Exhausters, coexhausters and converters in nonsmooth analysis

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December 2011, 31(4): 1273-1292. doi: 10.3934/dcds.2011.31.1273

Exhausters, coexhausters and converters in nonsmooth analysis

Vladimir F. Demyanov ^1, and Julia A. Ryabova ^1,

1.	Saint Petersburg State University, 7-9, Universitetskaya nab., St.Petersburg, Russian Federation, Russian Federation

Received March 2010 Revised October 2010 Published September 2011

Abstract
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Usually, positively homogeneous functions are studied by means of exhaustive families of upper and lower approximations and their duals - upper and lower exhausters. Upper exhausters are used to find minimizers while lower exhausters are employed to find maximizers. In the paper, some properties of the so-called conversion operator (which converts an upper exhauster into a lower one, and vice versa) are discussed. The notions of cycle of exhausters, minimal cycle of exhausters and equivalent exhausters are introduced. A conjecture is formulated claiming that in the case of polyhedral exhausters only 1-cycle minimal exhausters exist.

Keywords: Nonsmooth analysis, Converters, Cycles of Exhausters., Subdifferential, Exhausters, Coexhausters.

Mathematics Subject Classification: Primary: 90C30; Secondary: 65K0.

Citation: Vladimir F. Demyanov, Julia A. Ryabova. Exhausters, coexhausters and converters in nonsmooth analysis. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1273-1292. doi: 10.3934/dcds.2011.31.1273

References:

[1]	M. Castellani, A dual characterization for proper positively homogeneous functions, Journal of Global Optimization, 16 (2000), 393-400. doi: 10.1023/A:1008394516838.
[2]	V. F. Demyanov, Exhausters of a positively homogeneous function, Dedicated to the Memory of Professor Karl-Heinz Elster, Optimization, 45 (1999), 13-29. doi: 10.1080/02331939908844424.
[3]	V. F. Demyanov, Exhausters and convexificators - new tools in nonsmooth analysis, in "Quasidifferentiability and Related Topics" (eds. V. F. Demyanov and A. M. Rubinov), Nonconvex Optim. Appl., 43, Kluwer Acad. Publ., Dordrecht, (2000), 85-137.
[4]	V. F. Demyanov and V. A. Roshchina, Optimality conditions in terms of upper and lower exhausters, Optimization, 55 (2006), 525-540. doi: 10.1080/02331930600815777.
[5]	V. F. Demyanov and A. M. Rubinov, Elements of quasidifferential calculus, in "Nonsmooth Problems of Optimization Theory and Control" (ed. V. F. Demyanov), Leningrad University Press, Leningrad, (1982), 5-127.
[6]	V. F. Demyanov and A. M. Rubinov, "Quasidifferential Calculus," Springer - Optimization Software, New York, 1986.
[7]	V. F. Demyanov and A. M. Rubinov, "Constructive Nonsmooth Analysis," Approximation and Optimization, 7, Peter Lang, Frankfurt a/M., 1995.
[8]	V. F. Demyanov and A. M. Rubinov, Exhaustive families of approximations revisited, in "From Convexity to Nonconvexity" (eds. R. P. Gilbert, P. D. Panagiotopoulos and P. M. Pardalos), Nonconvex Optim. Appl., 55, Kluwer Acad. Publ., Dordrecht, (2001), 43-50.
[9]	B. N. Pschenichnyi, "Convex Analysis and Extremal Problems," Nauka, Moscow, 1980.
[10]	R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.
[11]	A. Uderzo, Convex approximators, convexificators and exhausters: Applications to constrained extremum problems, in "Quasidifferentiability and Related Topics" (eds. V. F. Demyanov and A. M. Rubinov), Nonconvex Optim. Appl., 43, Kluwer Acad. Publ., Dordrecht, (2000), 297-327.

show all references

References:

[1]	M. Castellani, A dual characterization for proper positively homogeneous functions, Journal of Global Optimization, 16 (2000), 393-400. doi: 10.1023/A:1008394516838.
[2]	V. F. Demyanov, Exhausters of a positively homogeneous function, Dedicated to the Memory of Professor Karl-Heinz Elster, Optimization, 45 (1999), 13-29. doi: 10.1080/02331939908844424.
[3]	V. F. Demyanov, Exhausters and convexificators - new tools in nonsmooth analysis, in "Quasidifferentiability and Related Topics" (eds. V. F. Demyanov and A. M. Rubinov), Nonconvex Optim. Appl., 43, Kluwer Acad. Publ., Dordrecht, (2000), 85-137.
[4]	V. F. Demyanov and V. A. Roshchina, Optimality conditions in terms of upper and lower exhausters, Optimization, 55 (2006), 525-540. doi: 10.1080/02331930600815777.
[5]	V. F. Demyanov and A. M. Rubinov, Elements of quasidifferential calculus, in "Nonsmooth Problems of Optimization Theory and Control" (ed. V. F. Demyanov), Leningrad University Press, Leningrad, (1982), 5-127.
[6]	V. F. Demyanov and A. M. Rubinov, "Quasidifferential Calculus," Springer - Optimization Software, New York, 1986.
[7]	V. F. Demyanov and A. M. Rubinov, "Constructive Nonsmooth Analysis," Approximation and Optimization, 7, Peter Lang, Frankfurt a/M., 1995.
[8]	V. F. Demyanov and A. M. Rubinov, Exhaustive families of approximations revisited, in "From Convexity to Nonconvexity" (eds. R. P. Gilbert, P. D. Panagiotopoulos and P. M. Pardalos), Nonconvex Optim. Appl., 55, Kluwer Acad. Publ., Dordrecht, (2001), 43-50.
[9]	B. N. Pschenichnyi, "Convex Analysis and Extremal Problems," Nauka, Moscow, 1980.
[10]	R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.
[11]	A. Uderzo, Convex approximators, convexificators and exhausters: Applications to constrained extremum problems, in "Quasidifferentiability and Related Topics" (eds. V. F. Demyanov and A. M. Rubinov), Nonconvex Optim. Appl., 43, Kluwer Acad. Publ., Dordrecht, (2000), 297-327.

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