Vector variational-like inequalities with constraints: separation and alternative. (English) Zbl 1322.49010
Summary: Based on the oriented distance function, a linear weak separation function and three nonlinear regular weak separation functions are introduced in reflexive Banach spaces. Particularly, a nonlinear regular weak separation function does not involve any parameters. Moreover, theorems of the weak alternative for vector variational-like inequalities with constraints are derived by the separation functions without any convexity. Saddle-point conditions, which show the equivalence between the existence of a saddle point and a (linear) nonlinear separation of two suitable subsets of the image space, are established for the linear and nonlinear regular weak separation functions, respectively. Necessary and sufficient optimality conditions for vector variational-like inequalities with constraints are also obtained via the saddle-point conditions. Finally, two gap functions for vector variational-like inequalities with constraints and their continuity are derived by using the image space analysis.
MSC:
| 49J40 | Variational inequalities |
| 49J27 | Existence theories for problems in abstract spaces |
| 49K27 | Optimality conditions for problems in abstract spaces |
| 49J53 | Set-valued and variational analysis |
| 47J20 | Variational and other types of inequalities involving nonlinear operators (general) |
Keywords:
vector variational-like inequalities; constraints; image space analysis; saddle points; weak alternative; optimality conditions; oriented distance functionReferences:
| [1] | Giannessi, F.; Cottle, RW (ed.); Giannessi, F. (ed.); Lions, JL (ed.), Theorems of alternative, quadratic programs and complementarity problems, 151-186 (1980), New York · Zbl 0484.90081 |
| [2] | Chen, G.Y., Yang, X.Q.: The vector complementary problem and its equivalences with vector minimal elements in ordered spaces. J. Math. Anal. Appl. 153, 136-158 (1990) · Zbl 0719.90078 · doi:10.1016/0022-247X(90)90223-3 |
| [3] | Facchinei, F., Pang, J.S.: Finite Dimensional Variational Inequalities and Complementarity Problems, vol. I, II. Springer, New York (2003) · Zbl 1062.90002 |
| [4] | Giannessi, F.: Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0952.00009 · doi:10.1007/978-1-4613-0299-5 |
| [5] | Giannessi, F.; Mastroeni, G.; Pellegrini, L.; Giannessi, F. (ed.), On the theory of vector optimization and variational inequalities. Image space analysis and seperation, 153-215 (2000), Dordrech · Zbl 0985.49005 · doi:10.1007/978-1-4613-0299-5_11 |
| [6] | Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42-58 (1997) · Zbl 0878.49006 · doi:10.1006/jmaa.1997.5192 |
| [7] | Li, J., Huang, N.J.: Image space analysis for vector variational inequalities with matrix inequality constraints and applications. J. Optim. Theory Appl. 145, 459-477 (2010) · Zbl 1213.90243 · doi:10.1007/s10957-010-9691-4 |
| [8] | Li, J., Huang, N.J.: Image space analysis for variational inequalities with cone constraints and applications to traffic equilibria. Sci. China Math. 55, 851-868 (2012) · Zbl 1266.90182 · doi:10.1007/s11425-011-4287-5 |
| [9] | Ansari, Q.H., Rezaei, M., Zafarani, J.: Generalized vector variational-like inequalities and vector optimization. J. Glob. Optim. 53, 271-284 (2012) · Zbl 1274.90421 · doi:10.1007/s10898-011-9686-1 |
| [10] | Fang, Y.P., Huang, N.J.: Variational-like inequalities with generalized monotone mappings in Banach spaces. J. Optim. Theory Appl. 118, 327-338 (2003) · Zbl 1041.49006 · doi:10.1023/A:1025499305742 |
| [11] | Giannessi, F.: Semidifferentiable functions and necessary optimality conditions. J. Optim. Theory Appl. 60, 191-241 (1989) · Zbl 0632.90061 · doi:10.1007/BF00940005 |
| [12] | Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 60, 331-365 (1984) · Zbl 0504.49012 · doi:10.1007/BF00935321 |
| [13] | Giannessi, F.: Constrained Optimization and Image Space Analysis, Seperation of Sets and Optimality Conditions, vol. 1. Springer, Berlin (2005) · Zbl 1082.49001 |
| [14] | Li, S.J., Xu, Y.D., Zhu, S.K.: Nonlinear seperation approach to constrained extremum problems. J. Optim. Theory Appl. 154, 842-856 (2012) · Zbl 1267.90140 · doi:10.1007/s10957-012-0027-4 |
| [15] | Xu, Y.D., Li, S.J.: Nonlinear seperation functions and constrained extremum problems. Optim. Lett. 8, 1149-1160 (2014) · Zbl 1321.90133 · doi:10.1007/s11590-013-0644-3 |
| [16] | Chinaie, M., Zafarani, J.: Image space analysis and scalarization of multivalued optimization. J. Optim. Theory Appl. 142, 451-467 (2009) · Zbl 1188.90231 · doi:10.1007/s10957-009-9531-6 |
| [17] | Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Glob. Optim. 42, 401-412 (2008) · Zbl 1172.90382 · doi:10.1007/s10898-008-9301-2 |
| [18] | Giannessi, F., Mastroeni, G., Yao, J.C.: On maximum and variational principles via image space analysis. Positivity 16, 405-427 (2012) · Zbl 1334.49028 · doi:10.1007/s11117-012-0160-1 |
| [19] | Guu, S.M., Li, J.: Vector quasi-equilibrium problems: separation, saddle points and error bounds for the solution set. J. Glob. Optim. 58, 751-767 (2014) · Zbl 1317.90295 · doi:10.1007/s10898-013-0073-y |
| [20] | Mastroeni, G., Pellegrini, L.: On the image space analysis for vector variational inequalities. J. Ind. Manag. Optim. 1, 123-132 (2005) · Zbl 1106.49017 · doi:10.3934/jimo.2005.1.123 |
| [21] | Mastroeni, G., Panicucci, B., Passacantando, M., Yao, J.C.: A separation approach to vector quasiequilibrium problems: saddle point and gap function. Taiwanese J. Math. 13, 657-673 (2009) · Zbl 1213.90223 |
| [22] | Zhu, S.K., Li, S.J.: Unified duality theory for constrained extremum problems. Part I: image space analysis. J. Optim. Theory Appl. 161, 738-762 (2014) · Zbl 1307.90198 · doi:10.1007/s10957-013-0468-4 |
| [23] | Hiriart-Urruty, J.B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79-97 (1979) · Zbl 0409.90086 · doi:10.1287/moor.4.1.79 |
| [24] | Xu, Y.D., Li, S.J.: Gap functions and error bounds for weak vector variational inequalities. Optim. 63, 1339-1352 (2014) · Zbl 1293.49025 · doi:10.1080/02331934.2012.721115 |
| [25] | Liu, C.G., Ng, K.F., Yang, W.H.: Merit functions in vector optimization. Math. Program. 119, 215-237 (2009) · Zbl 1279.90153 · doi:10.1007/s10107-008-0208-y |
| [26] | Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071-1086 (2003) · Zbl 1046.90084 · doi:10.1137/S0363012902411532 |
| [27] | Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984) · Zbl 0641.47066 |
| [28] | Jeyakumar, V., Oettli, W., Natividad, M.: A solvability theorem for a class of quasiconvex mappings with applications to optimization. J. Math. Anal. Appl. 179, 537-546 (1993) · Zbl 0791.46002 · doi:10.1006/jmaa.1993.1368 |
| [29] | Wu, H.X., Luo, H.Z., Yang, J.F.: Nonlinear seperation approach for the augmented Lagrangian in nonlinear semidefinite programming. J. Glob. Optim. 59, 695-727 (2014) · Zbl 1330.90071 · doi:10.1007/s10898-013-0093-7 |
| [30] | Antoni, C.; Giannessi, F.; Demyanov, VF (ed.); etal., Some remarks on bi-level vector extremum problems, No. 87, 137-157 (2014), New York · Zbl 1282.90161 |
| [31] | Mastroeni, G., Pappalardo, L., Yen, D.N.: Image of a parametric optimization problem and continuity of the perturbation function. J. Optim. Theory Appl. 81, 193-202 (1994) · Zbl 0807.90110 · doi:10.1007/BF02190319 |
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