## Levitin-Polyak well-posedness of variational inequality problems with functional constraints.(English)Zbl 1191.90083

Summary: We introduce several types of (generalized) Levitin-Polyak well-posednesses for a variational inequality problem with abstract and functional constraints. Criteria and characterizations for these types of well-posednesses are given. Relations among these types of well-posednesses are also investigated.

### MSC:

 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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### References:

 [1] Auslender A.: Optimization: Methodes Numeriques. Masson, Paris (1976) [2] Auslender A.: Asymptotic analysis for penalty and barrier methods in variational inequalities. SIAM J. Control Optim. 37, 653–671 (1999) · Zbl 0926.90075 [3] Auslender A.: Variational inequalities over the cone of semidefinite positive symmetric matrices and over the Lorentz cone, the Second Japanese-Sino Optimization Meeting, Part II, Kyoto, 2002. Optim. Methods Softw. 18, 359–376 (2003) · Zbl 1061.90500 [4] Bednarczuk E.: Well-posedness of Vector Optimization Problems. Lecture Notes in Economics and Mathematical Systems, vol. 294, pp. 51–61. Berlin, Springer-Verlag (1987) · Zbl 0643.90083 [5] Bednarczuk E., Penot P.J.: Metrically well-set minimization problems. Appl. Math. Optim. 26, 273–285 (1992) · Zbl 0762.90073 [6] Beer G., Lucchetti R.: The epi-distance topology: continuity and stability results with application to convex optimization problems. Math. Oper. Res. 17, 715–726 (1992) · Zbl 0767.49011 [7] Deng S.: Coercivity properties and well-posedness in vector optimization. RAIRO Oper. Res. 37, 195–208 (2003) · Zbl 1070.90095 [8] Dontchev A.L., Zolezzi T.: Well-posed Optimization Problems. Lecture Notes in Mathematics, vol.1543. Springer, Berlin (1993) · Zbl 0797.49001 [9] Fukushima M.: Equivalent differentiable optimization problem and descent method for symmteric variational inequalities. Math. Program. 53, 99–110 (1992) · Zbl 0756.90081 [10] Furi M., Vignoli A.: About well-posed minimization problems for functionals in metric spaces. J. Optim. Theory Appl. 5, 225–229 (1970) · Zbl 0188.48802 [11] Harker P.T., Pang J.S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. 48, 161–220 (1990) · Zbl 0734.90098 [12] He B.S., Yang H., Zhang C.S.: A modified augmented Lagrangian method for a class of monotone variational inequalities. Eur. J. Oper. Res. 159, 35–51 (2004) · Zbl 1067.90152 [13] Huang X.X.: Extended well-posed properties of vector optimization problems. J. Optim. Theory Appl. 106, 165–182 (2000) · Zbl 1028.90067 [14] Huang X.X.: Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 53, 101–116 (2001) · Zbl 1018.49019 [15] Huang X.X., Yang X.Q.: Generalized Levitin–Polyak well-posedness in constrained optimization. SIAM J. Optim. 17, 243–258 (2006) · Zbl 1137.49024 [16] Huang, X.X., Yang, X.Q.: Levitin–Polyak well-posedness of constrained vector optimization problems. J. Glob. Optim. Published Online, 06 July (2006) · Zbl 1149.90133 [17] Konsulova A.S., Revalski J.P.: Constrained convex optimization problems-well-posedness and stability. Num. Funct. Anal. Optim. 15, 889–907 (1994) · Zbl 0830.90119 [18] Levitin E.S., Polyak B.T.: Convergence of minimizing sequences in conditional extremum problems. Soviet Math. Dokl. 7, 764–767 (1966) · Zbl 0161.07002 [19] Lignola M.B., Morgan J.: Approxiamte solutions and {$$\alpha$$}-well-posedness for variational inequalities and Nash equilibria. In: Zaccour, G. (eds) Decision and control in management science, pp. 369–378. Kluwer Academic Publishers, Dordrecht (2001) [20] Lignola, M.B., Morgan, J.: {$$\alpha$$}-well-posedness for Nash equilibria and for optimization with Nash equilibrium constraints. J. Glob. Optim. Published Online, 27 June (2006) · Zbl 1105.49029 [21] Loridan P.: Well-posed vector optimization, recent developments in well-posed variational problems, mathematics and its applications, vol. 331. Kluwer Academic Publishers, Dordrecht (1995) · Zbl 0848.49017 [22] Lucchetti, R.: Well-posedness towards vector optimization. Lecture notes in economics and mathematical systems, vol. 294. Springer-Verlag, Berlin (1987) [23] Lucchetti, R., Revalski, J. (eds): Recent developments in well-posed variational problems. Kluwer Academic Publishers, Dordrecht (1995) · Zbl 0823.00006 [24] Margiocco M., Patrone F., Chicco P.: A new approach to Tikhonov well-posedness for Nash equilibria. Optimization 40, 385–400 (1997) · Zbl 0881.90136 [25] Nguyen S., Dupuis C.: An efficient method for computing traffic equilibria in networks with asymmetric transportation costs. Transport. Sci. 18, 185–202 (1984) [26] Patrone, F.: Well-posedness for Nash equilibria and related topics, recent developments in well-posed variational problems, Math. Appl., vol. 331, pp. 211–227. Kluwer Academic Publishers, Dordrecht (1995) · Zbl 0849.90131 [27] Revalski J.P.: Hadamard and strong well-posedness for convex programs. SIAM J. Optim. 7, 519–526 (1997) · Zbl 0873.49011 [28] Tykhonov A.N.: On the stability of the functional optimization problem. USSR Comput. Math. Math. Phys. 6, 28–33 (1966) · Zbl 0212.23803 [29] Yamashita N., Fukushima M.: On the level-boundedness of the natural residual function for variational inequality problems. Pac. J. Optim. 1, 625–630 (2005) · Zbl 1126.47051 [30] Yang H., Yu J.: Unified approaches to well-posedness with some applications. J. Glob. Optim. 31, 371–381 (2005) · Zbl 1080.49021 [31] Zhu D.L.: Augmented Lagrangian theory, duality and decomposition methods for variational inequality problems. J. Optim. Theory Appl. 117, 195–216 (2003) · Zbl 1046.49008 [32] Zolezzi T.: Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. Theory Methods Appl. 25, 437–453 (1995) · Zbl 0841.49005 [33] Zolezzi T.: Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91, 257–266 (1996) · Zbl 0873.90094 [34] Zolezzi T.: Well-posedness and optimization under perturbations. Ann. Oper. Res. 101, 351–361 (2001) · Zbl 0996.90081
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