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Constrained optimality conditions in terms of proper and adjoint exhausters. (English) Zbl 1209.90349
Summary: The notions of exhaustive families of upper convex and lower concave approximations (in the sense of B.N. Pschenichnyi) were introduced by A.M. Rubinov. 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 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. It has been shown earlier how to use a lower (upper) exhauster to get conditions for a minimum (maximium) without converting the lower (upper) exhauster into an upper (lower) one in the unconstrained case. In the present paper the constrained case is described in details.
MSC:
90C46Optimality conditions, duality
90C30Nonlinear programming
49J52Nonsmooth analysis (other weak concepts of optimality)