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A proximal point algorithm for DC fuctions on Hadamard manifolds

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Abstract

An extension of a proximal point algorithm for difference of two convex functions is presented in the context of Riemannian manifolds of nonposite sectional curvature. If the sequence generated by our algorithm is bounded it is proved that every cluster point is a critical point of the function (not necessarily convex) under consideration, even if minimizations are performed inexactly at each iteration. Application in maximization problems with constraints, within the framework of Hadamard manifolds is presented.

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References

  1. Martinet, B.: Regularisation d’inéquations variationelles par approximations succesives. Rev. Franaise d’Inform. Recherche Oper. 4, 154–159 (1970)

    MATH  MathSciNet  Google Scholar 

  2. Moreau, J.J.: Proximité et dualité dans un espace Hilbertien. Bull. Soc. Math. France 93, 273–299 (1965)

    MATH  MathSciNet  Google Scholar 

  3. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bento, G.C., Ferreira, O.P., Oliveira, P.R.: Local convergence of the proximal point method for a special class of nonconvex functions on Hadamard manifolds. Nonlinear Anal. 73, 564–572 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Papa Quiroz, E.A., Oliveira, P.R.: Proximal point method for minimization quasiconvex locally Lipschitz functions on Hadamard manifolds. Nonlinear Anal. 75, 5924–5932 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)

  7. da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R., Németh, S.Z.: Convex-and monotone-transformable mathematical programming problems and a proximal-like point algorithm. J. Glob. Optim. 35, 53–69 (2006)

    Article  MATH  Google Scholar 

  8. Ferreira, O.P., Oliveira, P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim. Theory Appl. 97, 93–104 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. da Cruz Neto, J.X., de Lima, L.L., Oliveira, P.R.: Geodesic algorithms in Riemannian geometry. Balk. J. Geom. Appl. 3, 89–100 (1998)

    MATH  Google Scholar 

  10. Kritály, A.: Nash-type equilibria on Riemannian manifolds: a variational approach. J. Math. Pures Appl. 101, 660–688 (2014)

    Article  MathSciNet  Google Scholar 

  11. Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, S.L., Li, C., Liou, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71, 5695–5706 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52, 1491–1498 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim. Theory Appl. 146, 691–708 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Li, C., Wang, J.H.: Newton’s method for sections on Riemannian manifolds: generalized covariant \(\alpha \)-theory. J. Complex. 24, 423–451 (2008)

    Article  MATH  Google Scholar 

  16. Li, C., Mordukhovich, B.S., Wang, J.H., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21(4), 1523–1560 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. 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)

    Article  MATH  MathSciNet  Google Scholar 

  18. Li, C., Yao, J.C.: Variational inequalities for set-valued vector fields on Riemannian manifolds: convexity of the solution set and the proximal point algorithm. SIAM J. Control Optim. 50(4), 2486–2514 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Li, C., López, G., Wang, J.H., Yao, J.C.: Convergence analysis of inexact proximal point algorithms on Hadamard manifolds. J. Glob. Optim. (2014). doi:10.1007/s10898-014-0182-2

  20. Huang, N., Tang, G.: An inexact proximal point algorithm for maximal monotone vector fields on Hadamard manifolds. Oper. Res. Lett. 41, 586–591 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Absil, P.A., Baker, C.G.: Trust-region methods on Riemannian manifolds. Found. Comput. Math. 7, 303–330 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. 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)

    Article  MATH  MathSciNet  Google Scholar 

  23. Lee, P.Y.: Geometric Optimization for Computer Vision. PhD thesis, Australian National University (2005)

  24. Riddell, R.C.: Minimax problems on Grassmann manifolds. Sums of eigenvalues. Adv. Math. 54, 107–199 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hiriart-Urruty, J.B.: From convex optimization to nonconvex optimization: necessary and sufficient conditions for global optimization. Nonsmooth optimization and related topics, pp. 219–239. Springer, US (1989)

  26. Hiriart-Urruty, J.B.: Generalized differentiabity, duality and optimization for problems dealing with diference of convex functions. In Convexity and duality in optimization. pp. 37–70. Springer, Berlin Hiedelberg (1985)

  27. Hiriart-Urruty, J.B., Tuy, H.: Essays on nonconvex optimization, Mathematical programming, vol. 41. North-Holland (1988)

  28. Elhilali Alaoui, A.: Caractrisation des fonctions D.C. (Characterization of D. C. functions). Ann. Sci. Math. Qu 20(1), 1–13 (1996)

    MATH  MathSciNet  Google Scholar 

  29. Toland, J.F.: Duality in nonconvex optimization. J. Math. Anal. Appl. 66, 399–415 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sun, W., Sampaio, R.J.B., Candido, M.A.B.: Proximal point algorithm for minimization of DC functions. J. Comput. Math. 21, 451–462 (2003)

    MATH  MathSciNet  Google Scholar 

  31. Moudafi, A., Maing, P.-E.: On the convergence of an approximate proximal method for DC functions. J. Comput. Math. 24, 475–480 (2006)

    MATH  MathSciNet  Google Scholar 

  32. do Carmo, M.P.: Riemannian Geometry. Birkhauser, Boston (1992)

    Book  MATH  Google Scholar 

  33. Sakai, T.: Riemannian geometry. Translations of mathematical monographs. Am. Math. Soc. Provid. 149 (1996)

  34. Udriste, C.: Convex functions and optimization algorithms on Riemannian manifolds, Mathematics and its applications, vol. 297. Kluwer Academic, Dordrecht (1994)

  35. Polyak, B.T.: Sharp Minima Institute of Control Sciences Lecture Notes, Moscow, USSR, 1979. Presented at the IIASA workshop on generalized Lagrangians and their applications, IIASA, Laxenburg, Austria (1979)

  36. Ferris, M.C.: Weak sharp minima and penalty functions in mathematical programming. Ph.D. Thesis. University of Cambridge, UK (1988)

  37. Bento G.C., Cruz Neto, J.X.: Finite termination of the proximal point method for convex functions on Hadamard manifolds. Optimization, 63, 1281–1288 (2014)

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Acknowledgments

The authors wish to express their gratitude to the anonymous referee for his helpful comments.

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Authors and Affiliations

  1. COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, Rio de Janeiro, RJ, CEP 21945-970, Brazil

    J. C. O. Souza & P. R. Oliveira

  2. CEAD, Universidade Federal do Piauí, Teresina, PI, Brazil

    J. C. O. Souza

Authors
  1. J. C. O. Souza
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  2. P. R. Oliveira
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Correspondence to J. C. O. Souza.

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This research was partially supported by CNPq, Brazil.

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Souza, J.C.O., Oliveira, P.R. A proximal point algorithm for DC fuctions on Hadamard manifolds. J Glob Optim 63, 797–810 (2015). https://doi.org/10.1007/s10898-015-0282-7

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  • DOI: https://doi.org/10.1007/s10898-015-0282-7

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