Abstract
This paper presents a descent direction method for finding extrema of locally Lipschitz functions defined on Riemannian manifolds. To this end we define a set-valued mapping \(x\rightarrow \partial _{\varepsilon } f(x)\) named ε-subdifferential which is an approximation for the Clarke subdifferential and which generalizes the Goldstein- ε-subdifferential to the Riemannian setting. Using this notion we construct a steepest descent method where the descent directions are computed by a computable approximation of the ε-subdifferential. We establish the global convergence of our algorithm to a stationary point. Numerical experiments illustrate our results.
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Absil, P.A., Baker, C.G.: Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7, 303–330 (2007)
Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithm on Matrix Manifolds. Princeton University Press (2008)
Adler, R.L., Dedieu, J.P., Margulies, J.Y., Martens, M., Shub, M.: Newton’s method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22, 359–390 (2002)
Afsari, B., Tron, R., Vidal, R.: On the convergence of gradient descent for finding the Riemannian center of mass. SIAM J. Control Optim. 51, 2230–2260 (2013)
Azagra, D., Ferrera, J., López-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220, 304–361 (2005)
Azagra, D., Ferrera, J.: Applications of proximal calculus to fixed point theory on Riemannian manifolds. Nonlinear. Anal. 67, 154–174 (2007)
Bagirov, A.M.: Continuous subdifferential approximations and their applications. J. Math. Sci. 115, 2567–2609 (2003)
Bento, G.C., Ferreira, O.P., Oliveira, P.R.: Unconstrained steepest descent method for multicriteria optimization on Riemannian manifolds. J. Optim. Theory Appl. 154, 88–107 (2012)
Bento, G.C., Melo, J.G.: A subgradient method for convex feasibility on Riemannian manifolds. J. Optim. Theory Appl. 152, 773–785 (2012)
Borckmans, P.B., Easter Selvan, S., Boumal, N., Absil, P.A.: A Riemannian subgradient algorithm for economic dispatch with valve-point effect. J. Comput. Appl. Math. 255, 848–866 (2014)
Burke, J.V., Lewis, A.S., Overton, M.L.: Approximating subdifferentials by random sampling of gradients. Math. Oper. Res. 27, 567–584 (2002)
Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15, 751–779 (2005)
Clarke, F.H.: Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations, Thesis, University of Washington, Seattle (1973)
da Cruz Neto, J.X., Lima, L.L., Oliveira, P.R.: Geodesic algorithm in Riemannian manifolds. Balkan J. Geom. Appl. 2, 89–100 (1998)
da Cruz Neto, J.X., Ferreira, O.P., Lucambio Perez, L.R.: A proximal regularization of the steepest descent method in Riemannian manifold. Balkan J. Geom. Appl. 2, 1–8 (1999)
Dirr, G., Helmke, U., Lageman, C.: Nonsmooth Riemannian optimization with applications to sphere packing and grasping In Lagrangian and Hamiltonian Methods for Nonlinear Control 2006: Proceedings from the 3rd IFAC Workshop, Nagoya, Japan, 2006, Lecture Notes in Control and Information Sciences, vol. 366. Springer, Berlin (2007)
Ferreira, O.P., Oliveira, P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim. Theory Appl. 97, 93–104 (1998)
Grohs, P., Hardering, H., Sander, O.: Optimal a priori discretization error bounds for geodesic finite elements, SAM Report 2013–16, ETH Zürich. Submitted (2013)
Goldstein, A.A.: Optimization of Lipschitz continuous functions. Math. Program. 13, 14–22 (1977)
Hosseini, S., Pouryayevali, M.R.: Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds. Nonlinear Anal. 74, 3884–3895 (2011)
Hosseini, S., Pouryayevali, M.R.: Euler characterization of epi-Lipschitz subsets of Riemannian manifolds. J. Convex. Anal. 20(1), 67–91 (2013)
Hosseini, S., Pouryayevali, M.R.: On the metric projection onto prox-regular subsets of Riemannian manifolds. Proc. Amer. Math. Soc. 141, 233–244 (2013)
Kiwiel, K.C.: Methods of descent for nondifferentiable optimization. Lecture Notes in Mathematics, vol. 1133. Springer, Berlin (1985)
Klingenberg, W.: Riemannian Geometry. Walter de Gruyter Studies in Mathematics, vol. 1. Walter de Gruyter, Berlin (1995)
Kressner, D., Steinlechner, M., Vandereycken, B.: Low-rank tensor completion by Riemannian optimization. BIT Numer. Math. 54, 447–468 (2014)
Lang, S.: Fundamentals of Differential Geometry. Graduate Texts in Mathematics, vol. 191. Springer, New York (1999)
Lee, P.Y.: Geometric Optimization for Computer Vision. PhD thesis, Australian National University (2005)
Lemarechal, C.: Nondifferentiable optimization. In: Nemhauser, G.L., et al. (eds.) Handbook in Operations Research and Management Science, vol. 1, pp 529–572. North Holland, Amsterdam (1989)
Li, C., Mordukhovich, B.S., Wang, J., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21(4), 1523–1560 (2011)
Mahony, R.E.: The constrained Newton method on a Lie group and the symmetric eigenvalue problem. Linear Algebra. Appl. 248, 67–89 (1996)
Mahdavi-Amiri, N., Yousefpour, R.: An effective nonsmooth optimization algorithm for locally Lipschitz functions. J. Optim. Theory Appl. 155, 180–195 (2012)
Moakher, M., Zerai, M.: The Riemannian geometry of the space of positive-definite matrices and its application to the regularization of positive-definite matrix-valued data. J. Math. Imaging Vision 40, 171–187 (2011)
Papa Quiroz, E.A., Quispe, E.M., Oliveira, P.R.: Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds. J. Math. Anal. Appl. 341, 467–477 (2008)
Ring, W., Wirth, B.: Optimization methods on Riemannian manifolds and their application to shape space. SIAM J. Optim. 22(2), 596–627 (2012)
Riddell, R.C.: Minimax problems on Grassmann manifolds. Sums of eigenvalues. Adv. Math. 54, 107–199 (1984)
Rockafellar, R.T.: Convex Functions and Dual Extremum Problems, Thesis, Harvard (1963)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D 60(1), 259–268 (1992)
Sander, O.: Geodesic finite elements for Cosserat rods. Internat. J. Numer. Methods Engrg. 82, 1645–1670 (2010)
Smith, S.T.: Optimization techniques on Riemannian manifolds. Fields Institute Communications 3, 113–146 (1994)
Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers, Dordrecht (1994)
Usevich, K, Markovsky, I.: Optimization on a Grassmann manifold with application to system identification. Submitted, ( http://homepages.vub.ac.be/imarkovs/t-abstracts.html)
Vandereycken, B., Vandewalle, S.: A Riemannian optimization approach for computing low-rank solutions of Lyapunov equations. SIAM. J. Matrix Anal. Appl. 31(5), 2553–2579 (2010)
Vandereycken, B.: Low-rank matrix completion by Riemannian optimization. SIAM J. Optim. 23(2), 1214–1236 (2013)
Yau, S.T.: Non-existence of continuous convex functions on certain Riemannian manifolds. Math. Ann. 207, 269–270 (1974)
Zhang, L.S., Sun, X.L.: An algorithm for minimizing a class of locally Lipschitz functions. J. Optim. Theory Appl. 90, 203–212 (1996)
Zhang, L.H.: Riemannian Newton method for the multivariate eigenvalue problem. SIAM J. Matrix Anal. Appl. 31, 2972–2996 (2010)
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Communicated by: A. Zhou
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Grohs, P., Hosseini, S. ε-subgradient algorithms for locally lipschitz functions on Riemannian manifolds. Adv Comput Math 42, 333–360 (2016). https://doi.org/10.1007/s10444-015-9426-z
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DOI: https://doi.org/10.1007/s10444-015-9426-z