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Relaxed \(\mu\)-quasimonotone variational inequalities in Hadamard manifolds. (English) Zbl 1426.49009

Summary: In this article, we introduce relaxed \(\mu\)-quasimonotone set-valued vector fields on Hadamard manifolds and prove the existence of solutions of the Stampacchia variational inequality for such mappings. We also present the notion of relaxed \(\mu\)-quasiconvexity and show that the Upper Dini and Clarke-Rockafellar subdifferentials of a relaxed \(\mu\)-quasiconvex function is relaxed \(\mu\)-quasimonotone. Under relaxed \(\mu\)-quasiconvexity in the nondifferentiable sense, we establish the connection between the Stampacchia variational inequality problem and a nonsmooth constrained optimization problem.

MSC:

49J40 Variational inequalities
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
47H04 Set-valued operators
49J52 Nonsmooth analysis
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[1] Aussel, D., Corvellec, J.-N., Lassonde, M.: Mean-value property and subdifferential criteria for lower semicontinuous functions. Trans. Am. Math. Soc. 347, 4147-4161 (1995) · Zbl 0849.49016 · doi:10.1090/S0002-9947-1995-1307998-0
[2] Bai, M.R., Hadjisavvas, N.: Relaxed quasimonotone operators and relaxed quasiconvex functions. J. Optim. Theory Appl. 138(3), 329-339 (2008) · Zbl 1162.47043 · doi:10.1007/s10957-008-9382-6
[3] Bai, M.R., Zhou, S.Z., Ni, G.Y.: On the generalized monotonicity of variational inequalities. Comput. Math. Appl. 53(6), 910-917 (2007) · Zbl 1122.49006 · doi:10.1016/j.camwa.2006.09.013
[4] Baiocchi, C., Capelo, A., Jayakar, L.: Variational and Quasivariational Inequalities: Applications to Free Boundary Problems. Wiley, New York (1984) · Zbl 0551.49007
[5] Barbagallo, A., Daniele, P., Maugeri, A.: Variational formulation for a general dynamic financial equilibrium problem: balance law and liability formula. Nonlinear Anal. 75(3), 1104-1123 (2012) · Zbl 1236.49015 · doi:10.1016/j.na.2010.10.013
[6] Baygorrea, N., Papa Quiroz, E.A., Maculan, N.: Inexact proximal point methods for quasiconvex minimization on Hadamard manifolds. J. Oper. Res. Soc. China 4(4), 397-424 (2016) · Zbl 1365.90211 · doi:10.1007/s40305-016-0133-3
[7] Baygorrea, N., Papa Quiroz, E.A., Maculan, N.: On the convergence rate of an inexact proximal point algorithm for quasiconvex minimization on Hadamard manifolds. J. Oper. Res. Soc. China 5(4), 457-467 (2017) · Zbl 1386.90111 · doi:10.1007/s40305-016-0129-z
[8] Browder, F.E.: Multivalued monotone nonlinear mappings and duality mappings in Banach spaces. Trans. Am. Math. Soc. 71, 780-785 (1965) · Zbl 0138.39902
[9] Chen, S.L., Fang, C.J.: Vector variational inequality with pseudoconvexity on Hadamard manifolds. Optimization 65, 2067-2080 (2016) · Zbl 1377.90084 · doi:10.1080/02331934.2016.1235161
[10] Chen, Guang-Ya; Cheng, Ging-Min, Vector Variational Inequality and Vector Optimization Problem, 408-416 (1987), Berlin, Heidelberg
[11] Colao, V., López, G., Marino, G., Martńn-Márquez, V.: Equilibrium problems in Hadamard manifolds. J. Math. Anal. Appl. 388, 61-77 (2012) · Zbl 1273.49015 · doi:10.1016/j.jmaa.2011.11.001
[12] Crouzeix, J.P.: Pseudomonotone variational inequality problems: existence of solutions. Math. Program. 78, 305-314 (1997) · Zbl 0887.90167
[13] Danildis, A., Hadjisavvas, N.: On the subdifferentials of pseudoconvex and quasiconvex functions and cyclic monotonicity. J. Math. Anal. Appl. 237(1), 30-42 (1999) · Zbl 0934.49015 · doi:10.1006/jmaa.1999.6437
[14] Debrunner, H., Flor, P.: Ein Erweiterungssatz für monotone Mengen. Arch. Math. 15, 445-447 (1964) · Zbl 0129.09203 · doi:10.1007/BF01589229
[15] do Carmo, M.P.: Riemannian Geometry. Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory Applications. Birkhäuser Boston, Inc., Boston (1992) · Zbl 0752.53001 · doi:10.1007/978-1-4757-2201-7
[16] Ferreira, O.P., Pérez, L.R., Németh, S.Z.: Singularities of monotone vector fields and an extragradienttype algorithm. J. Glob. Optim. 31, 133-151 (2005) · Zbl 1229.58007 · doi:10.1007/s10898-003-3780-y
[17] Fichera, G.: Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I 7(8), 91-140 (1963/1964) · Zbl 0146.21204
[18] Giannessi, F.: Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0952.00009 · doi:10.1007/978-1-4613-0299-5
[19] Giannessi, F.; Cottle, RW (ed.); Giannessi, F. (ed.); Lions, JC (ed.), Theorems of alternative, quadratic programs and complementary problems (1980), New York · Zbl 0484.90081
[20] Hadjisavvas, N., Schaible, S.: Quasimonotone variational inequalities in Banach spaces. J. Optim. Theory Appl. 90, 95-111 (1996) · Zbl 0904.49005 · doi:10.1007/BF02192248
[21] Hosseini, S.: Characterization of lower semicontinuous convex functions on Riemannian manifolds. Set-Valued Var. Anal. 25(2), 297-311 (2017) · Zbl 1371.58006 · doi:10.1007/s11228-016-0380-9
[22] Jost, J.: Nonpositive curvature: geometric and analytic aspects. In: Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1997) · Zbl 0896.53002
[23] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1984) · Zbl 0457.35001
[24] Konnov, I.V., Lee, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206(1), 42-58 (1998) · Zbl 0878.49006 · doi:10.1006/jmaa.1997.5192
[25] Ledyaev, Y.S., Zhu, Q.J.: Nonsmooth analysis on smooth manifolds. Trans. Am. Math. Soc. 359, 3687-3732 (2007) · Zbl 1157.49021 · doi:10.1090/S0002-9947-07-04075-5
[26] Lee, Gue Myung, On Relations between Vector Variational Inequality and Vector Optimization Problem, 167-179 (2000), Boston, MA · Zbl 0969.49003
[27] Lee, G.M., Kim, D.S., Kuk, S.: Existence of solutions for vector optimization problems. J. Math. Anal. Appl. 220, 90-98 (1998) · Zbl 0911.90290 · doi:10.1006/jmaa.1997.5821
[28] Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 34, 745-765 (1998) · Zbl 0956.49007 · doi:10.1016/S0362-546X(97)00578-6
[29] 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) · Zbl 1180.58012 · doi:10.1016/j.na.2009.04.048
[30] Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79(2), 663-683 (2009) · Zbl 1171.58001 · doi:10.1112/jlms/jdn087
[31] Li, C., Mordukhovich, B., Wang, J., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21(4), 1523-1560 (2011) · Zbl 1236.49089 · doi:10.1137/09075367X
[32] 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) · Zbl 1257.49011 · doi:10.1137/110834962
[33] Li, X.B., Zhou, L.W., Huang, N.J.: Gap functions and global error bounds for generalized mixed variational inequalities on Hadamard manifolds. J. Optim. Theory Appl. 168(3), 830-849 (2016) · Zbl 1341.58013 · doi:10.1007/s10957-015-0834-5
[34] Lions, J.-L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493-519 (1967) · Zbl 0152.34601 · doi:10.1002/cpa.3160200302
[35] Luc, D.T.: Existence results for densely psudomonotone variational inequalities. J. Math. Anal. Appl. 254, 291-308 (2001) · Zbl 0974.49006 · doi:10.1006/jmaa.2000.7278
[36] Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. 52(5), 1491-1498 (2003) · Zbl 1016.49012 · doi:10.1016/S0362-546X(02)00266-3
[37] Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with bregman distances on Hadamard manifolds. J. Convex Anal. 16(1), 49-69 (2009) · Zbl 1176.90361
[38] Papa Quiroz, E.A., Oliveira, P.R.: Full convergence of the proximal point methods for quasiconvex functions on Hadamard manifolds. ESAIM Control Optim. Calc. Var. 18(2), 483-500 (2012) · Zbl 1273.90162 · doi:10.1051/cocv/2011102
[39] Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for minimizing quasiconvex locally lipschitz functions on Hadamard manifolds. Nonlinear Anal. 75(15), 5924-5932 (2012) · Zbl 1244.49075 · doi:10.1016/j.na.2012.06.005
[40] Udriste, C.: Convex functions and optimization methods on Riemannian manifolds. In: Mathematics and its Applications, vol. 297. Kluwer Academic Publishers, Dordrecht (1994) · Zbl 0932.53003
[41] Smith, S.T.: Optimization Techniques on Riemannian Manifolds, Hamiltonian and Gradient Flows, Algorithms and Control. Fields Inst. Commun. 3, American Mathematical Society, Providence (1994)
[42] Zhou, L.W., Xiao, Y.B., Huang, N.J.: New characterization of geodesic convexity on Hadamard manifolds with applications. J. Optim. Theory Appl. 172(3), 824-844 (2017) · Zbl 1362.49007 · doi:10.1007/s10957-016-1012-0
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