A feasible method for optimization with orthogonality constraints

Minimization with orthogonality constraints (e.g., ) and/or spherical constraints (e.g., ) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are dif...

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Bibliographic Details
Published in:Mathematical programming Vol. 142; no. 1-2; pp. 397 - 434
Main Authors: Wen, Zaiwen, Yin, Wotao
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2013
Springer Nature B.V
Subjects:
Algorithms
Analysis
Applied mathematics
Assignment problem
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Eigenvalues
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Partial differential equations
Polynomials
Studies
Theoretical
Topological manifolds
Nearest correlation matrix
Invariant subspace
65K05
Curvilinear search
Maxcut SDP
90C26
Eigenvalue and eigenvector
Quadratic assignment problem
90C27
Orthogonality constraint
Cayley transformation
90C30
Polynomial optimization
Stiefel manifold
90C22
Spherical constraint
49Q99
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:Minimization with orthogonality constraints (e.g., ) and/or spherical constraints (e.g., ) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform—a Crank-Nicolson-like update scheme—to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 % to the best known solution on the largest problem “tai256c” in QAPLIB can be reached in 5 min on a typical laptop.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-012-0584-1