Abstract
Positively homogeneous (p.h.) functions play an outstanding role in Nonsmooth Analysis (NSA) and Nondifferentiable Optimization (NDO) since optimality conditions are usually expressed in terms of directional derivatives or their generalizations (the Dini and Hadamard upper and lower directional derivatives, the Clarke derivative, the Michel-Penot derivative etc.). All these derivatives are positively homogeneous functions of direction. In the convex case the directional derivative is convex (and p.h.) and, by the Minkowski duality, optimality conditions can be stated in geometric terms.
Attempts to reduce the problem of minimizing an arbitrary function to a sequence of convex problems were undertaken, among others, by Pschenichnyi (1980), who introduced the notions of upper convex and lower concave approximations (u.c.a’s and l.c.a.’s) and by Clarke (1983) who introduced generalized derivatives. Demyanov and Rubinov (1982) proposed to consider exhaustive families of upper convex and lower concave approximations. Recently some new tools — upper and lower exhausters and convexificators — closely related to exhaustive families of approximations were used. They represent dual objects and allow to reduce the original optimization problem to a sequence of convex optimization problems. This paper is a survey of some results related to these new tools. Since conditions for a minimum are expressed in terms of an upper exhauster and conditions for a maximum are described by means of a lower exhauster, a conversion operator is introduced to convert upper exhausters into lower ones, and vice versa. A characterization of the convexity and concavity of a p. h. function is given in terms of minimal convexificators . Representations of a p.h. function in terms of its minimal convexificators are also derived. Examples illustrating the theory are described.
The research was supported by the Russian Foundation for Fundamental Studies ( grant RFFI No. 97-01-00499)
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References
Aban’kin, A.E. (1997), H-quasidifferentiable functions. Newsletter of Mathematical Programming Association. Ekaterinburg: Russian Academy of Sci-ences, Ural Branch. No. 7, pp. 15–16. (In Russian)
Castellani, M. (1998), A dual characterization of proper positively homogeneous functions. Technical Report, 3.235.1111, Department of Mathematics, University of Pisa.
Castellani, M. and Romeo, F. (1998), On constraint qualifications in Nonlinear Programming. Technical Report, 3.236.1147, Department of Mathematics, University of Pisa. Submitted to Numerical Functional Analysis and Optimization.
Clarke, F.H. (1983), Optimization and Nonsmooth Analysis, Wiley Interscience, New York.
Demyanov, V.F. (1994) , Convexification and concavification of a positively homogeneous function by the same family of linear functions, Universita di Pisa , Report 3,208,802.
Demyanov, V.F. Minimal convexificators of a positively homogeneous function and a characterization of its convexity and concavity. Forthcoming in Nonlinear Optimization and Applications. 2. G. Di Pillo, F.Giannessi Editors. Kluwer Academic Publishers.
Demyanov, V.F. and Rubinov, A.M. (1980) On quasidifferentiable functional. Doklady of USSR Acad, of Sei, vol. 250, No.1, pp. 21–25.
Demyanov, V.F. and Rubinov, A.M. (1981), On some approaches to the Non-smooth Optimization problem. Economics and Mathematical Methods. Vol. 17, No. 6, pp. 1153–1174. (In Russian)
Demyanov, V.F. and Rubinov, A.M. (1982), Elements of Quasidifferential Calculus. In “Nonsmooth Problems of Optimization Theory and Control”. Ed. V.F. Demyanov. Leningrad, Leningrad University Press. Ch.1, pp.5–127. (In Russian)
Demyanov, V.F. and Vasiliev, L.V. (1986), Nondifferentiable Optimization. New York, Springer-Optimization Software.
Demyanov, V.F. and Rubinov, A.M. (1995), Constructive Nonsmooth Analysis. Frankfurt a/M, Verlag Peter Lang.
Demyanov, V.F. and Jeyakumar, V. (1997), Hunting for a smaller convex sub-differential, J. of Global Optimization , Vol, 10, No. 3, pp.305–326.
Demyanov, V.F., Di Pillo, G. and Facchinei, F. (1998), Exact penalization via Dini and Hadamard conditional derivatives. Optimization Methods and Software, Vol. 9, pp. 19–36.
Demyanov, V.F. and Rubinov, A.M. (1986), Quasidifferential Calculus. New York, Optimization Software.
Demyanov, V.F. (1999a), Exhausters of a positively homogeneous function. Optimization, Vol. 45, pp. 13–29.
Demyanov, V.F. (1999b), Convexification of a positively homogeneous function. Doklady of the Russian Academy of Sciences, Vol. 366, N. 6, pp. 734–737.
Demyanov, V.F. Conditional derivatives and exhausters in Nonsmooth Analysis. Forthcoming in Doklady of the Russian Academy of Sciences.
Demyanov, V.F. and Murzabekova, G.E. (1999), Convexificators and Implicit functions in Nonsmooth Systems. Computational Mathematics and Mathematical Physics. Vol. 39, No. 2, pp. 212–223.
Demyanov, V.F. and Rubinov, A.M. Exhaustive families of approximations revisited. Forthcoming.
Glover, B.M., Ishizuka, Y., Jeyakumar, V. and Tuan, H.D. (1996), Complete Characterizations of Global Optimality for problems involving the pointwise minimum of sublinear functions. SIAM J. Optimization. Vol.6, No.2, pp. 362–372.
Ishizuka, Yo.(1988), Optimality conditions for quasidifferentiable programs with application to two-level optimization. SIAM J. Control and Optimization. Vol. 26, No. 6, pp. 1388–1398.
Jeyakumar, V. and Demyanov, V.F. (1996), A mean-value Theorem and a characterization of convexity using convexificators, Applied Math. Research Report AMR 96/13, Univ. of New South Wales, Sydney, Australia.
Jeyakumar, V., Luc, D.T. and Schaible, S. (1998), Characterizations of generalized monotone nonsmooth continuous maps using approximate Jacobians, J. of Convex Analysis, Vol. 5, No. 1, pp. 119–132.
Kutateladze, S.S. and Rubinov, A.M.(1976) Minkowski duality and its applications. Novosibirsk, Nauka. (In Russian)
Michel, P. and Penot, J.-P. (1984), Calcus sous-differential pour les functions lipschitzienness et non-lipschitziennes, CR. Acad. Sc. Paris, ser. I 298, pp. 269–272.
Pallaschke, D., Scholtes, S. and Urbanski, R. (1991), On minimal pairs of compact convex sets. Bull. Acad. Polon. Sei. Ser. Math. y Vol. 39, pp. 1–5.
Pschenichnyi, B.N.(1980), Convex analysis and Extremal Problems. Moscow, Nauka Publishers. (In Russian)
Rockafellar, R.T.(1970), Convex Analysis. Princeton, N.J., Princeton Univer-sity Press.
Uderzo, A. Convex approximators, convexificators and exhausters: applications to constrained extremum problems. See this volume.
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Demyanov, V.F. (2000). Exhausters and Convexificators — New Tools in Nonsmooth Analysis. In: Demyanov, V., Rubinov, A. (eds) Quasidifferentiability and Related Topics. Nonconvex Optimization and Its Applications, vol 43. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3137-8_4
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DOI: https://doi.org/10.1007/978-1-4757-3137-8_4
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