Two adaptive scaled gradient projection methods for Stiefel manifold constrained optimization
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Two adaptive scaled gradient projection methods for Stiefel manifold constrained optimization Harry Oviedo1 · Oscar Dalmau1
· Hugo Lara2
Received: 17 June 2019 / Accepted: 10 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This article is concerned with the problem of minimizing a smooth function over the Stiefel manifold. In order to address this problem, we introduce two adaptive scaled gradient projection methods that incorporate scaling matrices that depend on the stepsize and a parameter that controls the search direction. These iterative algorithms use a projection operator based on the QR factorization to preserve the feasibility in each iteration. However, for some particular cases, the proposals do not require the use of any projection operator. In addition, we consider a Barzilai and Borweinlike step-size combined with the Zhang–Hager nonmonotone line-search technique in order to accelerate the convergence of the proposed procedures. We proved the global convergence for these schemes, and we evaluate their effectiveness and efficiency through an extensive computational study, comparing our approaches with other state-of-the-art gradient-type algorithms. Keywords Nonlinear programming · Equality-constrained optimization · Riemannian constrained optimization · Orthogonality constraint · Spherical constraint · Stiefel manifold · Gradient projection method · Barzilai-Borwein-like method · Nonmonotone line search Mathematics Subject Classification (2010) 49Q99 · 65K05 · 90C30 · 90C22 · 90C26 · 90C27
Oscar Dalmau
[email protected] Harry Oviedo [email protected] Hugo Lara [email protected] 1
Mathematics Research Center, CIMAT A.C. Guanajuato, Guanajuato, Mexico
2
Universidade Federal de Santa Catarina, Campus Blumenau, Blumenau, Brazil
Numerical Algorithms
1 Introduction In this work, we are interested in numerical methods for solving the following optimization problem with orthogonality constraints, min F (X)
X∈Rn×p
s.t.
X X = Ip ,
(1)
where F : Rn×p → R is a given continuously differentiable function. It is wellknown that the feasible set M := {X ∈ Rn×p : X X = Ip } is a compact Riemannian sub-manifold embedded in the Euclidean space Rn×p [1], and is called the Stiefel manifold. This fact turns problem (1) into a special case of manifold constrained optimization problems. This model appears regularly in a significant number of applications such as eigenvalue problems and subspace tracking [2, 3], 1-bit compressive sensing [4, 5], sparse principal component analysis [6], quadratic assignment problems [3], Kohn– Sham total energy minimization [7], pattern recognition (dimensionality reduction techniques) [8], linear eigenvalue problem [9], joint diagonalization [10, 11], the orthogonal Procrustes problem [12], color image restoration, and conformal mapping construction [13], which makes (1) a problem of great interest to the optimization community. Our focus is in gradient-type methods for solving the Stiefel manifold constrained minimizat
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