A Splitting Method for Orthogonality Constrained Problems

• Published in: Journal of scientific computing Vol. 58; no. 2; pp. 431 - 449
• Main Authors: Lai, Rongjie, Osher, Stanley
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-02-2014; Springer Nature B.V
• Subjects: Algorithms; Color; Color imagery; Computational Mathematics and Numerical Analysis; Conformal mapping; Constraints; Exact solutions; Image restoration; Iterative methods; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Optimization techniques; Orthogonality; Robustness (mathematics); Splitting; Theoretical; Spherical constraint; harmonic energy minimization; Bregman iteration; Conformal mapping; Orthogonality constraint
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-013-9740-x

Orthogonality constrained problems are widely used in science and engineering. However, it is challenging to solve these problems efficiently due to the non-convex constraints. In this paper, a splitting method based on Bregman iteration is represented to tackle the optimization problems with orthogonality constraints. With the proposed method, the constrained problems can be iteratively solved by computing the corresponding unconstrained problems and orthogonality constrained quadratic problems with analytic solutions. As applications, we demonstrate the robustness of our method in several problems including direction fields correction, noisy color image restoration and global conformal mapping for genus-0 surfaces construction. Numerical comparisons with existing methods are also conducted to illustrate the efficiency of our algorithms.