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

show all references

##### References:
 [1] Majid E. Abbasov. Generalized exhausters: Existence, construction, optimality conditions. Journal of Industrial & Management Optimization, 2015, 11 (1) : 217-230. doi: 10.3934/jimo.2015.11.217 [2] Renato Huzak, Domagoj Vlah. Fractal analysis of canard cycles with two breaking parameters and applications. Communications on Pure & Applied Analysis, 2019, 18 (2) : 959-975. doi: 10.3934/cpaa.2019047 [3] Cédric Villani. Regularity of optimal transport and cut locus: From nonsmooth analysis to geometry to smooth analysis. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 559-571. doi: 10.3934/dcds.2011.30.559 [4] Sanming Liu, Zhijie Wang, Chongyang Liu. On convergence analysis of dual proximal-gradient methods with approximate gradient for a class of nonsmooth convex minimization problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 389-402. doi: 10.3934/jimo.2016.12.389 [5] E. Fossas-Colet, J.M. Olm-Miras. Asymptotic tracking in DC-to-DC nonlinear power converters. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 295-307. doi: 10.3934/dcdsb.2002.2.295 [6] Masahiro Kubo. Quasi-subdifferential operators and evolution equations. Conference Publications, 2013, 2013 (special) : 447-456. doi: 10.3934/proc.2013.2013.447 [7] Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91 [8] Shengji Li, Xiaole Guo. Calculus rules of generalized $\epsilon-$subdifferential for vector valued mappings and applications. Journal of Industrial & Management Optimization, 2012, 8 (2) : 411-427. doi: 10.3934/jimo.2012.8.411 [9] Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations & Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015 [10] Mi-Ho Giga, Yoshikazu Giga. A subdifferential interpretation of crystalline motion under nonuniform driving force. Conference Publications, 1998, 1998 (Special) : 276-287. doi: 10.3934/proc.1998.1998.276 [11] Noboru Okazawa, Tomomi Yokota. Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 311-341. doi: 10.3934/dcds.2010.28.311 [12] Mahdi Boukrouche, Grzegorz Łukaszewicz. On global in time dynamics of a planar Bingham flow subject to a subdifferential boundary condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 3969-3983. doi: 10.3934/dcds.2014.34.3969 [13] Sophie Guillaume. Evolution equations governed by the subdifferential of a convex composite function in finite dimensional spaces. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 23-52. doi: 10.3934/dcds.1996.2.23 [14] Anna M. Barry, Esther WIdiasih, Richard Mcgehee. Nonsmooth frameworks for an extended Budyko model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2447-2463. doi: 10.3934/dcdsb.2017125 [15] Mohamed Aly Tawhid. Nonsmooth generalized complementarity as unconstrained optimization. Journal of Industrial & Management Optimization, 2010, 6 (2) : 411-423. doi: 10.3934/jimo.2010.6.411 [16] Freddy Dumortier, Robert Roussarie. Birth of canard cycles. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 723-781. doi: 10.3934/dcdss.2009.2.723 [17] Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237 [18] Vladimir Georgiev, Eugene Stepanov. Metric cycles, curves and solenoids. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1443-1463. doi: 10.3934/dcds.2014.34.1443 [19] Jaume Llibre, Ana Rodrigues. On the limit cycles of the Floquet differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1129-1136. doi: 10.3934/dcdsb.2014.19.1129 [20] Freddy Dumortier, Robert Roussarie. Canard cycles with two breaking parameters. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 787-806. doi: 10.3934/dcds.2007.17.787

2018 Impact Factor: 1.143