# American Institute of Mathematical Sciences

January  2015, 11(1): 217-230. doi: 10.3934/jimo.2015.11.217

## Generalized exhausters: Existence, construction, optimality conditions

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

Received  May 2013 Revised  March 2014 Published  May 2014

In this work a generalization of the notion of exhauster is considered. Exhausters are new tools in nonsmooth analysis introduced in works of Demyanov V.F., Rubinov A.M., Pshenichny B.N. In essence, exhausters are families of convex compact sets, allowing to represent the increments of a function at a considered point in an $\inf\max$ or $\sup\min$ form, the upper exhausters used for the first representation, and the lower one for the second representation. Using this objects one can get new optimality conditions, find descent and ascent directions and thus construct new optimization algorithms. Rubinov A.M. showed that an arbitrary upper or lower semicontinuous positively homogenous function bounded on the unit ball has an upper or lower exhausters respectively. One of the aims of the work is to obtain the similar result under weaker conditions on the function under study, but for this it is necessary to use generalized exhausters - a family of convex (but not compact!) sets, allowing to represent the increments of the function at a considered point in the form of $\inf\sup$ or $\sup\inf$. The resulting existence theorem is constructive and gives a theoretical possibility of constructing these families. Also in terms of these objects optimality conditions that generalize the conditions obtained by Demyanov V.F., Abbasov M.E. are stated and proved. As an illustration of obtained results, an example of $n$-dimensional function, that has a non-strict minimum at the origin, is demonstrated. A generalized upper and lower exhausters for this function at the origin are constructed, the necessary optimality conditions are obtained and discussed.
Citation: Majid E. Abbasov. Generalized exhausters: Existence, construction, optimality conditions. Journal of Industrial and Management Optimization, 2015, 11 (1) : 217-230. doi: 10.3934/jimo.2015.11.217
##### References:
 [1] M. E. Abbasov, Extremality conditions in terms of adjoint exhausters, (In Russian) Vestnik of Saint-Petersburg University; Applied mathematics, informatics, control processes. 10 (2011), 3-8. [2] M. E. Abbasov and V. F. Demyanov, Extremum Conditions for a Nonsmooth Function in Terms of Exhausters and Coexhausters, (In Russian) Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2009, Vol. 15, No. 4. English translation: Proceedings of the Steklov Institute of Mathematics, 2010, Suppl. 2, pp. S1-S10. Pleiades Publishing, Ltd. (2010). [3] M. E. Abbasov and V. F. Demyanov, Proper and adjoint exhausters in Nonsmooth analysis: Optimality conditions, Journal of Global Optimization, 56 (2013), 569-585. doi: 10.1007/s10898-012-9873-8. [4] M. Castellani, A dual representation for proper positively homogeneous functions, J. Global Optim., 16 (2000), 393-400. doi: 10.1023/A:1008394516838. [5] V. F. Demyanov, Optimality Conditions and Variational Calculus, (In Russian). Moscow, Higher School Publishing, 2005. [6] V. F. Demyanov, Exhausters and convexificators - new tools in nonsmooth analysis, Nonconvex Optim. Appl., Kluwer Acad. Publ., Dordrecht, Quasidifferentiability and related topics, 43 (2000), 85-137. doi: 10.1007/978-1-4757-3137-8_4. [7] V. F. Demyanov, Proper Exhausters and Coexhausters in Nonsmooth Analysis, Optimization, 61 (2012), 1347-1368. doi: 10.1080/02331934.2012.700929. [8] V. F. Demyanov, Exhausters of a positively homogeneous function, Optimization, 45 (1999), 13-29. doi: 10.1080/02331939908844424. [9] V. F. Demyanov and V. A. Roschina, Optimality conditions in terms of upper and lower exhausters, Optimization, 55 (2006), 525-540. doi: 10.1080/02331930600815777. [10] V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Approximation & Optimization, 7. Peter Lang, Frankfurt am Main, 1995. iv+416 pp. [11] V. F. Demyanov and A. M. Rubinov, Exhaustive families of approximations revisited,From convexity to nonconvexity, Nonconvex Optim. Appl., Kluwer Acad. Publ., Dordrecht, 55 (2001), 43-50. doi: 10.1007/978-1-4613-0287-2_4. [12] B. N. Pshenichny, Convex Analysis and Extremal Problems (in Russian). Nauka, Moscow, 1980, 320 pp. [13] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton University Press, Princeton, N.J. 1970. [14] V. A. Roshchina, Limited exhausters and optimality conditions, Control Processes and Stability: Proceedings of the 36-th international conference of students and graduate students, Saint-Petresburg, Saint-Petrsburg State University Press, (2005), 521-524.

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##### References:
 [1] M. E. Abbasov, Extremality conditions in terms of adjoint exhausters, (In Russian) Vestnik of Saint-Petersburg University; Applied mathematics, informatics, control processes. 10 (2011), 3-8. [2] M. E. Abbasov and V. F. Demyanov, Extremum Conditions for a Nonsmooth Function in Terms of Exhausters and Coexhausters, (In Russian) Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2009, Vol. 15, No. 4. English translation: Proceedings of the Steklov Institute of Mathematics, 2010, Suppl. 2, pp. S1-S10. Pleiades Publishing, Ltd. (2010). [3] M. E. Abbasov and V. F. Demyanov, Proper and adjoint exhausters in Nonsmooth analysis: Optimality conditions, Journal of Global Optimization, 56 (2013), 569-585. doi: 10.1007/s10898-012-9873-8. [4] M. Castellani, A dual representation for proper positively homogeneous functions, J. Global Optim., 16 (2000), 393-400. doi: 10.1023/A:1008394516838. [5] V. F. Demyanov, Optimality Conditions and Variational Calculus, (In Russian). Moscow, Higher School Publishing, 2005. [6] V. F. Demyanov, Exhausters and convexificators - new tools in nonsmooth analysis, Nonconvex Optim. Appl., Kluwer Acad. Publ., Dordrecht, Quasidifferentiability and related topics, 43 (2000), 85-137. doi: 10.1007/978-1-4757-3137-8_4. [7] V. F. Demyanov, Proper Exhausters and Coexhausters in Nonsmooth Analysis, Optimization, 61 (2012), 1347-1368. doi: 10.1080/02331934.2012.700929. [8] V. F. Demyanov, Exhausters of a positively homogeneous function, Optimization, 45 (1999), 13-29. doi: 10.1080/02331939908844424. [9] V. F. Demyanov and V. A. Roschina, Optimality conditions in terms of upper and lower exhausters, Optimization, 55 (2006), 525-540. doi: 10.1080/02331930600815777. [10] V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Approximation & Optimization, 7. Peter Lang, Frankfurt am Main, 1995. iv+416 pp. [11] V. F. Demyanov and A. M. Rubinov, Exhaustive families of approximations revisited,From convexity to nonconvexity, Nonconvex Optim. Appl., Kluwer Acad. Publ., Dordrecht, 55 (2001), 43-50. doi: 10.1007/978-1-4613-0287-2_4. [12] B. N. Pshenichny, Convex Analysis and Extremal Problems (in Russian). Nauka, Moscow, 1980, 320 pp. [13] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton University Press, Princeton, N.J. 1970. [14] V. A. Roshchina, Limited exhausters and optimality conditions, Control Processes and Stability: Proceedings of the 36-th international conference of students and graduate students, Saint-Petresburg, Saint-Petrsburg State University Press, (2005), 521-524.
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