Abstract
The concept of metric regularity and its role in deriving the optimality conditions for optimization problems is not new. This chapter presents the notion of metric regularity and explores the relationship between a modified version of the well-known basic constraint qualification with that of metric regularity.We also study its application in obtaining the Karush—Kuhn—Tucker optimality conditions for nonsmooth optimization problems with set inclusion and abstract constraints by converting the constrained problem into an unconstrained problem.
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References
Borwein, J.M.: Stability and regularity points of inequality systems. J. Optim. Theory Appl. 48, 9–52 (1986).
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983).
Clake, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth analysis and control theory. In: Axler, S., Gehring, F.W., Ribet, K.A. (eds.) Graduate Texts in Mathematics, 178. Springer, New York (1998).
Cominetti, R.: Metric regularity, tangent sets and second-order optimality conditions. Appl. Math. Optim. 21, 265–287 (1990).
Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms (I). Springer-Verlag, New York (1993).
Jourani, A.: Metric regularity and second-order necessary optimality conditions for minimization problems under inclusion constraints. J. Optim. Theory Appl. 81, 97–120 (1994).
Jourani, A.: Constraint qualifications and Lagrange multipliers in nondifferentiable programming problems. J. Optim. Theory Appl. 81, 533–548 (1994).
Jourani, A., Thibault, L.: Approximate subdifferential and metric regularity: the finite dimensional case. Math. Program. 47, 203–218 (1990).
Jourani, A., Thibault, L.: Metric regularity and subdifferential calculus in Banach spaces. Set-Valued Anal. 3, 87–100 (1995).
Jourani, A., Thibault, L.: Metric regularity for strongly compactly Lipschitzian mappings. Nonlinear Anal. 24, 229–240 (1995).
Li, W.: Abadie’s constraint qualification, metric regularity, and error bounds for differentiable convex inequalities. SIAM J. Optim. 7, 966–978 (1997).
Mordukhovich, B.S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340, 1–35 (1993).
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Springer-Verlag, Berlin (2006).
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton, NJ (1970).
Rockafellar, R.T., Wets, R.J-B.: Variational Analysis. Springer-Verlag, Berlin (1998).
Zheng, X.Y., Ng, K.F.: Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14, 757–772 (2004).
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Dhara, A., Mehra, A. (2011). Metric Regularity and Optimality Conditions in Nonsmooth Optimization. In: Mishra, S. (eds) Topics in Nonconvex Optimization. Springer Optimization and Its Applications(), vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9640-4_7
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DOI: https://doi.org/10.1007/978-1-4419-9640-4_7
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