Abstract
A proximal point method for nonsmooth multiobjective optimization in the Riemannian context is proposed, and an optimality condition for multiobjective problems is introduced. This allowed replacing the classic approach, via “scalarization,” by a purely vectorial and considering the method without any assumption of convexity over the constraint sets that determine the vectorial improvement steps. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Moreover, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto optimal is obtained.
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References
Jahn, J.: Vector Optimization-Theory, Applications, and Extensions. Springer, Berlin (2004)
Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)
Bento, G.C., Ferreira, O.P., Pereira, Y.R.: Proximal point method for vector optimization on Hadamard manifolds. Oper. Res. Lett. 46(1), 13–18 (2018)
Bento, G.C., Cruz Neto, J.X., Soubeyran, A.: A proximal point-type method for multicriteria optimization. Set Valued Var. Anal. 22(3), 557–573 (2014)
Rockafellar, R.T.: Favorable classes of Lipschitz continuous functions in subgradient optimization. In: Nurminski, E. (ed.) Nondifferentiable Optimization. Pergamon Press, New York (1982)
Spingarn, J.E.: Submonotone mappings and the proximal point algorithm. Numer. Funct. Anal. Optim. 4, 123–150 (1981–1982)
Kaplan, A., Tichatschke, R.: Proximal point methods and nonconvex optimization. J. Glob. Optim. 13, 389–406 (1998)
Pennanen, T.: Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27, 193–202 (2002)
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)
Bento, G.C., Ferreira, O.P., Oliveira, P.R.: Proximal point method for a special class of nonconvex functions on Hadamard manifolds. Optimization 64(2), 289–319 (2015)
Grohs, P., Hosseini, S.: \(\varepsilon \)-subgradient algorithms for locally Lipschitz functions on Riemannian manifolds. Adv. Comput. Math. 42(2), 333–360 (2016)
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)
Minami, M.: Weak Pareto-optimal necessary conditions in a nondifferentiable multiobjective program on a Banach space. J. Optim. Theory Appl. 41(3), 451–461 (1983)
Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359(8), 3687–3732 (2007)
Li, C., Mordukhovich, B.S., Wang, J., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21(4), 1523–1560 (2011)
Vinter, R.B.: Optimal Control. Birkhauser, Basel (2000)
Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc. (N.S.) 1(3), 443–474 (1979)
Bento, G.C., Cruz Neto, J.X.: A subgradient method for multiobjective optimization on Riemannian manifolds. J. Optim. Theory Appl. 159(1), 125–137 (2013)
Huang, X.X., Yang, X.Q.: Duality for multiobjective optimization via nonlinear Lagrangian functions. J. Optim. Theory Appl. 120(1), 111–127 (2004)
Luc, D.T.: Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319. Springer, Berlin (1989)
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)
Ceng, L.C., Yao, J.C.: Approximate proximal methods in vector optimization. Eur. J. Oper. Res. 183(1), 1–19 (2007)
Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51(2), 257–270 (2002)
Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and Its Applications, vol. 297. Kluwer Academic Publishers Group, Dordrecht (1994)
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Communicated by Sándor Zoltán Németh.
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Bento, G.d.C., da Cruz Neto, J.X. & de Meireles, L.V. Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization of Hadamard Manifolds. J Optim Theory Appl 179, 37–52 (2018). https://doi.org/10.1007/s10957-018-1330-5
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DOI: https://doi.org/10.1007/s10957-018-1330-5