Abstract
In this paper we select two tools of investigation of the classical metric regularity of set-valued mappings, namely the Ioffe criterion and the Ekeland Variational Principle, which we adapt to the study of the directional setting. In this way, we obtain in a unitary manner new necessary and/or sufficient conditions for directional metric regularity. As an application, we establish stability of this property at composition and sum of set-valued mappings. In this process, we introduce directional tangent cones and the associated generalized primal differentiation objects and concepts. Moreover, we underline several links between our main assertions by providing alternative proofs for several results.
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Acknowledgements
Radek Cibulka was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports. The work of Marius Durea was supported by a grant of Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0188, within PNCDI III. The work of Marian Panţiruc and Radu Strugariu was supported by a research grant of TUIASI, project number TUIASI-GI-2018-0647.
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Appendix
Appendix
In this section, we illustrate some other connections between our results and several well-known tools in variational analysis. In this sense, we provide some different proofs for two of the key results in this work.
On one hand, we present a direct and constructive proof for the general criterion for the directional regularity of single-valued maps. This underlines again the fact that the use of the Ekeland Variational Principle is an alternative for explicit iterative procedures. On the other hand, we provide as well another proof for the result about the stability at composition of the directional regularity. For this, we employ now, instead of Proposition 13, a variant of the directional Ekeland Variational Principle, formulated on product spaces on the basis of Lemma 12.
1.1 A.1 Proof of the Criterion for the Directional Regularity by an Iterative Procedure
Let us present next the announced constructive proof of the criterion for the directional regularity of single-valued maps.
Proof (of Proposition 11 by Iterative Procedure)
Let \(\lambda :=\text {dirsur}_{L\times M}g(\overline {x})\) and s be the supremum from the statement. We only prove that s ≤ λ, since the opposite inequality is straightforward, as shown in the proof given in Section 3.
Define a function φ : X × X → [0, +∞] by φ(u, v) = TL(u, v), (u, v) ∈ X × X, and a function ψ : Y × Y → [0, +∞] by ψ(y, z) = TM(y, z), (y, z) ∈ Y × Y. Observe that the convexity of coneL implies that
To show that s ≤ λ, fix an arbitrary c ∈ (0, s) (if there is any) for which there is r > 0 such that for all \((x,y)\in (B[\overline {x} ,r]\cap \text {Dom} g)\times B[g(\overline {x}),r]\), with 0 ≠ g(x) − y ∈coneM, there is a point x′ ∈ Dom g such that
Make r > 0 smaller, if necessary, so that the set \(B[\overline {x} ,r]\cap \text {Dom} g\) is complete and g is continuous on this set. By the continuity of g, there is ε ∈ (0, r) such that
Fix any t ∈ (0, ε) and any \(u\in B[\overline {x},\varepsilon ]\cap \text {Dom} g\). Let Λ := B[u, t] ∩ (u + coneL) ∩ Dom g. As L ⊂ SX is closed, so is coneL. Consequently, Λ is complete. We have to show that
Consider any fixed y ∈ B[g(u), ct] ∩ (g(u) −coneM); we will find x ∈Λ such that y = g(x). If y = g(u), take x := u and we are done. Assume further that y ≠ g(u). We will construct a sequence x1, x2, … in Λ satisfying
As φ(u, u) = 0 and g(u) − y ∈coneM (thus ψ(y, g(u)) is finite), the point x1 := u satisfies (A.4) with m = 1. Let \(n\in \mathbb {N}\) and assume that xn ∈Λ satisfying (A.4) with m = n was already found. If g(xn) = y, then take x := xn, and stop the construction. Assume further that g(xn) ≠ y. Then (A.4), with m := n, implies that ψ(y, g(xn)) is finite, meaning that g(xn) − y ∈coneM. Using (A.3) and (A.2), we find xn+ 1 ∈ Dom g such that
where
Note that \(0\leq s_{n}\leq \frac {1}{c}\psi (y,g(x_{n}))<+\infty \). Using (A.1), the first inequality in (A.5), and (A.4) with m := n, we get
which is (A.4) with m := n + 1. In particular, we have cφ(u, xn+ 1) ≤ ψ(y, g(u)) = ∥y − g(u)∥≤ ct; thus xn+ 1 ∈ u + coneL and φ(u, xn+ 1) = ∥u − xn+ 1∥. Consequently, xn+ 1 ∈Λ. If the process stops at some \(n\in \mathbb {N}\), we are done. Assume that this was not the case, that is, g(xn) ≠ y for every \(n\in \mathbb {N}\). From (A.5) and (A.1) we have, for all 1 ≤ n < m, that
and so, ψ(y, g(xn)) > ψ(y, g(xm)). Thus \(\ell :=\lim _{n\rightarrow +\infty }\psi (y,g(x_{n}))\) exists and is finite. By (A.6), for all 1 ≤ n < m, we have φ(xn, xm) < +∞, and hence φ(xn, xm) = ∥xn − xm∥. Consequently, (xn) is a Cauchy sequence in Λ (which is a complete metric space). Put \(x:=\lim _{n\rightarrow +\infty }x_{n}\). Then x ∈Λ and ψ(y, g(x)) ≤ ℓ < +∞ because ψ(y,⋅) is lower semicontinuous and g is continuous. Moreover, for any \(n\in \mathbb {N}\), using the lower semi-continuity of φ(xn,⋅) and (A.6) we get that
Consequently, \(\lim _{n\rightarrow +\infty }\varphi (x_{n},x)= 0\). Suppose that y ≠ g(x). By (A.2), there is x′ ∈ Dom g such that
Then (A.1) implies that \(\limsup _{n\rightarrow +\infty }\varphi (x_{n},x^{\prime })\leq \lim _{n\rightarrow +\infty }\varphi (x_{n},x)+\varphi (x,x^{\prime })=\varphi (x,x^{\prime })\). This and (A.7) imply that, for each \(n\in \mathbb {N}\) sufficiently large, we have
As x ≠ x′ by (A.7), we have φ(x, x′) > 0. The lower semicontinuity of φ(⋅, x′), the choice of sn, and (A.5) yield that
a contradiction. Therefore y = g(x). We proved that c ≤ λ, and thus s ≤ λ.
1.2 A.2 Proof of Directional Openness Stability at Composition by Directional EVP
As mentioned before, in the second part of this appendix, we discuss the possibility to give an alternative proof of the main result of the paper, namely Theorem 16, by the use of the next variant of the directional Ekeland Variational Principle.
Theorem 32
Let (X1, ∥⋅∥), … , (Xn,∥⋅∥) be Banach spaces andA ⊂ X1 × ... × Xnbe a nonempty closed set.Consider nonempty closed sets\(L_{i}\subset S_{X_{i}}\), i = 1, … , nsuchthat coneLiare convex. Then, for every lower semicontinuous bounded from below function\(f:A\rightarrow {\mathbb {R}\cup \{+\infty \}}\), everya0 := (x01, ... , x0n) ∈ Asuch thatf(a0) < +∞, and everyδ, α1, ... , αn > 0, thereexistsaδ := (xδ1, ... , xδn) ∈ Asuch that
and, for every a := (x1, ... , xn) ∈ A ∖{aδ},
Proof
Take \(\widetilde {L}\) as in the Lemma 12 and observe that \(\text {cone}\widetilde {L}=\text {cone} L_{1}\times ...\times \text {cone} L_{n}\) is convex. Apply Theorem 10 with X := X1 ×⋯ × Xn and \(M:=\widetilde {L}\) to get the statement. □
Now, we are ready to provide the announced proof of the main (and the essential) part of Theorem 16.
Proof (of Theorem 16 by Directional EVP)
Again, as in the proof of Theorem 16, we only have to consider the case where the right-hand side of the inequality (4.2) is positive. We find again positive constants α, β, β′, γ, and δ such that c := αγ − βδ > 0, and inequalities (4.4) and (4.5) hold. Moreover, keeping the notation of Theorem 16, there is ε > 0 such that (4.6), (4.7) and (4.9) hold. Also, taking into account Proposition 3, we may suppose that for any \(z\in B(\overline {z},\varepsilon ),\) the mapping \(G_{z}^{-1}\) is directionally Aubin continuous around \((\overline {w},\overline {y})\) with respect to P and − M with modulus γ− 1, i.e.,
for any \(z\in B(\overline {z},\varepsilon ),\) and any \(w,w^{\prime }\in B(\overline {w},\varepsilon )\).
Also, in view of the local closedness of the graphs of F1, F2 and G, we can consider that GrF1 ∩ (B [x, ε] × B [y, αε]), GrF2 ∩ (B [x, ε] × B [z, βε]) and GrG ∩ (B [y, αε] × B [z, βε] × B [w, (αγ + βδ)ε]) are closed, for any \((x,y,z,w)\in B(\overline {x},\varepsilon )\times B(\overline {y} ,\alpha \varepsilon )\times B(\overline {z},\beta \varepsilon )\times B\left (\overline {w},\left (\alpha \gamma +\beta \delta \right ) \varepsilon \right ) \) with y ∈ F1(x), z ∈ F2(x) and w ∈ G(y, z).
Take
Fix t ∈ (0, ρ) and \((x,y,z,w)\in B(\overline {x},\rho )\times B(\overline {y},\alpha \rho )\times B(\overline {z},\beta \rho )\times B\left (\overline {w},\left (\alpha \gamma +\beta \delta \right ) \rho \right ) \) with y ∈ F1(x), z ∈ F2(x) and w ∈ G(y, z). We want to prove that
where the norm on X × Y × Z and \(\widetilde {L}\subset S_{X\times Y\times Z}\) are as in the proof of Theorem 16.
Denote
Take an arbitrary v ∈ w − [0, ct) ⋅ P. We must prove that \(v\in \mathcal {E}_{G,(F_{1},F_{2})}((x,y,z)+[0,t)\cdot \widetilde {L})\).
We can find τ ∈ (0, 1) such that ∥v − w∥ < τct. Remark that \({\Omega }\cap \overline {A}\) is closed (since 2ρ < ε). Define
and observe that it is lower semicontinuous and bounded from below. Thus, we can apply Theorem 32, for τc > 0 instead of δ, and a0 = (x, y, z, w) and − L, M,−N, and SW as sets in X, Y, Z and W, respectively, to find \((\widetilde {a},\widetilde {b},\widetilde {c},\widetilde {d})\in {\Omega }\cap \overline {A}\) satisfying
for every \((p,q,r,s)\in {\Omega }\cap \overline {A}\). As an immediate consequence, \(\widetilde {b}\in F_{1}(\widetilde {a})\), \(\widetilde {c}\in F_{2}(\widetilde {a})\), \(\widetilde {d}\in G(\widetilde {b},\widetilde {c}),\) and
which implies the following:
Moreover, since v ∈ w −coneP, we also have
so
Hence, \((\widetilde {a},\widetilde {b},\widetilde {c},\widetilde {d})\in A\). Now, if \(v=\widetilde {d},\) then
which is exactly what we need. We will prove that \(v=\widetilde {d}\) is the only possibility.
Assume, on the contrary, that \(v\neq \widetilde {d}\). Remark that \(T_{P} (v,\widetilde {d})\leq T_{P}(v,w)<\infty ,\) which means that \(v-\widetilde {d} \in -\text {cone} P\). Then
since its norm equals 1 and it belongs to −coneP. Fix σ ∈ (0, αγ) such that
and choose \(\zeta \in \left (0,\min \left \{ 3^{-1}\rho ,(\alpha \gamma -\sigma )^{-1}\left \Vert v-\widetilde {d}\right \Vert \right \} \right ) \).
We have that
hence by (A.8),
hence there exists m ∈coneM with ∥m∥ < 1 such that \(\widetilde {b}-\gamma ^{-1}(\alpha \gamma -2^{-1}\sigma )\zeta m\in G_{\widetilde {c}}^{-1}(\widetilde {d}+(\alpha \gamma -\sigma )\zeta v^{\prime })\) or, equivalently,
Now, since ζ < ε and \(\widetilde {b}-\gamma ^{-1}(\alpha \gamma -2^{-1}\sigma )\zeta m\in \widetilde {b}-[0,\alpha \zeta )\cdot M,\) it follows using (4.6) that
hence there exists ℓ ∈coneL with ∥ℓ∥ < 1 such that \(\widetilde {b}-\gamma ^{-1}(\alpha \gamma -2^{-1}\sigma )\zeta m\in F_{1}(\widetilde {a}+\zeta \ell )\). But we have
and since \(\widetilde {c}\in B(\overline {z},\varepsilon ),\) we can apply the directional Aubin property of F2 (4.8) to find that
It follows that we can find n ∈coneN with ∥n∥ < 1 such that \(\widetilde {c}+\beta \zeta n\in F_{2}(\widetilde {a}+\zeta \ell )\).
Finally, since
we can use the directional Aubin property of G with respect to z (4.9) to get that
hence there exists p ∈ coneP with ∥p∥ < 1 such that
Observe that
hence
and we can use the second relation in the Ekeland variational principle to find that
Remark that
Then the previous relation becomes
Using this and (A.9), we get
a contradiction. This finishes the proof. □
Taking into account that the proof of the directional EVP is based on an iterative procedure, we can summarize the implications between the assertions in this work as follows:
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Cibulka, R., Durea, M., Panţiruc, M. et al. On the Stability of the Directional Regularity. Set-Valued Var. Anal 28, 209–237 (2020). https://doi.org/10.1007/s11228-019-00507-2
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DOI: https://doi.org/10.1007/s11228-019-00507-2