Proto-derivative formulas for basic subgradient mappings in mathematical programming R. A. Poliquin R. T. Rockafellar Article Received: 01 December 1993 DOI :
10.1007/BF01027106

Cite this article as: Poliquin, R.A. & Rockafellar, R.T. Set-Valued Anal (1994) 2: 275. doi:10.1007/BF01027106
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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.

Mathematics Subjects Classifications (1991) Primary 49J52, 58C06, 58C20 Secondary 90C30

Key words Proto-derivatives generalized second derivatives nonsmooth analysis epi-derivatives subgradient mappings amenable functions 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.

References 1.

Rockafellar, R. T.: Proto-differentiability of set-valued mappings and its applications in optimization, in H. Attouch

et al. (eds.),

Analyse Non Linéaire , Gauthier-Villars, Paris 1989, pp. 449–482.

Google Scholar 2.

Poliquin R. A. and Rockafellar, R. T.: Amenable functions in optimization, F. Giannessi (ed.) in

Nonsmooth Optimization Methods and Applications , Gordon & Breach, Philadelphia, 1992, pp. 338–353.

Google Scholar 3.

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.

Google Scholar 4.

Rockafellar, R. T.: Perturbation of generalized Kuhn-Tucker points in finite dimensional optimization, in F. H. Clarke

et al. (eds.),

Nonsmooth Optimization and Related Topics , Plenum Press, New York, 1989, pp. 393–402.

Google Scholar 5.

Poliquin, R. A.: Proto-differentiation of subgradient set-valued mappings,

Canad. J. Math.
42 (1990), 520–532.

Google Scholar 6.

Rockafellar, R. T.: First- and second-order epi-differentiability in nonlinear programming,

Trans. Amer. Math. Soc.
307 (1988), 75–107.

Google Scholar 7.

Rockafellar, R. T.: Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives,

Math. Oper. Research
14 (1989), 462–484.

Google Scholar 8.

Poliquin R. A. and Rockafellar, R. T.: A calculus of epi-derivatives applicable to optimization,Canad. J. Math. , to appear.

9.

Clarke, F. H.:

Optimization and Nonsmooth Analysis , Wiley, New York, 1983.

Google Scholar 10.

Mordukhovich, B. S.:

Approximation Methods in Problems of Optimization and Control , Nauka, Moscow (1988) (in Russian); English translation: Wiley-Interscience, to appear.

Google Scholar 11.

Rockafellar, R. T.: Lagrange multipliers and optimality,

SIAM Rev.
35 (1993), 183–238.

Google Scholar 12.

Auslender, A. and Cominetti, R.: A comparative study of multifunction differentiability with applications in mathematical programming,

Math. Oper. Res.
16 (1991), 240–258.

Google Scholar 13.

Penot, J. P.: On the differentiability of the subdifferential of a maximum function, preprint, 1992.

14.

Rockafellar, R. T.: Maximal monotone relations and the second derivatives of nonsmooth functions,

Ann. Inst. H. Poincaré: Analyse Nonlinéaire
2 (1985), 167–184.

Google Scholar 15.

Rockafellar, R. T.: Generalized second derivatives of convex functions and saddle functions,

Trans. Amer. Math. Soc.
322 (1990), 51–77.

Google Scholar 16.

Do, C.: Generalized second derivatives of convex functions in reflexive Banach spaces,

Trans. Amer. Math. Soc.
334 (1992), 281–301.

Google Scholar © Kluwer Academic Publishers 1994

Authors and Affiliations R. A. Poliquin R. T. Rockafellar 1. Department of Mathematics University of Alberta Edmonton Canada 2. Department of Mathematics University of Washington Seattle USA