Abstract
We investigate stability (in terms of metric regularity) for the specific class of cone increasing constraint mappings. This class is of interest in problems with additional knowledge on some nondecreasing behavior of the constraints (e.g. in chance constraints, where the occurring distribution function of some probability measure is automatically nondecreasing). It is demonstrated, how this extra information may lead to sharper characterizations. In the first part, general cone increasing constraint mappings are studied by exploiting criteria for metric regularity, as recently developed by Mordukhovich. The second part focusses on genericity investigations for global metric regularity (i.e. metric regularity at all feasible points) of nondecreasing constraints in finite dimensions. Applications to chance constraints are given.
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References
Aubin, J. P.: Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87–111.
Auslender, A.: Stability in mathematical programming with nondifferentiable data, SIAM J. Control Optim. 22 (1984), 29–41.
Borwein, J. M.: Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), 9–52.
Borwein, J. M. and Zhuang, D. M.: Verifiable necessary and sufficient conditions for regularity of set-valued and single-valued maps, J. Math. Anal. Appl. 134 (1988), 441–459.
Borwein, J. M., Moors, W. B. and Xianfu, W.: Lipschitz functions with prescribed derivatives and subderivatives, Nonlinear Anal. 29 (1997), 53–63.
Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
Clarke, F. H., Stern, R. J. and Wolenski, P. R.: Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993), 1167–1183.
Combari, C., Laghdir, M. and Thibault, L.: Sous-Différentiels de fonctions convexes compos ées, Ann. Sci. Math. Québec 18 (1994), 119–148.
Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions, Appl. Math. Optim. 21 (1990), 265–287.
Henrion, R. and Klatte, D.: Metric regularity of the feasible set mappingin semi-infinite optimization, Appl. Math. Optim. 30 (1994), 103–109.
Henrion, R. and Römisch, W.: Metric regularity and quantitative stability in stochastic programming with probabilistic constraints, Preprint 96-2, Humboldt University, Berlin, submitted to Math. Programming. SVAN310.tex; 24/02/1998; 10:08; v.7; p.26
Ioffe, A. D.: Approximate subdifferentials and applications. I: the finite-dimensional theory, Trans. Amer. Math. Soc. 281 (1984), 389–416.
Ioffe, A. D.: Approximate subdifferentials and applications. 3: the metric theory, Mathematika 36 (1989), 1–38.
Jongen, H. T., Jonker, P. and Twilt, F.: Nonlinear Optimization in ℝn, II: Transversality, Flows, Parametric Aspects, Lang, Frankfurt a. M., 1986.
Jourani, A. and Thibault, L.: Approximate subdifferentials and metric regularity: the finite dimensional case, Math. Programming 47 (1990), 203–218.
Jourani, A. and Thibault, L.: Verifiable conditions for openness and regularity of multivalued mappings in Banach spaces, Trans. Amer. Math. Soc. 347 (1995), 1255–1268.
Jourani, A. and Thibault, L.: Metric regularity for strongly compactly Lipschitzian mappings, Nonlinear Anal. 24 (1995), 229–240.
Jourani, A. and Thibault, L.: Coderivatives of multivalued mappings, locally compact cones and metric regularity, Manuscript.
Katriel, G.: Are the approximate and the Clarke subgradients generically equal? J. Math. Anal. Appl. 193 (1995), 588–593.
Loewen, P. D.: Limits of Fréchet normals in nonsmooth analysis, in: A. D. Ioffe et al. (eds), Optimization and Nonlinear Analysis, Pitman Research Notes Math. Ser. 244, Longman, Harlow, 1992, pp. 178–188.
Loewen, P. D.: A mean value theorem for Fréchet subgradients, Nonlinear Anal. 23 (1994), 1365–1381.
Mordukhovich, B. S.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), 1–35.
Mordukhovich, B. S. and Shao, Y.: Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), 1235–1270.
Mordukhovich, B. S. and Shao, Y.: Stability of set-valued mappings in infinite dimensions: point criteria and applications, SIAM J. Control Optim. 35 (1997), 285–314.
Penot, J. P.: On regularity conditions in mathematical programming, Math. Programming 19 (1982), 167–199.
Penot, J. P.: Metric regularity, openness and Lipschitzian behavior of multifunctions, Nonlinear Anal. 13 (1989), 629–643.
Robinson, S. M.: Stability theorems for systems of inequalities, Part II: Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), 497–513.
Robinson, S. M.: Regularity and stability for convex multifunctions, Math. Oper. Res. 1 (1976), 130–143.
Rockafellar, R. T.: Lipschitzian properties of multifunctions, Nonlinear Anal. 9 (1985), 867–885.
Rockafellar, R. T.: Favorable classes of Lipschitz-continuous functions in sugradient optimization, in: E. A. Nurminski (ed.), Progress in Nondifferentiable Optimization, IIASA Collaborative Proceedings Series CP-82-S8, 1982, IIASA, Laxenburg, pp. 125–143.
Römisch, W. and Schultz, R.: Distribution sensitivity for certain classes of chance-constrained models with applications to power dispatch, J. Optim. Theory Appl. 71 (1991), 569–588.
Zowe, J. and Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces, Appl. Math. Optim. 5 (1979), 49–62.
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Henrion, R. Characterization of Stability for Cone Increasing Constraint Mappings. Set-Valued Analysis 5, 323–349 (1997). https://doi.org/10.1023/A:1008629709451
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DOI: https://doi.org/10.1023/A:1008629709451