Directional openness for epigraphical mappings and optimality conditions for directional efficiency

Exact Penalization and Optimality Conditions for Approximate Directional Minima

“…While it cannot be possible to have a directional openness at an approximate efficiency point, after making a careful choice of some constants, we focus on proving that F can be directional open at an arbitrary point (x, y) ∈ Gr F . The Theorem 3.10 from [9] is the key to the proof of the next result. Before moving forward, we will briefly recall some construction introduced by Mordukhovich and his collaborators (see [12]).…”

“…Proof. The hypotheses (i) and (ii), together with our fixed framework, allow us to obtain the conclusion of Theorem 3.10 from [9], that is, for every a ∈ (0, d), there exists θ > 0 such that, for every ρ ∈ (0, θ) and for all (x, y) ∈ Gr for some r > 0. In particular, the above inclusion holds for (x, y) ∈ Gr F .…”

“…Firstly, we take a brief look at the proof of Theorem 3.10 from [9], in order to note how the constants were fixed. Take Note that the constants µ and ν can be chosen arbitrarily small.…”