Optimality conditions in terms of upper and lower exhausters.

*(English)*Zbl 1156.90458Summary: The notions of exhaustive families of upper convex and lower concave approximations (in the sense of Pschenichnyi) were introduced by the first author and A. M. Rubinov [Nonconvex Optim. Appl. 55, 43–50 (2001; Zbl 1043.49021)]. For some classes of nonsmooth functions, these tools appeared to be very productive and constructive (e.g., in the case of quasidifferentiable functions). Dual tools – the upper exhauster and the lower exhauster – can be employed to describe optimality conditions and to find directions of the steepest ascent and descent. If a proper exhauster is known (for minimality conditions we need an upper exhauster, while for maximality ones a lower exhauster is required), the above problems are reduced to the problems of finding the nearest points to convex sets. If we study, e.g., the minimization problem and a lower exhauster is available, it is required to convert it into an upper one. In the present article it is shown how to use a lower exhauster to get conditions for a minimum without converting the lower exhauster into an upper one.

##### MSC:

90C46 | Optimality conditions and duality in mathematical programming |

90C48 | Programming in abstract spaces |

49J52 | Nonsmooth analysis |

##### Keywords:

positively homogeneous function; optimality conditions; upper and lower exhausters; proper and adjoint exhausters; steepest ascent and descent directions; unconstrained optimization problems
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\textit{V. F. Demyanov} and \textit{V. A. Roshchina}, Optimization 55, No. 5--6, 525--540 (2006; Zbl 1156.90458)

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