zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Well-posedness of mixed variational inequalities, inclusion problems and fixed point problems. (English) Zbl 1149.49009
Summary: 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.

MSC:
49J40Variational methods including variational inequalities
49K40Sensitivity, stability, well-posedness of optimal solutions
90C31Sensitivity, stability, parametric optimization
47H10Fixed point theorems for nonlinear operators on topological linear spaces
References:
[1]Bednarczuk E. and Penot J.P. (1992). Metrically well-set minimization problems. Appl. Math. Optim. 26(3): 273–285 · Zbl 0762.90073 · doi:10.1007/BF01371085
[2]Brezis H. (1973). Operateurs maximaux monotone et semigroups de contractions dans les espaces de hilbert. North-Holland, Amsterdam
[3]Brøndsted A. and Rockafellar R.T. (1965). On the subdifferentiability of convex functions. Proc. Am. Math. Soc 16: 605–611
[4]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
[5]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)
[6]Dontchev A.L. and Zolezzi T. (1993). Well-posed optimization problems. Lecture Notes in Math, vol. 1543. Springer, Berlin
[7]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
[8]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)
[9]Glowinski R., Lions J.L. and Tremolieres R. (1981). Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam
[10]Huang X.X. (2001). Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 53: 101–116 · Zbl 1018.49019 · doi:10.1007/s001860000100
[11]Klein E. and Thompson A.C. (1984). Theory of Correspondences. Wiley, New York
[12]Kuratowski K. (1968). Topology, vols. 1 and 2. Academic, New York, NY
[13]Lemaire B. (1998). Well-posedness, conditioning and regularization of minimization, inclusion, and fixed-point problems. Pliska Studia Mathematica Bulgaria 12: 71–84
[14]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 · Zbl 1047.90067 · doi:10.1023/A:1020840322436
[15]Lignola M.B. (2006). Well-posedness and L-well-posedness for quasivariational inequalities. J. Optim. Theory Appl 128(1): 119–138 · Zbl 1093.49005 · doi:10.1007/s10957-005-7561-2
[16]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 · Zbl 0960.90079 · doi:10.1023/A:1008370910807
[17]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)
[18]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 · Zbl 0479.49025 · doi:10.1080/01630568108816100
[19]Lucchetti R. and Patrone F. (1982). Hadamard and Tykhonov well-posedness of certain class of convex functions. J. Math. Anal. Appl. 88: 204–215 · Zbl 0487.49013 · doi:10.1016/0022-247X(82)90187-1
[20]Lucchetti, R., Revalski, J. (eds.): (1995). Recent Developments in Well-Posed Variational Problems. Kluwer Academic Publishers, Dordrecht, Holland
[21]Margiocco M., Patrone F. and Pusillo L. (1997). A new approach to Tikhonov well-posedness for Nash equilibria. Optimization 40(4): 385–400 · Zbl 0881.90136 · doi:10.1080/02331939708844321
[22]Margiocco M., Patrone F. and Pusillo L. (1999). Metric characterizations of Tikhonov well-posedness in value. J. Optim. Theory Appl. 100(2): 377–387 · Zbl 0915.90271 · doi:10.1023/A:1021738420722
[23]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 · Zbl 1011.91004 · doi:10.1023/A:1013658023971
[24]Miglierina E. and Molho E. (2003). Well-posedness and convexity in vector optimization. Math. Methods Oper. Res. 58: 375–385 · Zbl 1083.90036 · doi:10.1007/s001860300310
[25]Morgan J. (2005). Approximations and well-posedness in multicriteria games. Ann. Oper. Res. 137: 257–268 · Zbl 1138.91407 · doi:10.1007/s10479-005-2260-9
[26]Tykhonov A.N. (1966). On the stability of the functional optimization problem. USSR J. Comput. Math. Math. Phys. 6: 631–634
[27]Yang H. and Yu J. (2005). Unified approaches to well-posedness with some applications. J. Glob. Optim. 31: 371–381 · Zbl 1080.49021 · doi:10.1007/s10898-004-4275-1
[28]Yuan G.X.Z. (1999). KKM Theory and Applications to Nonlinear Analysis. Marcel Dekker, New York
[29]Zolezzi T. (1995). Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal. TMA 25: 437–453 · Zbl 0841.49005 · doi:10.1016/0362-546X(94)00142-5
[30]Zolezzi T. (1996). Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91: 257–266 · Zbl 0873.90094 · doi:10.1007/BF02192292