Abstract
We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some of which are new even in the single objective setting.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Amer. Math. Soc. 357, (2004)
Aze, D., Corvellec, J.N.: On the sensitivity analysis of Hoffman constants for systems of linear inequalities. SIAM J. Optim. 12, 913–927 (2002)
Burke, J.V., Deng, S.: Weak sharp minima revisited Part I: basic theory. Control Cybernet. 31, 439–469 (2002)
Burke, J.V., Ferris, M.C.: A Gauss–Newton method for convex composite optimization. Math. Programming 71, 179–194 (1995)
Burke, J.V., Ferris, M.C., Qian, M.: On the Clarke subdifferential of the distance function of a closed set. J. Math. Anal. Appl. 166, 199–213 (1992)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Clarke, F.H., Stern, R., Wolenski, P.: Proximal smoothness and the lower-C2 property. J. Convex Anal. 2, 117–144 (1995)
Cromme, L.: Strong uniqueness, a far-reaching criterion for the convergence analysis of iterative procedures. Numer. Math. 29, 179–193 (1978)
Jeyakumar, V.: Composite nonsmooth programming with Gateaux differentiability. SIAM J. Optim. 1, 30–41 (1991)
Jeyakumar, V., Luc, D.T., Tinh, P.N.: Convex composite non-Lipschitz programming. Math. Programming 92, 177–195 (2002)
Jeyarkumar, V., Yang, X.Q.: Convex composite multiobjective nonsmooth programming. Math. Programming 59, (1993)
Jimenez, B.: Strict efficiency in vector optimization. J. Math. Anal. Appl. 265, 264–284 (2002)
Jimenez, B.: Strict minimality conditions in nondifferentiable multiobjective programming. J. Optim. Theory Appl. 116, 99–116 (2003)
Jittorntrum, K., Osborne, M.R.: Strong uniqueness and second order convergence in nonlinear discrete approximation. Numer. Math. 34, 439–455 (1980)
Lewis, A., Pang, J.S.: Error bounds for convex inequality systems. In: Crouzeix, J.-P., Martinez-Legaz, J.-E., Volle, M. (eds.) Generalized Convexity, Generalized Monotonicity: Recent Results, Proceedings of the Fifth Symposium on Generalized Convexity, Luminy, 1996, p. 75–110. Kluwer Academic Publishers, Dordrecht, The Netherlands (1997)
Li, W.: Sharp Lipschitz constants for basic optimal solutions and basic feasible solutions of linear programs. SIAM J. Control Optim. 32, 140–153 (1994)
Li, C., Wang, X.: On convergence of the Gauss–Newton method for convex composite optimization. Math. Programming 91, 349–356 (2002)
Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I/II. Springer, Berlin Heidelberg New York (2006)
Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc. 348, 1235–1280 (1996)
Osborne, M.R., Womersley, R.S.: Strong uniqueness in sequential linear programming. J. Austral. Math. Soc. Ser. B 31, 379–384 (1990)
Pang, J.S.: Error bounds in mathematical programming. Math. Programming Ser. B 79, 299–332 (1997)
Penot, J.P.: Optimality conditions in mathematical programming and composite optimization. Math. Programming 67, 225–245 (1994)
Poliquin, R., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Amer. Math. Soc. 352, 5231–5249 (2000)
Polyak, B.T.: Introduction to Optimization. Optimization Software, New York (1987)
Rockafellar, R.T.: Directional Lipschitzian functions and subdifferential calculus. Proc. London Math. Soc. 39, 331–355 (1979)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin Heidelberg New York (1998)
Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)
Shapiro, A.: On a class of nonsmooth composite functions. Math. Oper. Res. 28, 677–692 (2003)
Studniarski, M., Ward, D.E.: Weak sharp minima: Characterizations and sufficient conditions. SIAM J. Control Optim. 38, 219–236 (1999)
Womersley, R.S.: Local properties of algorithms for minimizing nonsmooth composite functions. Math. Programming 32, 69–89 (1985)
Zalinescu, C.: Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces. Proc. 12th Baikal Internat. Conf. on Optimization Methods and Their Appl. Irkutsk, Russia, pp. 272–284 (2001)
Zalinescu, C.: Sharp estimates for Hoffman's constant for systems of linear inequalities and equalities. SIAM J. Optim. 14, 517–533 (2003)
Zheng, X.Y., Ng, K.F.: Hoffmans least error bounds for systems of linear inequalities. J. Global Optim. 30, 391–403 (2004)
Zheng, X.Y., Ng, K.F.: Error bound moduli for conic convex systems on Banach spaces. Math. Oper. Res. 29, 213–228 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by a Central Research Grant of The Hong Kong Polytechnic University (Grant No. G-T 507). Research of the first author was also supported by the National Natural Science Foundation of PR China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, China (Grant No. 2003A002M).
Rights and permissions
About this article
Cite this article
Zheng, X.Y., Yang, X.M. & Teo, K.L. Sharp Minima for Multiobjective Optimization in Banach Spaces. Set-Valued Anal 14, 327–345 (2006). https://doi.org/10.1007/s11228-006-0023-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-006-0023-7