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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Podrobná bibliografia
Vydané v:	Mathematical programming Ročník 142; číslo 1-2; s. 397 - 434
Hlavní autori:	Wen, Zaiwen, Yin, Wotao
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2013 Springer Nature B.V
Predmet:	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
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Popis
Shrnutí:	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.
Bibliografia:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-2 content type line 23
ISSN:	0025-5610 1436-4646
DOI:	10.1007/s10107-012-0584-1