# American Institute of Mathematical Sciences

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

## Exhausters, coexhausters and converters in nonsmooth analysis

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

Received  March 2010 Revised  October 2010 Published  September 2011

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.
Citation: Vladimir F. Demyanov, Julia A. Ryabova. Exhausters, coexhausters and converters in nonsmooth analysis. Discrete & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [6] V. F. Demyanov and A. M. Rubinov, "Quasidifferential Calculus," Springer - Optimization Software, New York, 1986.  Google Scholar [7] V. F. Demyanov and A. M. Rubinov, "Constructive Nonsmooth Analysis," Approximation and Optimization, 7, Peter Lang, Frankfurt a/M., 1995.  Google Scholar [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.  Google Scholar [9] B. N. Pschenichnyi, "Convex Analysis and Extremal Problems," Nauka, Moscow, 1980. Google Scholar [10] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [6] V. F. Demyanov and A. M. Rubinov, "Quasidifferential Calculus," Springer - Optimization Software, New York, 1986.  Google Scholar [7] V. F. Demyanov and A. M. Rubinov, "Constructive Nonsmooth Analysis," Approximation and Optimization, 7, Peter Lang, Frankfurt a/M., 1995.  Google Scholar [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.  Google Scholar [9] B. N. Pschenichnyi, "Convex Analysis and Extremal Problems," Nauka, Moscow, 1980. Google Scholar [10] R. T. Rockafellar, "Convex Analysis," Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970.  Google Scholar [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.  Google Scholar
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