Abstract
Exhausters are families of convex compact sets. They allow one to represent the principal part of the increment of a studied function in the form of minimax or maximin of linear functions. The calculus of exhausters was developed in the last decade. It gives formulas for building these families for a wide class of functions. There have been developed a number of optimality conditions that are described in terms of exhausters. This led to emergence of new optimizations algorithms. So exhausters became an effective tool in the study of nonsmooth functions. Since exhausters are not uniquely defined an important problems of their minimality and reduction arise. These problems were studied by researchers for decades. In this paper we propose new conditions for the verification of exhauster minimality and develop procedures for their reduction. The main advantage of our approach is its transparent geometric meaning.
Similar content being viewed by others
References
Demyanov, V.F.: Exhausters of a positively homogeneous function. Optimization 45, 13–29 (1999)
Demyanov, V.F.: Exhausters and convexificators—new tools in nonsmooth analysis. In: Demyanov, V., Rubinov, A. (eds.) Quasidifferentiability and Related Topics, pp. 85–137. Kluwer Academic Publishers, Dordrecht (2000)
Demyanov, V.F., Rubinov, A.M.: Constructive Nonsmooth Analysis. Verlag Peter Land, Frankfurt (1995)
Castellani, M.: A dual representation for proper positively homogeneous functions. J. Glob. Optim. 16(4), 393–400 (2000)
Abbasov, M.E., Demyanov, V.F.: Proper and adjoint exhausters in nonsmooth analysis: optimality conditions. J. Glob. Optim. 56, 569–585 (2013)
Demyanov, V.F., Roshchina, V.A.: Optimality conditions in terms of upper and lower exhausters. Optimization 55, 525–540 (2006)
Abbasov, M.E.: Generalized exhausters: existence, construction, optimality conditions. J. Ind. Manag. Optim. 11(1), 217–230 (2015)
Küçük, M., Urbanski, R., Grzybowski, J., et al.: Reduction of weak exhausters and optimality conditions via reduced weak exhausters. J. Optim. Theory Appl. 165(3), 693–707 (2015)
Küçük, M., Urbanski, R., Grzybowski, J., et al.: Weak subdifferential/superdifferential weak exhausters and optimality conditions. Optimization 64(10), 2199–2212 (2015)
Gorokhovik, V.V., Trafimovich, M.A.: On methods for converting exhausters of positively homogeneous functions. Optimization 65(3), 589–608 (2016)
Demyanov, V.F., Ryabova, J.A.: Exhausters, coexhausters and convertors in nonsmooth analysis. Discrete Contin. Dyn. Syst. 31(4), 1273–1292 (2011)
Muzarbekova, G.Y.: Exhausters and implicit functions in nonsmooth systems. Optimization 59, 105–113 (2010)
Sang, T.: On the conjecture by Demyanov–Ryabova in converting finite exhausters (2016). arXiv:1601.06382 21/23
Grzybowski, J., Pallaschke, D., Urbanski, R.: Reduction of finite exhausters. J. Glob. Optim. 46(4), 589–601 (2010)
Grzybowski, J., Küçük, M., Küçük, Y., Urbanski, R.: On minimal representations by a family of sublinear functions. J. Glob. Optim. 61(2), 279–289 (2015)
Roshchina, V.A.: Reducing exhausters. J. Optim. Theory Appl. 136(2), 261–273 (2008)
Roshchina, V.A.: On conditions for minimality of exhausters. J. Convex Anal. 15(4), 859–868 (2008)
Roshchina, V.A.: Topics in Optimization: Solving Second-Order Conic Systems with Finite Precision; Calculus of Generalized Subdifferentials for Nonsmooth Functions. Supervisor - Prof. Felipe Cucker, 229 pp. City University of Hong Kong (2009)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Author information
Authors and Affiliations
Corresponding author
Additional information
The reported study was partially supported by RFBR, Research Project No. 18-31-00014.
Rights and permissions
About this article
Cite this article
Abbasov, M.E. Geometric conditions of reduction of exhausters. J Glob Optim 74, 737–751 (2019). https://doi.org/10.1007/s10898-018-0683-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-018-0683-5