Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds

This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function Φ : TM → R on the tangent bundle TM , and at k-th iteration, using the restricted function TxkM where TxkM is the tangent space at xk, a local model function Qk that carries both first and second order information for the locally Lipschitz objective function f : M → R on a Riemannian manifold M , is defined and minimized over a trust region. We establish the global convergence of the proposed algorithm. Moreover, using the Riemannian esubdifferential, a suitable model function is defined. Numerical experiments illustrate our results.

[1]  I. Holopainen.  Riemannian Geometry , 1927, Nature.

[2]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[3]  R. Riddell,et al.  Minimax problems on Grassmann manifolds. Sums of eigenvalues , 1984 .

[4]  P. Toint,et al.  Global convergence of a class of trust region algorithms for optimization with simple bounds , 1988 .

[5]  李幼升,et al.  Ph , 1989 .

[6]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[7]  P. Basser,et al.  MR diffusion tensor spectroscopy and imaging. , 1994, Biophysical journal.

[8]  Liqun Qi,et al.  A trust region algorithm for minimization of locally Lipschitzian functions , 1994, Math. Program..

[9]  C. Udriste,et al.  Convex Functions and Optimization Methods on Riemannian Manifolds , 1994 .

[10]  Richard A. Tapia,et al.  A unified approach to global convergence of trust region methods for nonsmooth optimization , 1995, Math. Program..

[11]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[12]  R. Mahony.  The constrained newton method on a Lie group and the symmetric eigenvalue problem , 1996 .

[13]  O. P. Ferreira,et al.  Subgradient Algorithm on Riemannian Manifolds , 1998 .

[14]  O. P. Ferreira,et al.  A proximal regularization of the steepest descent method in riemannian manifolds , 1999 .

[15]  Thomas de Quincey.  [C] , 2000, The Works of Thomas De Quincey, Vol. 1: Writings, 1799–1820.

[16]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[17]  O. P. Ferreira,et al.  Proximal Point Algorithm On Riemannian Manifolds , 2002 .

[18]  R. Adler,et al.  Newton's method on Riemannian manifolds and a geometric model for the human spine , 2002 .

[19]  J. Ferrera,et al.  Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds , 2003, math/0305427.

[20]  Xavier Pennec,et al.  A Riemannian Framework for Tensor Computing , 2005, International Journal of Computer Vision.

[21]  Pei Yean Lee.  Geometric optimization for computer vision , 2005 .

[22]  D. Epstein,et al.  Fundamentals of hyperbolic geometry : selected expositions , 2006 .

[23]  U. Helmke,et al.  Nonsmooth Riemannian Optimization with Applications to Sphere Packing and Grasping , 2007 .

[24]  K. G. Murty,et al.  Nonsmooth optimization , 2007 .

[25]  Daniel Azagra Rueda,et al.  Applications of proximal calculus to fixed point theory on Riemannian manifolds , 2007 .

[26]  Pierre-Antoine Absil,et al.  Trust-Region Methods on Riemannian Manifolds , 2007, Found. Comput. Math..

[27]  P. Roberto Oliveira,et al.  Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds , 2008 .

[28]  Paulo Roberto Oliveira,et al.  Proximal point method for a special class of nonconvex functions on Hadamard manifolds , 2008, 0812.2201.

[29]  G. C. Bento,et al.  Proximal point method for a special class of nonconvex functions on Hadamard manifolds , 2008, 0809.2594.

[30]  Samir Elhedhli,et al.  Nondifferentiable Optimization , 2009, Encyclopedia of Optimization.

[31]  Oliver Sander,et al.  Geodesic finite elements for Cosserat rods , 2009 .

[32]  A. Robert Calderbank,et al.  Grassmannian Packings From Operator Reed–Muller Codes , 2010, IEEE Transactions on Information Theory.

[33]  Pierre-Antoine Absil,et al.  Riemannian BFGS Algorithm with Applications , 2010 .

[34]  Stefan Vandewalle,et al.  A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations , 2010, SIAM J. Matrix Anal. Appl..

[35]  Lei-Hong Zhang,et al.  Riemannian Newton Method for the Multivariate Eigenvalue Problem , 2010, SIAM J. Matrix Anal. Appl..

[36]  Chong Li,et al.  Mathematics 4-1-2009 Weak Sharp Minima on Riemannian Manifolds , 2014 .

[37]  M. R. Pouryayevali,et al.  Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds , 2011 .

[38]  Klaus Diepold,et al.  Camera-Pose Estimation via Projective Newton Optimization on the Manifold , 2012, IEEE Transactions on Image Processing.

[39]  Benedikt Wirth,et al.  Optimization Methods on Riemannian Manifolds and Their Application to Shape Space , 2012, SIAM J. Optim..

[40]  Jefferson G. Melo,et al.  Subgradient Method for Convex Feasibility on Riemannian Manifolds , 2011, Journal of Optimization Theory and Applications.

[41]  M. R. Pouryayevali,et al.  On the metric projection onto prox-regular subsets of Riemannian manifolds , 2012 .

[42]  Orizon Pereira Ferreira,et al.  Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds , 2012, Journal of Optimization Theory and Applications.

[43]  Nezam Mahdavi-Amiri,et al.  An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions , 2012, J. Optim. Theory Appl..

[44]  René Vidal,et al.  On the Convergence of Gradient Descent for Finding the Riemannian Center of Mass , 2011, SIAM J. Control. Optim..

[45]  Berna Ozbek,et al.  Feedback Strategies for Wireless Communication , 2013 .

[46]  Bart Vandereycken,et al.  Low-Rank Matrix Completion by Riemannian Optimization , 2013, SIAM J. Optim..

[47]  M. R. Pouryayevali,et al.  Euler Characteristic of Epi-Lipschitz Subsets of Riemannian Manifolds , 2013 .

[49]  Bart Vandereycken,et al.  Low-rank tensor completion by Riemannian optimization , 2014 .

[50]  Steven Thomas Smith,et al.  Optimization Techniques on Riemannian Manifolds , 2014, ArXiv.

[51]  Pierre-Antoine Absil,et al.  A Riemannian subgradient algorithm for economic dispatch with valve-point effect , 2014, J. Comput. Appl. Math..

[52]  Andreas Weinmann,et al.  Total Variation Regularization for Manifold-Valued Data , 2013, SIAM J. Imaging Sci..

[53]  Ivan Markovsky,et al.  Optimization on a Grassmann manifold with application to system identification , 2014, Autom..

[54]  Philipp Grohs,et al.  Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements , 2014, Foundations of Computational Mathematics.

[55]  M. Reza Peyghami,et al.  A New Nonsmooth Trust Region Algorithm for Locally Lipschitz Unconstrained Optimization Problems , 2015, J. Optim. Theory Appl..

[56]  Philipp Grohs,et al.  ε-subgradient algorithms for locally lipschitz functions on Riemannian manifolds , 2015, Advances in Computational Mathematics.

[57]  Brigitte Maier,et al.  Fundamentals Of Differential Geometry , 2016 .

[58]  Marie Faerber,et al.  Recent Advances In Optimization And Its Applications In Engineering , 2016 .