the proximal alternating linearized minimization method (EPALM) to deal with the generalized orthogonality constraints appeared in the subproblem of the approximate augmented Lagrangian method. Moreover, to accelerate the EPALM method, an inertial proximal alternating linearized minimization method (IPALM) is proposed to deal with unconstrained nonconvex, nonsmooth problems with coupled objective functions.;Keywords: Generalized Orthogonality Constraints; Stiefel Manifold; Tangent Space;Gradient Flow; Approximate Augmented Lagrangian Method; Proximal Alternating Linearized Minimization Method.

Song, Haifeng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 67-69). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminary --- p.8 / Chapter 2.1 --- Introduction --- p.8 / Chapter 2.2 --- Notations and fundamental properties --- p.8 / Chapter 2.3 --- Properties of polyhedra --- p.14 / Chapter 3 --- Results on Efficient Point Sets --- p.23 / Chapter 3.1 --- Introduction --- p.23 / Chapter 3.2 --- Geometric results on efficient point sets --- p.24 / Chapter 3.3 --- Density of positive proper efficient point sets --- p.33 / Chapter 4 --- Pareto Solutions of Polyhedral-valued Vector Optimization --- p.42 / Chapter 4.1 --- Introduction --- p.42 / Chapter 4.2 --- The structure of weak Pareto solution sets --- p.43 / Chapter 4.2.1 --- The general ordering cone case --- p.46 / Chapter 4.2.2 --- The polyhedral ordering cone case --- p.54 / Chapter 4.3 --- Connectedness of solution sets and optimal value sets --- p.55 / Chapter 4.4 --- Optimality conditions of piecewise linear mappings --- p.60 / Bibliography --- p.67