Abstract
Subgradient mappings associated with various convex and nonconvex functions are a vehicle for stating optimality conditions, and their proto-differentiability plays a role therefore in the sensitivity analysis of solutions to problems of optimization. Examples of special interest are the subgradients of the max of finitely manyC 2 functions, and the subgradients of the indicator of a set defined by finitely manyC 2 constraints satisfying a basic constraint qualification. In both cases the function has a property called full amenability, so the general theory of existence and calculus of proto-derivatives of subgradient mappings associated with fully amenable functions is applicable. This paper works out the details for such examples. A formula of Auslender and Cominetti in the case of a max function is improved in particular.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Rockafellar, R. T.: Proto-differentiability of set-valued mappings and its applications in optimization, in H. Attouchet al. (eds.),Analyse Non Linéaire, Gauthier-Villars, Paris 1989, pp. 449–482.
Poliquin R. A. and Rockafellar, R. T.: Amenable functions in optimization, F. Giannessi (ed.) inNonsmooth Optimization Methods and Applications, Gordon & Breach, Philadelphia, 1992, pp. 338–353.
Rockafellar, R. T.: Nonsmooth analysis and parametric optimization, in A. Cellina (ed.),Methods of Nonconvex Analysis Lecture Notes in Math. 1446 Springer-Verlag, New York, 1990, pp. 137–151.
Rockafellar, R. T.: Perturbation of generalized Kuhn-Tucker points in finite dimensional optimization, in F. H. Clarkeet al. (eds.),Nonsmooth Optimization and Related Topics, Plenum Press, New York, 1989, pp. 393–402.
Poliquin, R. A.: Proto-differentiation of subgradient set-valued mappings,Canad. J. Math. 42 (1990), 520–532.
Rockafellar, R. T.: First- and second-order epi-differentiability in nonlinear programming,Trans. Amer. Math. Soc. 307 (1988), 75–107.
Rockafellar, R. T.: Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives,Math. Oper. Research 14 (1989), 462–484.
Poliquin R. A. and Rockafellar, R. T.: A calculus of epi-derivatives applicable to optimization,Canad. J. Math., to appear.
Clarke, F. H.:Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
Mordukhovich, B. S.:Approximation Methods in Problems of Optimization and Control, Nauka, Moscow (1988) (in Russian); English translation: Wiley-Interscience, to appear.
Rockafellar, R. T.: Lagrange multipliers and optimality,SIAM Rev. 35 (1993), 183–238.
Auslender, A. and Cominetti, R.: A comparative study of multifunction differentiability with applications in mathematical programming,Math. Oper. Res. 16 (1991), 240–258.
Penot, J. P.: On the differentiability of the subdifferential of a maximum function, preprint, 1992.
Rockafellar, R. T.: Maximal monotone relations and the second derivatives of nonsmooth functions,Ann. Inst. H. Poincaré: Analyse Nonlinéaire 2 (1985), 167–184.
Rockafellar, R. T.: Generalized second derivatives of convex functions and saddle functions,Trans. Amer. Math. Soc. 322 (1990), 51–77.
Do, C.: Generalized second derivatives of convex functions in reflexive Banach spaces,Trans. Amer. Math. Soc. 334 (1992), 281–301.
Author information
Authors and Affiliations
Additional information
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983 for the first author and by the National Science Foundation under grant DMS-9200303 for the second author.
Rights and permissions
About this article
Cite this article
Poliquin, R.A., Rockafellar, R.T. Proto-derivative formulas for basic subgradient mappings in mathematical programming. Set-Valued Anal 2, 275–290 (1994). https://doi.org/10.1007/BF01027106
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01027106