Variational Conditions Under the Constant Rank Constraint Qualification

Published Online:https://doi.org/10.1287/moor.1090.0431

References

  • Bonnans J. F., Shapiro A. Perturbation analysis of optimization problems. Springer Series in Operations Research (2000) (Springer-Verlag, New York) Google Scholar
  • Dempe S. Directional differentiability of optimal solutions under Slater's condition. Math. Programming (1993) 59:49–69CrossrefGoogle Scholar
  • Dontchev A. L., Rockafellar R. T. Implicit functions and solution mappings: A view from variational analysis. Springer Monographs in Mathematics (2009) (Springer, Berlin/Heidelberg, Germany) Google Scholar
  • Facchinei F., Pang J.-S. Finite-dimensional variational inequalities and complementarity problems. Springer Series in Operations Research (published in two volumes, paginated continuously) (2003) (Springer-Verlag, New York) Google Scholar
  • Fonseca I., Gangbo W. Degree theory in analysis and applications. Oxford Lecture Series in Mathematics and Its Applications (1995) 2(Oxford University Press, Oxford) Google Scholar
  • Guddat J., Vasquez F. G., Jongen H. Th.Parametric Optimization: Singularities, Pathfollowing and Jumps (1990) (B. G. Teubner and John Wiley & Sons, Stuttgart, Germany, and Chichester, UK) CrossrefGoogle Scholar
  • Haraux A. How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Society Japan (1977) 29:615–631CrossrefGoogle Scholar
  • Janin R. Directional derivative of the marginal function in nonlinear programming. Math. Programming Stud. (1984) 21:110–126CrossrefGoogle Scholar
  • Klatte D., Kummer B.Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications (2002) (Kluwer Academic Publishers, Dordrecht, The Netherlands) Google Scholar
  • Kojima M., Stephen M. R. Strongly stable stationary solutions in nonlinear programs. Analysis and Computation of Fixed Points. (1980) (Academic Press, New York) 93–138Google Scholar
  • Kyparisis J. Sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers. Math. Oper. Res. (1990) 15:286–298LinkGoogle Scholar
  • Liu J. Sensitivity analysis in nonlinear programs and variational inequalities via continuous selections. SIAM J. Control Optim. (1995) 33:1040–1060CrossrefGoogle Scholar
  • LLoyd N. G.Degree Theory (1978) (Cambridge University Press, Cambridge, UK) Google Scholar
  • Lu S. Implications of the constant rank constraint qualification. Math. Programming (2009) . Published online at DOI 10.1007/s10107-009-0288-3Google Scholar
  • Lu S., Robinson S. M. Normal fans of polyhedral convex sets: Structures and connections. Set-Valued Anal. (2008a) 16:281–305CrossrefGoogle Scholar
  • Lu S., Robinson S. M. Variational inequalities over perturbed polyhedral convex sets. Math. Oper. Res. (2008b) 33:689–711LinkGoogle Scholar
  • Luo Z.-Q., Pang J.-S., Ralph D.Mathematical Programs with Equilibrium Constraints (1996) (Cambridge University Press, Cambridge, UK) CrossrefGoogle Scholar
  • Ortega J. M., Rheinboldt W. C.Iterative Solution of Nonlinear Equations in Several Variables (1970) (Academic Press, New York) Google Scholar
  • Pang J.-S. A degree-theoretic approach to parametric nonsmooth equations with multivalued solution sets. Math. Programming (1993) 62:359–383CrossrefGoogle Scholar
  • Pang J.-S., Ralph D. Piecewise smoothness, local invertibility, and parametric analysis of normal maps. Math. Oper. Res. (1996) 21:401–426LinkGoogle Scholar
  • Ralph D. On branching numbers of normal manifolds. Nonlinear Anal.: Theory, Methods, Appl. (1994) 22:1041–1050CrossrefGoogle Scholar
  • Ralph D., Dempe S. Directional derivatives of the solution of a parametric nonlinear program. Math. Programming (1995) 70:159–172CrossrefGoogle Scholar
  • Robinson S. M. Stability theory for systems of inequalities. Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. (1976) 13:497–513CrossrefGoogle Scholar
  • Robinson S. M. Generalized equations and their solutions. Part II: Applications to nonlinear programming. Math. Programming Stud. (1982) 19:200–221CrossrefGoogle Scholar
  • Robinson S. M. Local structure of feasible sets in nonlinear programming. Part II: Nondegeneracy. Math. Programming Stud. (1984) 22:217–230CrossrefGoogle Scholar
  • Robinson S. M. Local structure of feasible sets in nonlinear programming. Part III: Stability and sensitivity. Math. Programming Stud. (1987) 30:45–66CrossrefGoogle Scholar
  • Robinson S. M. Normal maps induced by linear transformations. Math. Oper. Res. (1992) 17:691–714LinkGoogle Scholar
  • Robinson S. M. Constraint nondegeneracy in variational analysis. Math. Oper. Res. (2003a) 28:201–232LinkGoogle Scholar
  • Robinson S. M. Variational conditions with smooth constraints: Structure and analysis. Math. Programming Ser. B (2003b) 97:245–265CrossrefGoogle Scholar
  • Robinson S. M. Localized normal maps and the stability of variational conditions. Set-Valued Anal. (2004) 12:259–274See Erratum, Set-Valued Anal. 14 207CrossrefGoogle Scholar
  • Rockafellar R. T.Convex Analysis (1970) (Princeton University Press, Princeton, NJ) CrossrefGoogle Scholar
  • Rockafellar R. T., Wets R. J.-B.Variational Analysis (1998) (Springer-Verlag, Berlin) CrossrefGoogle Scholar
  • Scholtes S. Introduction to piecewise differentiable equations. (1994) (Habilitationsschrift, Institut für Statistik und mathematische Wirtschaftstheorie, Universität Fridericiana Karlsruhe, Karlsruhe, Germany) Google Scholar
  • Scholtes S. A proof of the branching number bound for normal manifolds. Linear Algebra Appl. (1996) 246:83–95CrossrefGoogle Scholar
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