Geodesic
A method of construction of exhaustive family of upper convex approximations
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2013), pp. 37-51 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown how to construct Exhausters for a Lipschitz function $f$ at a point $x$ which is an important problem for optimization of such functions. At first the function $f$ is modified to some function $\tilde{f}$ and an exhaustive set of upper convex approximations is constructed for it whose subdifferentials at zero define the upper Exhauster of the function $\tilde{f}$ at the point $x$. A family $\Im$ of convex compact set pairs for the function $f$ is constructed. $\Im$ is called BiExhauster of the function $f$ at the point $x$. The exhaustive sets of upper and lower convex approximations of the function $f$ at the point $x$ are defined with the help of the set $\Im$. Convex compact sets from the upper Exhauster of the function $\tilde{f}$ are constructed as limit values of average integrals from gradients of the function $f$ along curves from a defined set of curves along which $\tilde{f}$ is almost everywhere differentiable. Bibliogr. 12. Il. 8.
Keywords: Lipschitz function, directional derivative, upper and lower convex approximations, upper and lower exhausters, extremum points, extremum condition.
Mots-clés : BiExhauster 
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I. M. Proudnikov. A method of construction of exhaustive family of upper convex approximations. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 1 (2013), pp. 37-51. http://geodesic.mathdoc.fr/item/VSPUI_2013_1_a4/

[1] Pshenichnyj B. N., Danilin Ju. M., Chislennye metody i jekstremal'nye zadachi (The calculation methods and extreem problems), Nauka, Moscow, 1975, 319 pp. | MR | Zbl

[2] Nocedal J., Wright S. J., Numerical optimization, Springer, New York, 1999, 634 pp. | MR | Zbl

[3] Clarke F., “Generalized gradients and applications”, Trans. Amer. Math. Soc., 205:2 (1975), 247–262 | MR | Zbl

[4] Pshenichnyj B. N., Vypuklyj analiz i jekstremal'nye zadachi (Convex analysis and extreem problems), Nauka, Moscow, 1980, 320 pp. | MR | Zbl

[5] Dem'janov V. F., Rubinov A. M., Jelementy kvazidifferencial'nogo ischislenija. Negladkie zadachi teorii optimizacii i upravlenija (Quasidifferential calculation. Nonsmooth problems of optimization and control theory), Izd-vo Leningr. un-ta, Leningrad, 1982, 322 pp. | MR

[6] Dem'janov V. F., Roshhina V. A., “Obobshhennye subdifferencialy i jekzostery (General subdifferentials and exhausters)”, Vladikavkaz. matem. zhurn., 8:4 (2006), 19–31 | MR

[7] Castellani M., “A Dual Representation for Proper Positively Homogeneous Functions”, J. of Global Opt., 16 (2000), 393–400 | MR | Zbl

[8] Proudnikov I. M., “New constructions for local approximation of Lipschitz functions, I”, Nonlinear analysis, 53:3 (2003), 373–390 | MR

[9] Proudnikov I. M., “New constructions for local approximation of Lipschitz functions, II”, Nonlinear analysis, 70:2 (2007), 1443–1453 | MR

[10] Prudnikov I. M., “Pravila postroenija nizhnih vypuklyh approksimacij (The rules of construction of lower convex approximations)”, Zhurn. vychisl. matematiki i matem. fiziki, 43:7 (2003), 939–950 | MR | Zbl

[11] Demyanov V. F., “Exhausters of a positively homomegeneous function”, Optimization, 45 (1999), 13–29 | MR | Zbl

[12] Dem'janov V. F., “Uslovnye proizvodnye i jekzostery v negladkom analize (Conditional derivatives and exhausters in nonsmooth analysis)”, Dokl. RAN, 338:6 (1999), 730–733 | MR