Convergence of nonsmooth descent methods via Kurdyka-Lojasiewicz inequality on Riemannian manifolds

We develop a subgradient-oriented descent method in nonsmooth optimization on Riemannian manifolds and prove convergence of the method in the sense of subsequences for nonsmooth functions whose standard models are strict. Moreover, we present a nonsmooth version of the Kurdyka-Lojasiewicz inequality and show that a locally Lipschitz C-function defined on an analytic manifold satisfies this inequality. Finally, we prove that if the objective function satisfies the Kurdyka-Lojasiewicz inequality and its standard model is strict, then the sequence of iterates of the subgradient-oriented descent algorithm converges to a singular critical point.

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

[2]  E. Bierstone,et al.  Semianalytic and subanalytic sets , 1988 .

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

[4]  S. Lang.  Fundamentals of differential geometry , 1998 .

[5]  K. Kurdyka.  On gradients of functions definable in o-minimal structures , 1998 .

[6]  Adrian S. Lewis,et al.  The [barred L]ojasiewicz Inequality for Nonsmooth Subanalytic Functions with Applications to Subgradient Dynamical Systems , 2006, SIAM J. Optim..

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

[8]  Adrian S. Lewis,et al.  Clarke Subgradients of Stratifiable Functions , 2006, SIAM J. Optim..

[9]  Aude Rondepierre,et al.  A Proximity Control Algorithm to Minimize Nonsmooth and Nonconvex Functions , 2008 .

[10]  Hédy Attouch,et al.  Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Lojasiewicz Inequality , 2008, Math. Oper. Res..

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

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

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

[14]  Dominikus Noll,et al.  Convergence of Non-smooth Descent Methods Using the Kurdyka–Łojasiewicz Inequality , 2014, J. Optim. Theory Appl..

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

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

[17]  P. Grohs,et al.  Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds , 2016 .

[18]  André Uschmajew,et al.  A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds , 2017, SIAM J. Optim..