Journal of Global Optimization

, Volume 40, Issue 1–3, pp 71–85 | Cite as

Exhausters, optimality conditions and related problems

  • V. F. Demyanov
  • V. A. Roshchina


The notions of exhausters were introduced in (Demyanov, Exhauster of a positively homogeneous function, Optimization 45, 13–29 (1999)). These dual tools (upper and lower exhausters) can be employed to describe optimality conditions and to find directions of steepest ascent and descent for a very wide range of nonsmooth functions. What is also important, exhausters enjoy a very good calculus (in the form of equalities). In the present paper we review the constrained and unconstrained optimality conditions in terms of exhausters, introduce necessary and sufficient conditions for the Lipschitzivity and Quasidifferentiability, and also present some new results on relationships between exhausters and other nonsmooth tools (such as the Clarke, Michel-Penot and Fréchet subdifferentials).


Positively homogeneous function Optimality conditions Upper and lower exhausters Proper and adjoint exhausters Unconstrained optimization problems Quasidifferentiability The Michel-Penot subdifferential The Clarke subdifferential The Fréchet subdifferential 


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© Springer Science+Business Media LLC 2007

Authors and Affiliations

  1. 1.Applied Mathematics DepartmentSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Department of MathematicsCity University of Hong KongKowloon TongHong Kong S.A.R

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