Introduction to the Theory of Nonlinear Optimization by Johannes Jahn

Author:Johannes Jahn
Language: eng
Format: epub, pdf
ISBN: 9783030427603
Publisher: Springer International Publishing

and

(6.18)

For x = 0X we get especially

From the inequality (6.18) we also obtain

(because the assumption that (c − u ∘ A)(x) < 0 for some x ∈ C X leads to a contradiction to the inequality (6.18)). Consequently we have

resulting in

This proves the first part of the assertion.

Next, we choose an arbitrary pair with and λ ≤ u(b). Then we conclude

and therefore it follows (λ, u) ∈ S ∗. □

With Lemma 6.14 the equivalent dual problem (6.16) is also equivalent to the problem

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