Nonmonotone algorithm for minimization on closed sets with applications to minimization on Stiefel manifolds

A nonmonotone Levenberg-Marquardt-based algorithm is proposed for minimization problems on closed domains. By preserving the feasible set's geometry throughout the process, the method generates a feasible sequence converging to a stationary point independently of the initial guess. As an application, a specific algorithm is derived for minimization on Stiefel manifolds and numerical results involving a weighted orthogonal Procrustes problem are reported.

[1]  Nicholas J. Higham,et al.  The symmetric procrustes problem , 1988 .

[2]  Yuhong Dai.  On the Nonmonotone Line Search , 2002 .

[3]  William W. Hager,et al.  A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization , 2004, SIAM J. Optim..

[4]  Haesun Park,et al.  A Procrustes problem on the Stiefel manifold , 1999, Numerische Mathematik.

[5]  Tamás Rapcsák,et al.  On minimization on Stiefel manifolds , 2002, Eur. J. Oper. Res..

[6]  J. Borwein,et al.  Two-Point Step Size Gradient Methods , 1988 .

[7]  Wotao Yin,et al.  A feasible method for optimization with orthogonality constraints , 2013, Math. Program..

[8]  Gene H. Golub,et al.  Matrix computations , 1983 .

[9]  Philippe L. Toint,et al.  An Assessment of Nonmonotone Linesearch Techniques for Unconstrained Optimization , 1996, SIAM J. Sci. Comput..

[10]  Wenyu Sun,et al.  Nonmonotone trust region method for solving optimization problems , 2004, Appl. Math. Comput..

[11]  Xiangsun Zhang,et al.  A nonmonotone trust-region algorithm with nonmonotone penalty parameters for constrained optimization , 2004 .

[12]  R. Janin.  Directional derivative of the marginal function in nonlinear programming , 1984 .

[13]  Paul Van Dooren,et al.  Recursive Calculation of Dominant Singular Subspaces , 2003, SIAM J. Matrix Anal. Appl..

[14]  J. M. Martínez,et al.  Inexact spectral projected gradient methods on convex sets , 2003 .

[15]  José Mario Martínez,et al.  Globally convergent trust-region methods for self-consistent field electronic structure calculations. , 2004, The Journal of chemical physics.

[16]  Wenyu Sun,et al.  Global convergence of nonmonotone descent methods for unconstrained optimization problems , 2002 .

[17]  Robert E. Mahony,et al.  Optimization Algorithms on Matrix Manifolds , 2007 .

[18]  Adam W. Bojanczyk,et al.  The Procrustes Problem for Orthogonal Stiefel Matrices , 1999, SIAM J. Sci. Comput..

[19]  L. Grippo,et al.  A nonmonotone line search technique for Newton's method , 1986 .

[20]  Zhenyue Zhang,et al.  Successive projection method for solving the unbalanced Procrustes problem , 2006 .

[21]  Alan Edelman,et al.  The Geometry of Algorithms with Orthogonality Constraints , 1998, SIAM J. Matrix Anal. Appl..