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Exhausters and subdifferentials in non-smooth analysis. (English) Zbl 1220.49007
Summary: Non-smooth analysis manifested itself in the 1960s of the last century and is still gaining momentum developing new tools and harnesses and covering new areas of application. One of the notions of non-smooth analysis is that of the exhauster. The exhauster represents a dual construction in nonsmooth analysis. The relationships between upper and lower exhausters and various subdifferentials of non-smooth functions are discussed in this article. It is shown that exhausters are closely related to other non-smooth tools, such as the Michel-Penot, Clarke, Gâteaux and Fréchet subdifferentials. Formulae for computing these subdifferentials by means of exhausters are obtained. The discovered relations are all in the form of equalities, i.e. a calculus for computing the mentioned subdifferentials by means of exhausters is provided.

MSC:
49J52 Nonsmooth analysis
49J53 Set-valued and variational analysis
26B25 Convexity of real functions of several variables, generalizations
90C46 Optimality conditions and duality in mathematical programming
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References:
[1] DOI: 10.1007/BF00934938 · Zbl 0259.90037
[2] DOI: 10.1016/S0362-546X(98)00142-4 · Zbl 0933.49006
[3] DOI: 10.1023/A:1008394516838 · Zbl 1028.90080
[4] DOI: 10.1090/S0002-9947-1975-0367131-6
[5] Clarke FH, Optimization and Non-smooth Analysis (1983)
[6] Demyanov VF, Vestnik of Leningrad Univ. 13 pp 18– (1980)
[7] DOI: 10.1080/02331939908844424 · Zbl 0954.90050
[8] Demyanov VF, Dokl. Russian Acad. Sci. 338 pp 730– (1999)
[9] Demyanov V.F, Quasidifferentiability and Related Topics pp 85– (2000)
[10] DOI: 10.1023/A:1008246130864 · Zbl 0872.90083
[11] Demyanov VF, Introduction to Minimax (1972)
[12] Demyanov VF, Exhausters, optimality conditions and related problems (2006)
[13] Demyanov VF, Appl. Comput. Math. 4 pp 25– (2005)
[14] Demyanov VF, Optimality conditions in terms of upper and lower exhausters · Zbl 1156.90458
[15] Demyanov VF, Non-smooth Problems of Optimization Theory and Control, Ch. 1 pp 5– (1982)
[16] Demyanov VF, Constructive Non-smooth Analysis (1995)
[17] Demyanov VF, Quasidifferential Calculus (1986)
[18] Demyanov VF, From Convexity to Non-convexity, Non-convex Optimization and Its Applications 55 pp 43– (2001)
[19] DOI: 10.1007/BF00131092 · Zbl 0874.46029
[20] DOI: 10.1007/BF01581262 · Zbl 0782.90095
[21] Ioffe AD, Upsekhi Mat. Nauk 55 333 pp 103– (2000)
[22] DOI: 10.1023/A:1023673105317 · Zbl 1039.49021
[23] Michel P, C.R. Acad. Sc. Paris, ser. I. 298 pp 269– (1984)
[24] DOI: 10.1016/0021-8928(76)90136-2 · Zbl 0362.49017
[25] Mordukhovich BS, Approximations Methods in Problems of Optimization and Control (1988)
[26] DOI: 10.1023/B:SVAN.0000023398.73288.82 · Zbl 1046.49011
[27] Mordukhovich BS, Grundlehren der Mathematischen Wissenschaften 330 (2006)
[28] DOI: 10.1090/S0002-9947-96-01543-7 · Zbl 0881.49009
[29] Pallaschke D, Bull. Acad. Polon. Sci. Ser. Math. 39 pp 1– (1991)
[30] Pallaschke D, Pairs of Compact Convex Sets (2002)
[31] Pschenichnyi BN, Convex Analysis and Extremal Problems (In Russian) (1980)
[32] Rockafellar RT, Convex Analysis (1970) · Zbl 0932.90001
[33] Roshchina V, International Workshop on Optimization: Theory and Algorithms (2006)
[34] DOI: 10.1016/j.jmaa.2006.12.059 · Zbl 1124.49021
[35] DOI: 10.1007/BF00940933 · Zbl 0682.49015
[36] Uderzo A, Quasidifferentiability and Related Topics pp 297– (2000)
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