Abstract
We generalize the concept of well-posedness to a mixed variational inequality and give some characterizations of its well-posedness. Under suitable conditions, we prove that the well-posedness of a mixed variational inequality is equivalent to the well-posedness of a corresponding inclusion problem. We also discuss the relations between the well- posedness of a mixed variational inequality and the well-posedness of a fixed point problem. Finally, we derive some conditions under which a mixed variational inequality is well-posed.
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References
Bednarczuk E. and Penot J.P. (1992). Metrically well-set minimization problems. Appl. Math. Optim. 26(3): 273–285
Brezis H. (1973). Operateurs maximaux monotone et semigroups de contractions dans les espaces de hilbert. North-Holland, Amsterdam
Brøndsted A. and Rockafellar R.T. (1965). On the subdifferentiability of convex functions. Proc. Am. Math. Soc 16: 605–611
Cavazzuti E. and Morgan J. (1983). Well-posed saddle point problems. In: Hirriart-Urruty, J.B., Oettli, W. and Stoer, J. (eds) Optimization, Theory and Algorithms, pp 61–76. Marcel Dekker, New York, NY
Del Prete, I., Lignola, M.B., Morgan, J.: New concepts of well-posedness for optimization problems with variational inequality constraints. JIPAM. J. Inequal. Pure Appl. Math. 4(1), Article 5 (2003)
Dontchev A.L. and Zolezzi T. (1993). Well-posed optimization problems. Lecture Notes in Math, vol. 1543. Springer, Berlin
Fang Y.P. and Deng C.X. (2004). Stability of new implicit iteration procedures for a class of nonlinear set-valued mixed variational inequalities. Z. Angew. Math. Mech. 84(1): 53–59
Fang, Y.P., Hu, R.: Parametric well-posedness for variational inequalities defined by bifunctions. Comput. Math. Appl., doi:10.1016/j.camwa.2006.09.009 (2007)
Glowinski R., Lions J.L. and Tremolieres R. (1981). Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam
Huang X.X. (2001). Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 53: 101–116
Klein E. and Thompson A.C. (1984). Theory of Correspondences. Wiley, New York
Kuratowski K. (1968). Topology, vols. 1 and 2. Academic, New York, NY
Lemaire B. (1998). Well-posedness, conditioning and regularization of minimization, inclusion, and fixed-point problems. Pliska Studia Mathematica Bulgaria 12: 71–84
Lemaire B., Ould Ahmed Salem C. and Revalski J.P. (2002). Well-posedness by perturbations of variational problems. J. Optim. Theory Appl. 115(2): 345–368
Lignola M.B. (2006). Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl 128(1): 119–138
Lignola M.B. and Morgan J. (2000). Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J. Glob. Optim. 16(1): 57–67
Lignola, M.B., Morgan, J.: Approximating solutions and α-well-posedness for variational inequalities and Nash equilibria. In: Decision and Control in Management Science, pp. 367–378. Kluwer Academic Publishers, Dordrecht (2002)
Lucchetti R. and Patrone F. (1981). A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities. Numer. Funct. Anal. Optim. 3(4): 461–476
Lucchetti R. and Patrone F. (1982). Hadamard and Tykhonov well-posedness of certain class of convex functions. J. Math. Anal. Appl. 88: 204–215
Lucchetti, R., Revalski, J. (eds.): (1995). Recent Developments in Well-Posed Variational Problems. Kluwer Academic Publishers, Dordrecht, Holland
Margiocco M., Patrone F. and Pusillo L. (1997). A new approach to Tikhonov well-posedness for Nash equilibria. Optimization 40(4): 385–400
Margiocco M., Patrone F. and Pusillo L. (1999). Metric characterizations of Tikhonov well-posedness in value. J. Optim. Theory Appl. 100(2): 377–387
Margiocco M., Patrone F. and Pusillo L. (2002). On the Tikhonov well-posedness of concave games and Cournot oligopoly games. J. Optim. Theory Appl. 112(2): 361–379
Miglierina E. and Molho E. (2003). Well-posedness and convexity in vector optimization. Math. Methods Oper. Res. 58: 375–385
Morgan J. (2005). Approximations and well-posedness in multicriteria games. Ann. Oper. Res. 137: 257–268
Tykhonov A.N. (1966). On the stability of the functional optimization problem. USSR J. Comput. Math. Math. Phys. 6: 631–634
Yang H. and Yu J. (2005). Unified approaches to well-posedness with some applications. J. Glob. Optim. 31: 371–381
Yuan G.X.Z. (1999). KKM Theory and Applications to Nonlinear Analysis. Marcel Dekker, New York
Zolezzi T. (1995). Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. TMA 25: 437–453
Zolezzi T. (1996). Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91: 257–266
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This work was supported by the National Natural Science Foundation of China (10671135) and Specialized Research Fund for the Doctoral Program of Higher Education (20060610005). The research of the third author was partially support by NSC 95-2221-E-110-078.
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Fang, YP., Huang, NJ. & Yao, JC. Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems. J Glob Optim 41, 117–133 (2008). https://doi.org/10.1007/s10898-007-9169-6
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DOI: https://doi.org/10.1007/s10898-007-9169-6