# Subdifferential Properties of Minimal Time Functions Associated with Set-Valued Mappings with Closed Convex Graphs in Hausdorff Topological Vector Spaces

Authors: Messaoud Bounkhel
Journal: Journal of Function Spaces and Applications
 (2013)

---

## Abstract

For a set-valued mappingM defined between two Hausdorff topological vector spaces E and F and with closed convex graph and for a given point (x,y)∈E×F, we study the minimal time function associated with the images of M and a bounded set Ω⊂F defined by 𝒯M,Ω(x,y):=inf{t≥0:M(x)∩(y+tΩ)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of 𝒯M,Ω at those points of (x,y)∈E×F (both cases: points in the graph gphM and points outside the graph). These results are used to prove, in terms of the minimal time function, various new characterizations of the convex tangent cone and the convex normal cone to the graph of M at points inside gphM and to the graph of the enlargement set-valued mapping at points outside gphM. Our results extend many existing results, from Banach spaces and normed vector spaces to Hausdorff topological vector spaces (Bounkhel, 2012; Bounkhel and Thibault, 2002; Burke et al., 1992; He and Ng, 2006; and Jiang and He 2009). An application of the minimal time function 𝒯M,Ω to the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.


---

## Body

## 1. Introduction

LetE and F be two Hausdorff topological vector spaces and Ω a bounded nonempty subset of F. Let M:E⇉F be a set-valued mapping with nonempty closed values. We associate to with M and Ω the so-called minimal time function defined by
(1)𝒯M,Ω(x,y)∶=inf{t≥0:M(x)∩(y+tΩ)≠∅},mmmmmmmmmmmmmmm,,,∀(x,y)∈E×F.The caseM(x)=S is a fixed closed subset of F that coincides with the minimal time function associated with S and Ω and is defined by
(2)TS,Ω(u)∶=inf{t≥0:S∩(u+tΩ)≠∅},∀u∈F.This functionTS,Ω plays an important role in variational analysis, optimization, control theory, Hamilton-Jacobi partial differential equations, approximation theory, and so forth; the reader can find more discussions in [1–13] and the references therein. The value 𝒯M,Ω(x0,y0) can be seen as the minimal time t≥0 associated with the differential inclusion y˙(τ)∈Ω, almost everywhere on [0,t], with y(0)=y0 and y(t)∈M(x0); in other words,
(3)𝒯M,Ω(x0,y0)={inf{t>0:∃y(·)s.t.y˙(τ)∈Ω,infma.e.[0,t],y(0)=y0,infmy(t)∈M(x0)},∀y0∉M(x0),0,∀y0∈M(x0).We notice that the minimal time function𝒯M,Ω covers many crucial functions in variational analysis: the distance function to images ΔM, the indicator function, the usual distance function, and the Minkowski function, by taking some particular cases of M and Ω as follows.(1)
IfM(x)={0}, then 𝒯M,Ω(x,y)=T{0},Ω(-y)=ρΩ(y) which coincides with the Minkowski (gauge) function associated with Ω; that is,
(4)ρΩ(y)=inf{t≥0:y∈tΩ},∀y∈F.(2)
IfΩ={0}, then 𝒯M,Ω(x,y)=TM(x),{0}(y)=ψM(x)(y)=ψgphM(x,y), which coincides with the indicator function associated with gphM. Here, ψS(u)=0 if u∈S and ψS(u)=+∞, otherwise.(3)
IfF is normed, M(x)={0}, and Ω=𝔹F is the closed unit ball in F, then 𝒯M,Ω(x,y)=T{0},𝔹F(y)=∥y∥.(4)
IfF is normed and Ω=𝔹F, then
(5)𝒯M,Ω(x,y)=𝒯M,𝔹(x)=infs∈M(x)∥y-s∥=ΔM(x,y)
which coincides with the distance function associated with the images of M (which has been successfully used in optimization theory, first by Clarke in [14] for Lipschitz set-valued mappings and later by various authors (see for instance the book [15] and the references therein) for set-valued mappings that are not necessarily locally Lipschitz).(5)
IfF is normed, M(x)=S, and Ω=𝔹F, then
(6)𝒯M,Ω(x,y)=TS,𝔹(y)=infs∈S∥y-s∥=dS(y),
which coincides with the usual distance function associated with S in F.From the above cases, we can see the importance of the study of the minimal time function𝒯M,Ω in normed vector spaces as well as in Hausdorff topological vector spaces. This type of study will unify the study of all the above functions.The case of minimal time function associated with a closed setS has been the subject of many recent works [1–3, 5–10, 12, 13]. To the best of our knowledge, the unique work studying the function 𝒯M,Ω is [16], in which the author studied the Fréchet subdifferential of 𝒯M,Ω in Banach spaces. We mention that there are no results on the directional derivatives and subdifferentials of 𝒯M,Ω in the Hausdorff topological vector spaces. Starting from this point, as a goal, we will develop a thorough study of the minimal time function in Hausdorff topological vector spaces in the convex setting. The nonconvex case will be the subject of a series of forthcoming works by the author. In the present paper, we extend various existing results on directional derivatives and subdifferentials of 𝒯M,Ω and their relationships to tangent and normal cones in Hausdorff topological vector spaces. The paper is organized as follows. Section 2 is devoted to state the main notations and definitions used throughout the paper. In Section 3, we prove our main results for points on the graph of M. The case of points outside the graph of M is studied in Section 4. In the last section we state an application of the minimal time function 𝒯M,Ω to the study of the calmness property of optimization problems in Hausdorff topological vector spaces.


## 2. Notations and Preliminaries

Throughout the whole paper (unless otherwise specified), we assume thatE and F are two Hausdorff topological vector spaces. We will denote by E* and F* the topological dual of E and F, respectively, and by 〈(·,·),(·,·)〉 the pairing between the spaces E×F and E*×F*.LetS be a nonempty closed convex subset of E×F and (x-,y-) a point in S. The convex tangent cone T(S;(x-,y-)) is defined by T(S;(x-,y-))=cl[ℝ+(S-(x-,y-))], where ℝ+ denotes the set of all nonnegative real numbers. The convex normal cone to S at (x-,y-) is defined as the negative polar of T(S;(x-,y-)); that is, N(S;(x-,y-)):=(T(S;(x-,y-)))-, where A-∶={(x*,y*)∈E*×F*:〈(x*,y*),(h,k)〉≤0,  for  all  (h,k)∈A}, for any subset A⊂E×F. Recall also from [14] that N(S;(x-,y-)) is also characterized by N(S;(x-,y-))={(x*,y*)∈E*×F*:〈(x*,y*),(x,y)-(x-,y-)〉≤0,  for  all  (x,y)∈S}.Letf be a convex function from E into ℝ∪{+∞} with u-∈domf:={u∈E:f(u)<∞}; the directional derivative (resp., the convex subdifferential) of f at u- is defined by
(7)f′(u-;v)∶=limt↓0t-1[f(u-+tv)-f(u-)],∀v∈E.({u*∈E*:〈u*,v〉≤f′(u-;v),∀v∈E}resp.,∂f(x-)mmm∶={u*∈E*:〈u*,v〉≤f′(u-;v),∀v∈E}mmm={u*∈E*:〈u*,u-u-〉≤f(u)-f(u-),∀u∈E}{u*∈E*:〈u*,v〉≤f′(u-;v),∀v∈E}).Note that for l.s.c convex functions we havef′(u-;h)=σ∂f(u-)(h), whenever u-∈core(domf) and E is a Hausdorff locally convex topological vector space. Here, σA denotes the support function associated with a closed subset A⊂E* defined by σA(h):=sup{〈x*,h〉:x*∈A} and core(S):={x∈E:for  allh∈E,∃tx,h>0,such  thatx+[0,tx,h]h⊂S} is called the algebraic interior of S (for more details on the core, we refer the reader to [17]).


## 3. Points on the Graph of the Set-Valued Mapping

Before starting the study of minimal time functions for set-valued mappings with closed convex graphs, we need to prove some results for general set-valued mappingsM:E⇉F with nonempty values (with graph not necessarily closed nor convex). These results have their own interests. We start with the following lemma which is needed in all the proofs of our work.Lemma 1.
LetE and F be Hausdorff topological vector spaces. Assume that Ω is a bounded set in F and M is a set-valued mapping with nonempty values in F. (1)
If𝒯M,Ω(x,y)=0, then y∈cl(M(x)).(2)
Conversely, for anyy∈M(x), one has TM,Ω(x,y)=0.(3)
If in addition0∈intΩ, then for any y∈cl(M(x)) one has TM,Ω(x,y)=0.Proof.
(1) Letx∈E with 𝒯M,Ω(x,y)=0, then for all n≥1, ∃tn≥0 such that 𝒯M,Ω(x,y)+1/n>tn≥𝒯M,Ω(x,y) with M(x)∩(y+tnΩ)≠∅; that is, there exists un∈Ω such that y+tnun∈M(x). Let V be any neighborhood of 0 in F. There exists W as a balanced neighborhood of 0 such that W⊂V. Since Ω is bounded, there exists α>0 such that Ω⊂αW. Hence, there exists wn∈W such that un=αwn. Clearly, for any n>α, we have tnα∈(0,1), and so by the fact that W is balanced we obtain
(8)tnun=tnαwn∈tnαW⊂W⊂V.
SinceV is arbitrary, we get the convergence of the sequence tnun to zero, that is, the convergence of the sequence y+tnun to y, and since y+tnun∈M(x), we get y∈cl(M(x)).
(2) Lety∈M(x) and let w∈Ω. Clearly, we have y=y+0·w∈(y+0·Ω)∩M(x). This ensures by definition that 𝒯M,Ω(x,y)=0.
(3) Assume now that0∈intΩ. Let y∈cl(M(x)) and fix any ϵ>0. Clearly, ϵΩ∈𝒩(0), and hence, (y+ϵΩ)∩M(x)≠∅. It follows that 0≤𝒯M,Ω(x,y)≤ϵ which ensures that 𝒯M,Ω(x,y)=0.Note that the assumption0∈intΩ in (3) cannot be removed when the values of M are open sets. Take M(x) as any open set, and take Ω={0}. Clearly, for all y∈bd(M(x)):=cl(M(x))∖M(x), we have 𝒯M,Ω(x,y)=ψM(x)(y)=+∞.As a corollary of parts (1) and (2) in Lemma1 we have the following.Corollary 2.
LetE and F be Hausdorff topological vector spaces, and let M:E⇉F be a set-valued mapping with nonempty closed values. Assume that Ω is a bounded set in F. Then 𝒯M,Ω(x,y)=0 if and only if y∈M(x).The following lemma characterizes the convexity of the graph of set-valued mappings in terms of the convexity of its associated minimal time function in Hausdorff topological vector spaces. It extends the well known characterization of the convexity (see, for instance, the lemma on page 53 in [14]) in terms of the distance function in normed vector spaces as well as the one in terms of the indicator function.Lemma 3.
LetE and F be two Hausdorff topological vector spaces, M:E⇉F a set-valued mapping with nonempty closed values, and Ω a bounded convex set in F. Then, 𝒯M,Ω(·,·) is convex on its domain if and only if the graph gphM is convex.Proof.
Assume that𝒯M,Ω(·,·) is convex on its domain; that is,
(9)𝒯M,Ω(λ(x1,y1)+(1-λ)(x2,y2))≤λ𝒯M,Ω(x1,y1)+(1-λ)𝒯M,Ω(x2,y2),
for all (x1,y1),(x2,y2)∈dom𝒯M,Ω, and all λ∈[0,1]. Therefore, by Corollary 2, for any (x1,y1),(x2,y2)∈gphM, we have 𝒯M,Ω(x1,y1)=𝒯M,Ω(x2,y2)=0, and so by the previous inequality, we obtain 𝒯M,Ω(λ(x1,y1)+(1-λ)(x2,y2))=0, and hence by Corollary 2, once again we get λy1+(1-λ)y2∈M(λx1+(1-λ)x2); that is, λ(x1,y1)+(1-λ)(x2,y2)∈gphM, which means that gphM is convex. Conversely, let (xi,yi)∈dom𝒯M,Ω (i=1,2), and let any λ∈[0,1]. Fix any ϵ>0. Since 𝒯M,Ω(x1,y1)<∞ and 𝒯M,Ω(x2,y2)<∞, we can find that wi∈Ω, (i=1,2) and ti≥0 (i=1,2) such that ti<𝒯M,Ω(xi,yi)+ϵ and yi+tiwi∈M(xi) (i=1,2). So the convexity of gphM ensures that λ(x1,y1+t1w1)+(1-λ)(x2,y2+t2w2)∈gphM. Hence,
(10)[λy1+(1-λ)y2]+[λt1+(1-λ)t2]w-∈M(λx1+(1-λ)x2),
where w-:=[(λt1/(λt1+(1-λ)t2))w1+((1-λ)t2/(λt1+(1-λ)t2))w2]. On the other hand, by the convexity of Ω, we have
(11)w-=λt1λt1+(1-λ)t2w1+(1-λ)t2λt1+(1-λ)t2w2∈Ω,
which ensures that
(12)([λy1+(1-λ)y2]+[λt1+(1-λ)t2]Ω)∩M(λx1+(1-λ)x2)≠∅.
This ensures that(13)𝒯M,Ω(λ(x1,y1)+(1-λ)(x2,y2))=𝒯M,Ω(λx1+(1-λ)x2,λy1+(1-λ)y2)≤λt1+(1-λ)t2<λ𝒯M,Ω(x1,y1)+(1-λ)𝒯M,Ω(x2,y2)+ϵ,
for any ϵ>0. Thus, taking ϵ→0 completes the proof.Note that the convexity ofΩ, in the proof of Lemma 3, is needed only in one direction (reverse implication); that is, the convexity of 𝒯M,Ω ensures the convexity of the graph of M even when Ω is not convex.Now, we are looking for the lower semicontinuity of the minimal time function in Hausdorff topological vector spaces.Proposition 4.
LetE and F be two Hausdorff topological vector spaces. Assume that Ω is compact in F and gphM is closed in E×F. Then 𝒯M,Ω is lower semicontinuous at any (x-,y-)∈dom𝒯M,Ω.Proof.
Let(x-,y-)∈dom𝒯M,Ω. We have to prove that
(14)𝒯M,Ω(x-,y-)≤liminf(x,y)→(x-,y-)𝒯M,Ω(x,y).
The caseliminf(x,y)→(x-,y-)𝒯M,Ω(x,y)=∞ is obvious, so we assume that
(15)α∶=liminf(x,y)→(x-,y-)𝒯M,Ω(x,y)<∞.
Let(xj,yj)j∈J be a net satisfying the above liminf; that is, (xj,yj)→(x-,y-) and α=limj𝒯M,Ω(xj,yj). Let (tj)j∈J be a net of positive real numbers satisfying tj→0. By the definition of the minimal time function, we can find for any j∈J a real number δj such that
(16)0≤𝒯M,Ω(xj,yj)≤δj<𝒯M,Ω(xj,yj)+tj,(yj+δjΩ)∩M(xj)≠∅,∀j∈J.
Hence, for any j∈J, there exists wj∈Ω such that (xj,yj+δjwj)∈gphM. Using the compactness of Ω, we get the convergence of a subnet of (wj)j to some point w∈Ω. Without loss of generality the subnet is still denoted by (wj)j. Therefore, (xj,yj+δjwj)→(x-,y-+αw)∈gphM (by the closedness of gphM). This shows that (y-+αΩ)∩M(x-)≠∅, which ensures that 𝒯M,Ω(x-,y-)≤α, and hence, the proof is complete.An inspection of the proof of the previous proposition shows that the conclusion is still valid under the assumptions thatΩ is weakly compact and gphM is weakly closed. Consequently, we have the two following corollaries.Corollary 5.
LetE and F be two Hausdorff topological vector spaces. Assume that Ω is weakly compact in F and gphM is weakly closed in E×F. Then, 𝒯M,Ω is lower semicontinuous at any (x-,y-)∈dom𝒯M,Ω.Corollary 6.
Assume thatF is a reflexive Banach space, Ω is closed convex bounded set in F, and gphM is closed convex set in E×F. Then 𝒯M,Ω is lower semicontinuous at any (x-,y-)∈dom𝒯S,Ω.The following lemma is technical and is needed in some forthcoming proofs. Its proof is straightforward.Lemma 7.
LetE and F be two Hausdorff topological vector spaces, M:E⇉F a set-valued mapping, and Ω a nonempty set in F. Then,
(17)t-1𝒯M,Ω(x+th,y+tk)=Tt-1[M(x+th)-y],Ω(k),∀(x,y),(h,k)∈E×F,∀t>0.Now, we are going to establish our main results of this section. We prove some formulas and relationships between the directional derivative and the convex subdifferential of𝒯M,Ω and the convex tangent cone and the convex normal cone of gphM in Hausdorff topological vector spaces at points in gphM. We associate with the set-valued mapping M a new set-valued mapping T~M:E⇉F with graph gphT~M:=T(gphM;(x-,y-)); that is, k∈T~M(h) if and only if (h,k)∈T(gphM;(x-,y-)).Theorem 8.
LetE and F be two Hausdorff topological vector spaces and M:E⇉F a set-valued mapping. Assume that gphM is a nonempty closed convex set in E×F, (x-,y-)∈gphM, and Ω is a bounded convex set in F. Then, one has(1)
(18)𝒯M,Ω′((x-,y-);(h,k))=limt→0+t-1𝒯M,Ω(x-+th,y-+tk)=limt→0+Tt-1[M(x-+th)-y-],Ω(k),mmmmmmm∀(h,k)∈E×F;(2)
(19)∂𝒯M,Ω(x-,y-)=N(gphM;(x-,y-))∩{(x*,y*)∈E*×F*:σΩ(-y*)≤1};(3)
if, in addition,(x-,y-)∈core(dom𝒯M,Ω), E×F is locally convex, then
(20)𝒯M,Ω′((x-,y-);(h,k))=𝒯T~M,Ω(h,k),mmmmmmmmmm∀(h,k)∈E×F;(4)
(21)T(gphM;(x-,k-))={(h,k)∈E×F:𝒯M,Ω′((x-,y-);(h,k))=0};(5)
and(22)N(gphM;(x-,y-))=clw*(ℝ+∂𝒯M,Ω((x-,y-))).Proof.
(1) By Lemma3, we have the convexity of 𝒯M,Ω, and so 𝒯M,Ω′((x-,y-);(h,k)) is well defined and hence, Corollary 2 and Lemma 7 complete the proof of part (1).
(2) Let(x*,y*)∈N(gphM;(x-,y-)) with σΩ(-y*)≤1. Then, by definition of the convex normal cone and by Corollary 2, we have
(23)〈(x*,y*),(x-x-,y-y-)〉≤0=𝒯M,Ω(x,y)-𝒯M,Ω(x-,y-),mmmmmmmmmm∀(x,y)∈gphM.
Assume now that(x,y)∉gphM, and let ϵ>0. If 𝒯M,Ω(x,y)=∞, then the previous inequality holds obviously. Assume that 𝒯M,Ω(x,y)<∞. By definition of minimal time function, there exists δ>0 and w∈Ω such that
(24)0<𝒯M,Ω(x,y)≤δ<𝒯M,Ω(x,y)+ϵwithy+δw∈M(x).
Hence, we get by the definition of convex normal cones that(25)〈(x*,y*),(x,y+δw)-(x-,y-)〉≤0.
Therefore, combining this inequality with the assumptionσΩ(-y*)≤1, we obtain that
(26)〈(x*,y*),(x-x-,y-y-)〉=〈(x*,y*),(x-x-,y+δw-y-)〉+δ〈-y*,w〉≤δσΩ(-y*)≤δ<𝒯M,Ω(x,y)+ϵ,
which ensures, by taking ϵ→0, that
(27)〈(x*,y*),(x-x-,y-y-)〉≤𝒯M,Ω(x,y)-𝒯M,Ω(x-,y-),∀(x,y)∉gphM.
Thus, this inequality holds for any(x,y)∈E×F which means that (x*,y*)∈∂𝒯M,Ω(x-,y-).
Conversely, let(x*,y*)∈∂𝒯M,Ω(x-,y-). Then, by definition, we have
(28)〈(x*,y*),(x,y)-(x-,y-)〉≤𝒯M,Ω(x,y)-𝒯M,Ω(x-,y-),∀(x,y)∈E×F.
Using Corollary2, we have
(29)〈(x*,y*),(x,y)-(x-,y-)〉≤0∀(x,y)∈gphM.
This means that(x*,y*)∈N(gphM;(x-,y-)). We have to prove that σΩ(-y*)≤1. Let w∈Ω and δ>0. Then, y-=y--δw+δw∈M(x-)∩(y--δw+δΩ); that is, M(x-)∩(y--δw+δΩ)≠∅, which ensures that 𝒯M,Ω(x-,y--δw)≤δ. Therefore, by (28), we have
(30)〈-y*,δw〉=〈(x*,y*),(x-,y--δw)-(x-,y-)〉≤𝒯M,Ω(x-,y--δw)≤δ,
which ensures that
(31)〈-y*,w〉≤1,∀w∈Ω.
This givesσΩ(-y*)≤1, and hence, the proof of (2) is complete.
(3) Let(h,k)∈E×F. Assume first that (h,k)∈T(gphM;(x-,y-))=gphT~M. Then, by Corollary 2, we have 𝒯T~M,Ω(h,k)=0. Also, we have 〈(x*,y*),(h,k)〉≤0, for all (x*,y*)∈N(gphM,(x-,y-)). Hence, by (2), we have 〈(x*,y*),(h,k)〉≤0, for all (x*,y*)∈∂𝒯M,Ω(x-,y-), and so σ∂𝒯M,Ω(x-,y-)(h,k)≤0. Now we use the assumption (x-,y-)∈core(dom𝒯M,Ω) to write 𝒯M,Ω′((x-,y-);(h,k))=σ∂𝒯M,Ω(x-,y-)(h,k) and so 𝒯M,Ω′((x-,y-);(h,k))≤0. Hence by Part (1) we obtain the needed equality
(32)𝒯M,Ω′((x-,y-);(h,k))=0=𝒯T~M,Ω(h,k).
Assume now that (h,k)∉T(gphM;(x-,y-)). Without loss of generality, we assume that 𝒯T~M,Ω(h,k)<∞. Put α:=𝒯T~M,Ω(h,k). Fix any δ≥0 with (k+δΩ)∩T~M(h)≠∅. There exists w∈Ω such that k+δw∈T~M(h); that is, (h,k+δw)∈T(gphM;(x-,y-)). Therefore, by Part (2), for any (x*,y*)∈∂𝒯M,Ω(x-,y-), we have σΩ(-y*)≤1 and (x*,y*)∈N(gphM;(x-,y-)), and hence, we get the following
(33)〈(x*,y*),(h,k)〉=〈(x*,y*),(h,k+δw)〉+δ〈-y*,w〉≤δσΩ(-y*)≤δ,∀(x*,y*)∈∂𝒯M,Ω(x-,y-).
This ensures that(34)𝒯M,Ω′((x-,y-),(h,k))=σ∂𝒯M,Ω(x-,y-)(h,k)≤δ,
and since δ is taken arbitrarily, we take the infimum to obtain that
(35)𝒯′M,Ω((x-,y-),(h,k))≤inf{δ>0:such  that(k+δΩ)∩T~M(h)≠∅}=𝒯T~M,Ω(h,k).
Let us prove the reverse inequality(36)𝒯T~M,Ω(h,k)≤𝒯M,Ω′((x-,y-),(h,k)).
Obviously, we assume that𝒯M,Ω′((x-,y-),(h,k))<∞. Put α:=𝒯M,Ω′((x-,y-),(h,k)). Let (tj)j be a net of positive real numbers converging to zero and satisfying the limit in the definition of the directional derivative; that is, α=limjTtj-1[M(x-+tjh)-y-],Ω(k). Let δj>0 and wj∈Ω such that
(37)Ttj-1[M(x-+tjh)-y-],Ω(k)≤δj<Ttj-1[M(x-+tjh)-y-],Ω(k)+tj,k+δjwj∈tj-1[M(x-+tjh)-y-];that  is,(h,k+δjwj)∈tj-1[gphM-(x-,y-)].
SincegphM is closed convex, we have T(gphM;(x-,y-))=cl(ℝ+[gphM-(x-,y-)]), and so (h,k+δjwj)∈T(gphM;(x-,y-)), for all j; that is, k+δjwj∈T~M(h), for all j.
On the other hand we clearly have𝒯T~M,Ω(h,k)≤ρΩ(s-k), for all s∈T~M(h). Consequently, we obtain for all j that
(38)𝒯T~M,Ω(h,k)≤ρΩ((k+δjwj)-k)=δjρΩ(wj)𝒯T~M,Ω(h,k)≤δj<Ttj-1[M(x-+tjh)-y-],Ω(k)+tj,∀j.
Taking the limit on this inequality, we obtain that
(39)𝒯T~M,Ω(h,k)≤𝒯M,Ω′((x-,y-);(h,k)),
thus completing the proof of (3).
(4) The inclusionT(gphM,(x-,y-))⊂{(h,k)∈E×F:𝒯M,Ω′((x-,y-);(h,k))=0} follows from the assertion (3) and Corollary 2. Conversely, let (h,k)∈E×F with 𝒯M,Ω′((x-,y-);(h,k))=0. Let (tj) be a net of positive real numbers converging to zero and satisfying the limit 𝒯M,Ω′((x-,y-);(h,k))=limtj→0𝒯tj-1[M(x-+tjh)-y-],Ω(k). For any j∈J, we can find that δj>0 and wj∈Ω such that
(40)𝒯tj-1[M(x-+tjh)-y-],Ω(k)≤δj<𝒯tj-1[M(x-+tjh)-y-],Ω(k)+tj,k+δjwj∈tj-1[M(x-+tjh)-y-];that  is,(h,k+δjwj)∈tj-1[gphM-(x-,y-)].
Since the graph ofM is closed convex, we obtain that
(41)(h,k+δjwj)∈tj-1[gphM-(x-,y-)]⊂T(gphM;(x-,y-)).
On the other hand, we havelimjδj=limtj→0𝒯tj-1[M(x-+tjh)-y-],Ω(k)=0, and hence, by the boundedness of the set Ω, we get the convergence of δjwj to zero. This ensures the convergence of k+δjwj→k which ensures by the closedness of the tangent cone that (h,k)∈T(gphM,(x-,y-)), and hence, the proof of (4) is finished.
(5) The inclusionclw*(ℝ+∂𝒯M,Ω((x-,y-)))⊂N(gphM;(x-,y-)) follows directly from part (2). So we prove the reverse one. To do that, we start by proving the inclusion
(42)(∂𝒯M,Ω((x-,y-)))-⊂T(gphM;(x-,y-)).
Fix any(h,k)∈(∂𝒯M,Ω((x-,y-)))-. Then
(43)〈(x*,y*);(h,k)〉≤0,∀(x*,y*)∈∂𝒯M,Ω((x-,y-)).
Since∂𝒯M,Ω((x-,y-))≠∅ (because 0∈∂𝒯M,Ω((x-,y-))) and (x-,y-)∈core(dom𝒯M,Ω), then 𝒯M,Ω′((x-,y-);(h,k))=σ∂𝒯M,Ω((x-,y-))(h,k). Thus
(44)𝒯M,Ω′((x-,y-);(h,k))=σ∂𝒯M,Ω((x-,y-))(h,k)≤0.
Since the reverse inequality 𝒯M,Ω′((x-,y-);(h,k))≥0 follows directly from part (1), then we get the equality form 𝒯M,Ω′((x-,y-);(h,k))=0, and hence, part (4) ensures that (h,k)∈T(gphM;(x-,y-)). Therefore, (∂𝒯M,Ω((x-,y-)))-⊂T(gphM;(x-,y-)). Since clearly we have (∂𝒯M,Ω((x-,y-)))-=(clw*(ℝ+∂𝒯M,Ω((x-,y-))))-, then
(45)(clw*(ℝ+∂𝒯M,Ω((x-,y-))))-⊂T(gphM;(x-,y-)),
which ensures that
(46)N(gphM;(x-,y-))=(T(gphM;(x-,y-)))-⊂((clw*(ℝ+∂𝒯M,Ω((x-,y-))))-)-=clw*(ℝ+∂𝒯M,Ω((x-,y-))).
The last equality follows from the fact thatE×F is locally convex. Thus, the proof of the theorem is complete.Many corollaries can be deduced from this theorem. We state the following one [18] by taking Ω to be the closed unit ball of a normed vector space F and M to be a constant set-valued mapping; that is, M(x)=S.Corollary 9.
Assume thatF is a normed vector space, S is a closed convex subset in F, and y-∈S. Then, one has
(47)(1)dS′(y-;k)=limt→0+t-1dS(y-+tk)(1)dS′(y-;k)=limt→0+dt-1[S-y-](k)∀k∈F;(2)∂dS(y-)=N(S;y-)∩{y*∈F*:∥y*∥≤1};(3)dS′(y-;k)=dT(S;y-)(k),∀k∈F;(4)T(S;y-)={k∈F:dS′(y-;k)=0}.


## 4. Points outside the Graph of the Set-Valued Mapping

Now, we consider those points outside the graph of set-valued mappingsM with nonempty closed values. Let (x-,y-)∉gphM. Clearly, by Corollary 2, we have 𝒯M,Ω(x-,y-)>0. Denote r:=𝒯M,Ω(x-,y-)>0, and define the enlargement set-valued mapping
(48)Mr(x)∶={y∈F:𝒯M,Ω(x,y)≤r}.
Clearly, the graph of Mr is defined as gphMr:={(x,y)∈E×F:𝒯M,Ω(x,y)≤r}. The following lemma is needed. It extends many existing results from normed spaces to Hausdorff topological vector spaces and from the case of sets to the case of set-valued mapping (see for instance [15, 19]). Obviously, we have y-∈Mr(x-) and M(x-)⊂Mr(x-). We have also to point out that, due to Lemma 3, the convexity of the graph gphM ensures the convexity of the graph of Mr whenever Ω is bounded convex. However, the l.s.c. of 𝒯M,Ω ensures the closedness of the graph of gphMr. It follows then, from Lemma 3, the convexity of the minimal time function 𝒯Mr,Ω whenever 𝒯M,Ω is lower semicontinuous and Ω is bounded convex.Lemma 10.
LetE and F be two Hausdorff topological vector spaces. Assume that M has nonempty closed values, (x-,y-)∉gphM, and Ω is a bounded convex set. Then, for any (x,y)∈E×F we have
(49)𝒯M,Ω(x,y)≤𝒯Mr,Ω(x,y)+r
and for any (x,y)∉gphMr we have
(50)𝒯M,Ω(x,y)=𝒯Mr,Ω(x,y)+r.Proof.
First we prove the inequality(51)𝒯M,Ω(x-,y)≤𝒯Mr,Ω(x-,y)+r,foranyy∈F.
Assume that (x,y)∈E×F with 𝒯Mr,Ω(x,y)<∞. The case 𝒯Mr,Ω(x,y)=∞ is obvious. Fix any δ≥0 satisfying [y+δΩ]∩Mr(x)≠∅. Then, there exists w∈Ω such that y+δw∈Mr(x); that is, 𝒯M,Ω(x,y+δw)≤r. Then, for any ϵ>0, we can choose λ>0 and z∈Ω such that
(52)𝒯M,Ω(x,y+δw)≤λ<𝒯M,Ω(x,y+δw)+ϵ,y+δw+λz∈M(x).
By convexity of Ω, we have δw+λz∈(δ+λ)Ω, and hence,
(53)y+δw+λz∈M(x)∩[y+(δ+λ)Ω]≠∅.
Therefore,
(54)𝒯M,Ω(x,y)≤δ+λ<δ+𝒯M,Ω(x,y+δw)+ϵ<δ+r+ϵ.
Since δ is taken arbitrarily satisfying (y+δΩ)∩Mr(x)≠∅, we obtain by taking the infimum over all δ the inequality
(55)𝒯M,Ω(x,y)≤𝒯Mr,Ω(x,y)+r+ϵ,∀ϵ>0,
and by taking ϵ→0 we get the first desired inequality
(56)𝒯M,Ω(x,y)≤𝒯Mr,Ω(x,y)+r.
Assume now that y∉Mr(x), and let us prove the reverse inequality; that is, 𝒯Mr,Ω(x,y)+r≤𝒯M,Ω(x,y). The case 𝒯M,Ω(x,y)=∞ is obvious so we suppose that 𝒯M,Ω(x,y)<∞. Take arbitrarily a nonnegative number δ≥0 for which [y+δΩ]∩M(x)≠∅. Then, there exists w∈Ω such that y+δw∈M(x). Clearly, 𝒯M,Ω(x,y)≤δ, and since y∉Mr(x), we have δ>r. Therefore,
(57)y+(δ-r)w+rw∈M(x)∩[y+(δ-r)w+rΩ]≠∅,
and so
(58)𝒯M,Ω(x,y+(δ-r)w)≤r,thatis,y+(δ-r)w∈Mr(x).
Thus,(59)y+(δ-r)w∈Mr(x)∩[y+(δ-r)Ω]≠∅.
Then,(60)𝒯Mr,Ω(x,y)≤δ-r.
Sinceδ is taken arbitrarily with [y+δΩ]∩M(x)≠∅, we can take the infimum over all those δ, and hence, we obtain that
(61)𝒯Mr,Ω(x,y)+r≤𝒯M,Ω(x,y).
The proof is complete.The previous lemma extends Lemma  3.4 in [2] from the case of sets to the case of set-valued mappings in Hausdorff topological vector spaces. Also, it extends the inequality (4.41) on page 97 in [15] from the case where F is a normed vector space to the case where F is a Hausdorff topological vector space and from the case of distance function to images ΔM to the case of minimal time function 𝒯M,Ω.The first consequence of Lemma10 is the following proposition in which we establish a relationship between the directional derivatives of 𝒯M,Ω and 𝒯Mr,Ω.Proposition 11.
LetE and F be two Hausdorff topological vector spaces. Assume that M has closed convex graph, (x-,y-)∉gphM, and Ω is a bounded convex set. Let r:=𝒯M,Ω(x-,y-)>0. Then, for any (h,k)∈E×F, one has
(62)𝒯M,Ω′((x-,y-);(h,k))≤𝒯Mr,Ω′((x-,y-);(h,k)).Proof.
It follows, directly from the first part of the previous lemma, the fact that𝒯Mr,Ω(x-,y-)=0 and the definition of the directional derivative.In the following theorem, we characterize the convex tangent cone of the graph of the enlargement set-valued mappingMr as the set of all directions (h,k) in E×F for which the directional derivative of TM,Ω is nonnegative.Theorem 12.
LetE and F be two Hausdorff topological vector spaces. Assume that M has a closed convex graph, (x-,y-)∉gphM, and Ω is a bounded convex set. Assume that 𝒯M,Ω is l.s.c. at (x-,y-). Then, one has
(63)T(gphMr;(x-,y-))={(h,k)∈E×F:𝒯Mr,Ω′((x-,y-);(h,k))=0}={(h,k)∈E×F:𝒯M,Ω′((x-,y-);(h,k))≤0}.Proof.
The first equality follows directly from part (4) in Theorem8. The direct inclusion of the second equality follows directly from the previous proposition. So it remains to prove the inclusion
(64){(h,k)∈E×F:𝒯M,Ω′((x-,y-);(h,k))≤0}⊂T(gphMr;(x-,y-)).
Let(h,k)∈E×F with 𝒯M,Ω′((x-,y-);(h,k))≤0. Let (tj)j be a net in (0,∞) converging to 0 and satisfying the limit
(65)𝒯M,Ω′((x-,y-);(h,k))=limjtj-1[𝒯M,Ω(x-+tjh,y-+tjk)-𝒯M,Ω(x-,y-)].
Use the lower semicontinuity of𝒯M,Ω at (x-,y-) and the assumption y-∉M(x-) to find some j0∈J such that
(66)𝒯M,Ω(x-+tjh,y-+tjk)>2tj2>0,for  anyj≥j0.
Using now the assumption𝒯M,Ω′((x-,y-);(h,k))≤0 ensures the existence of some j1≥j0 such that
(67)𝒯M,Ω(x-+tjh,y-+tjk)≤r+tj2,for  anyj≥j1.
Thus, combining (66) and (67), we obtain
(68)0<2tj2<𝒯M,Ω(x-+tjh,y-+tjk)≤r+tj2,for  anyj≥j1.
Use now the definition of the minimal time function to choose for anyj≥j1 some wj∈Ω and δj≥0 such that
(69)𝒯M,Ω(x-+tjh,y-+tjk)≤δj<𝒯M,Ω(x-+tjh,y-+tjk)+tj2,y-+tjk+δjwj∈M(x-+tjh).
Hence,(70)0≤δj-2tj2<r,∀j≥j1.
Letzj:=y-+tjk+2tj2wj and kj:=k+2tjwj. Then,
(71)zj+(δj-2tj2)wj=y-+tjk+δjwj∈M(x-+tjh)∩(zj+(δj-2tj2)Ω)≠∅.
So𝒯M,Ω(x-+tjh,zj)≤δj-2tj2≤r; that is, zj∈Mr(x-+tjh).
Therefore,(72)y-+tjkj=y-+tjk+2tj2wj∈Mr(x-+tjh),
and so
(73)(h,kj)∈tj-1[gphMr-(x-,y-)]⊂T(gphMr;(x-,y-)).
Now, the boundedness ofΩ ensures the convergence of 2tjwj to zero, which gives the convergence of kj to k. Thus, by the closedness of the convex tangent cone, we get (h,k)∈T(gphMr,(x-,y-)), and hence, the proof is finished.Various corollaries can be deduced from the above theorem by taking special cases ofM and Ω and the spaces E and F. We state the following one which has been proved in Corollary  4.4 in [15] (see also [20]) by taking F to be a normed vector space, Ω to be the closed unit ball of F, and M to be a constant set-valued mapping; that is, M(x)=S. In this case, Mr(x) coincides with the enlargement set S(r) defined by S(r)={y∈F:dS(y)≤r}.Corollary 13.
LetF be a normed vector space, S a closed convex subset in F, and y-∈S. Then, one has
(74)dS′(y-;k-)≤dS(r)′(y-;k-)≤dT(S(r);y-)(k-),∀k-∈F,T(S(r);y-)={k∈F:dS(r)′(y-;k-)=0}T(S(r);y-)={k∈F:dS′(y-;k-)≤0}.Now, we turn to establish some relationships between the convex normal cone togphMr and the convex subdifferential of 𝒯M,Ω at (x-,y-)∉gphM.Theorem 14.
LetE and F Hausdorff topological vector spaces, M a set-valued mapping with nonempty closed values, Ω a bounded convex set in E, and (x-,y-)∉gphM. Assume that 𝒯M,Ω is l.s.c. at (x-,y-), E and F are locally convex, and ∂𝒯M,Ω((x-,y-))≠∅. Then,
(75)N(gphMr;(x-,y-))=clw*(ℝ+∂𝒯Mr,Ω((x-,y-)))=clw*(ℝ+∂𝒯M,Ω((x-,y-))).Proof.
The first equality follows directly from Theorem8. We will prove the equality N(gphMr;(x-,y-))=clw*(ℝ+∂𝒯M,Ω((x-,y-))). Assume that ∂𝒯M,Ω(x-,y-)≠∅. First, we prove that
(76)∂𝒯M,Ω(x-,y-)⊂N(gphMr;(x-,y-)).
Since∂C𝒯M,Ω(x-,y-)≠∅, we take some (x*,y*)∈E*×F* such that
(77)〈(x*,y*),(h,k)〉≤𝒯M,Ω′((x-,y-);(h,k)),∀(h,k)∈E×F.
Hence, for any(h,k)∈T(gphMr;(x-,y-)), we have 𝒯M,Ω′((x-,y-);(h,k))≤0 (by Theorem 12), and so
(78)〈(x*,y*),(h,k)〉≤𝒯M,Ω′((x-,y-);(h,k))≤0;
that is, (x*,y*)∈(T(gphMr;(x-,y-)))-=N(gphMr;(x-,y-)). Since N(gphMr;(x-,y-)) is a closed convex cone in E*×F*, we obtain that
(79)clw*(ℝ+∂𝒯M,Ω(x-,y-))⊂N(gphMr;(x-,y-)).
Conversely, we have to prove that(80)N(gphMr;(x-,y-))⊂clw*(ℝ+∂𝒯M,Ω(x-,y-)).
We will prove that the negative polar ofclw*(ℝ+∂𝒯M,Ω(x-,y-)) is included in T(gphMr;(x-,y-))=(N(gphMr;(x-,y-)))-; that is, (by Theorem 12)
(81)(clw*(ℝ+∂𝒯M,Ω(x-,y-)))-⊂{(h,k)∈E×F:𝒯M,Ω′((x-,y-);(h,k))≤0}.
Note first that(82)(clw*(ℝ+∂𝒯M,Ω(x-,y-)))-=(∂𝒯M,Ω(x-,y-))-.
So consider any(h,k)∈(∂𝒯M,Ω(x-,y-))-. Then, we have
(83)〈(x*,y*),(h,k)〉≤0,∀(x*,y*)∈∂𝒯M,Ω(x-,y-).
Hence, (since∂𝒯M,Ω(x-,y-))≠∅ and (x-,y-)∈core(domTM,Ω))
(84)𝒯M,Ω′((x-,y-);(h,k))=σ∂𝒯M,Ω(x-,y-)(h,k)=sup(x*,y*)∈∂𝒯M,Ω(x-,y-)〈(x*,y*),(h,k)〉≤0,
which ensures by Theorem 12 that (h-,k-)∈T(gphMr;(x-,y-)). Thus, completing the proof.The following corollary extends Corollary  1 in [14] from the case of sets to the case of set-valued mappings and from normed vector spaces to Hausdorff Topological vector spaces. It says that if we put the calmness of 𝒯M,Ω instead of its l.s.c. and the assumption 0∉∂𝒯M,Ω(x-,y-) instead of the assumption ∂𝒯M,Ω(x-,y-)≠∅ in Theorem 14 we may remove the weak star closedness in the second equation in Theorem 14. Recall that f is said to be calm at u- if there exists a closed balanced neighborhood V of zero and U∈𝒩(u-) such that
(85)|f(u)-f(u-)|≤ρV(u--u),∀u∈U.Corollary 15.
LetE and F Hausdorff topological vector spaces, M:E⇉F a set-valued mapping with a closed convex graph, Ω be a bounded convex set in F, and (x-,y-)∉gphM. If 𝒯M,Ω is calm at (x-,y-) with 0∉∂𝒯M,Ω(x-,y-), then one has
(86)N(gphMr;(x-,y-))=ℝ+∂𝒯M,Ω(x-,y-).Proof.
Using the calmness of𝒯M,Ω at (x-,y-) and Banach-Alaoglu theorem (see Theorem  3.5 in [21]), we get the weak star compactness of ∂C𝒯M,Ω(x-,y-) in E*×F*. Therefore, the assumption 0∉∂C𝒯M,Ω(x-,y-) with the weak star compactness of ∂C𝒯M,Ω(x-,y-) ensures the weak star closedness of the cone generated by ∂C𝒯M,Ω(x-,y-); that is, clw*(ℝ+∂C𝒯M,Ω(x-,y-))=ℝ+∂C𝒯M,Ω(x-,y-). Thus, the conclusion follows from Theorem 14.The next results depend on the nonemptiness of the minimum set for the minimal time function𝒯M,Ω defined as follows
(87)πM(x),Ω(u)∶={y∈M(x):𝒯M,Ω(x,y)=ρΩ(y-u)}.
We begin with the following lemma.Lemma 16.
LetE and F be two Hausdorff topological vector spaces, M:E⇉F a set-valued mapping with closed convex graph, Ω a bounded convex set in F, and (x-,y-)∉gphM with 𝒯M,Ω(x-,y-)<∞ and πM(x-),Ω(y-)≠∅. For every y∈πM(x-),Ω(y-) and for every t∈(0,1], one has
(88)𝒯M,Ω(x-,ty+(1-t)y-)=(1-t)𝒯M,Ω(x-,y-).
This ensures thaty∈πM(x-),Ω(ty+(1-t)y-), for any t∈(0,1].Proof.
Lety∈πM(x-),Ω(y-). Then y∈M(x-) with 𝒯M,Ω(x-,y-)=ρΩ(y-y-)<∞. First, we prove the inequality
(89)𝒯M,Ω(x-,ty+(1-t)y-)≤(1-t)𝒯M,Ω(x-,y-),∀t∈(0,1].
Putα∶=ρΩ(y-y-)>0, and fix any t∈(0,1]. Clearly, by definition of the Minkowski function, we can find for any ϵ>0 some α0≥0 such that y-y-∈α0Ω and α0∈[α,α+ϵ), and so
(90)y=ty+(1-t)y-+α0(1-t)(y-y-)α0∈M(x-)y∩[ty+(1-t)y-+α0(1-t)Ω]≠∅,
which ensures that
(91)𝒯M,Ω(x-,ty+(1-t)y-)≤α0(1-t)<(1-t)[ρΩ(y-y-)+ϵ]≤(1-t)𝒯M,Ω(x-,y-)+ϵ,∀ϵ>0.
Taking ϵ→0+ finishes the proof of the first inequality.
By the first part, we have𝒯M,Ω(x-,ty+(1-t)y-)<∞ whenever 𝒯M,Ω(x-,y-)<∞ and so for any ϵ>0, we can find some w∈Ω, some δ>0, and some δ0>0 such that
(92)𝒯M,Ω(x-,ty+(1-t)y-)≤δ<𝒯M,Ω(x-,ty+(1-t)y-)+ϵ,ty+(1-t)y-+δw∈M(x-),y-y-∈δ0Ωwith0<ρΩ(y-y-)≤δ0<ρΩ(y-y-)+ϵ.
Then, by convexity ofΩ, we have δ0t((y-y-)/δ0)+δw∈(δ0t+δ)Ω, and so
(93)y-+t(y-y-)+δw=y-+δ0t(y-y-)δ0+δw∈M(x-)∩[y-+(δ0t+δ)Ω]≠∅.
Hence,(94)𝒯M,Ω(x-,y-)≤δ0t+δ<t(ρΩ(y-y-)+ϵ)+𝒯M,Ω(x-,ty+(1-t)y-)+ϵ<t𝒯M,Ω(x-,y-)+𝒯M,Ω(x-,ty+(1-t)y-)+2ϵ,
and so by taking ϵ→0+, we obtain that
(95)(1-t)𝒯M,Ω(x-,y-)≤𝒯M,Ω(x-,ty+(1-t)y-).
This completes the proof of the lemma.We use this lemma to prove the following proposition on directional derivatives and convex subdifferentials of𝒯M,Ω at points outside the graph.Proposition 17.
Assume that the assumptions of Lemma16 are fulfilled. Then, for every (x-,y-)∉gphM and for every y∈πM(x-),Ω(y-), one has
(96)𝒯M,Ω′((x-,y-);(0,y-y-))=-𝒯M,Ω(x-,y-),∂𝒯M,Ω(x-,y-)⊂{(x*,y*)∈E*×F*:σΩ(-y*)=1}.Proof.
By Lemma16, we have
(97)TM,Ω((x-,y-);(0,y-y-))=limt→0+t-1[𝒯M,Ω(x-,y-+t(y-y-))-𝒯M,Ω(x-,y-)]=-𝒯M,Ω(x-,y-).
We turn now to show the inclusion(98)∂𝒯M,Ω(x-,y-)⊂{(x*,y*)∈E*×F*:σΩ(-y*)=1}.
Fix any (x*,y*)∈∂𝒯M,Ω(x-,y-). Then
(99)〈y*,k〉≤𝒯M,Ω′((x-,y-);(0,k)),∀k∈F.
By assumption, we haveπM(x-),Ω(y-)≠∅. Then, there exists y0∈πM(x-),Ω(y-); that is, y0∈M(x-) with 𝒯M,Ω(x-,y-)=ρΩ(y0-y-)≠0. By the first part, we obtain that
(100)〈y*,y0-y-〉≤𝒯M,Ω′((x-,y-);(0,y0-y-))=-𝒯M,Ω(x-,y-)=-ρΩ(y0-y-),
and so
(101)〈-y*,y0-y-ρΩ(y0-y-)〉≥1,
since ρΩ(y0-y-)≠0, because y-∉M(x-). Thus, for any ϵ>0, there exist w∈Ω and α0>0 such that
(102)0<ρΩ(y0-y-)≤α0<ρΩ(y0-y-)+ϵ
and y0-y-∈α0Ω. Thus,
(103)σΩ(-y*)≥〈-y*,y0-y-α0〉=ρΩ(y0-y-)α0〈-y*,y0-y-ρΩ(y0-y-)〉≥ρΩ(y0-y-)α0≥ρΩ(y0-y-)ρΩ(y0-y-)+ϵ,
and hence,
(104)σΩ(-y*)≥1-ϵρΩ(y0-y-)+ϵ,∀ϵ>0.
Takingϵ→0+ yields
(105)σΩ(-y*)≥1.
Let us prove the reverse inequality(106)σΩ(-y*)≤1.
Let(tj)j be a net in (0,∞) converging to 0 and satisfying
(107)𝒯M,Ω′((x-,y-);(0,k))=limjtj-1[𝒯M,Ω(x-,y-+tjk)-𝒯M,Ω(x-,y-)].
By Lemma10, we have, for r∶=𝒯M,Ω(x-,y-), the following inequality
(108)𝒯M,Ω(x,y)-𝒯M,Ω(x-,y-)≤𝒯Mr,Ω(x,y)-𝒯Mr,Ω(x-,y-),∀(x,y)∈E×F,
which ensures that
(109)tj-1[𝒯M,Ω(x-,y-+tjk)-𝒯M,Ω(x-,y-)]≤tj-1𝒯Mr,Ω(x-,y-+tjk)=tj-1infu∈Mr(x-)ρΩ(u-(y-+tjk))≤tj-1ρΩ(y--(y-+tjk))=ρΩ(-k),
and so
(110)tj-1[𝒯M,Ω(x-,y-+tjk)-𝒯M,Ω(x-,y-)]≤ρΩ(-k).
Taking the limit onj, we get that
(111)𝒯M,Ω′((x-,y-);(0,k))=limjtj-1[𝒯M,Ω(x-,y-+tjk)-𝒯M,Ω(x-,y-)]≤ρΩ(-k).
Using the fact that(x*,y*)∈∂𝒯M,Ω(x-,y-), we write
(112)〈-y*,-k〉=〈(x*,y*);(0,k)〉≤𝒯M,Ω((x-,y-);(0,k))〈-y*,-k〉≤ρΩ(-k),∀k∈F.
Fix anyv∈Ω, and take k:=-v in the previous inequality we get
(113)〈-y*,v〉=〈-y*,-k〉≤ρΩ(v)≤1.
This ensures that σΩ(-y*)≤1, and hence, the proof is finished.The following theorem establishes another relationship between the convex subdifferential of𝒯M,Ω and the convex normal cone of Mr at points outside the graph gphM.Theorem 18.
LetE and F be two Hausdorff topological vector spaces, M a set-valued mapping with nonempty closed values of F,Ω be a bounded convex set in F, and (x-,y-)∉gphM. Assume that πM(x-),Ω(y-)≠∅. Then, one has
(114)∂𝒯M,Ω(x-,y-)⊂N(gphMr;(x-,y-))∂𝒯M,Ω(x-,y-)∩{(x*,y*)∈E*×F*:σΩ(-y*)=1}.
If, in addition,𝒯M,Ω is calm at (x-,y-), then one has
(115)∂𝒯M,Ω(x-,y-)=N(gphMr;(x-,y-))∩{(x*,y*)∈E*×F*:σΩ(-y*)=1}.Proof.
The first inclusion follows directly from the proof of Theorem14 and Proposition 17. Let us prove the reverse inclusion. Fix any (x*,y*)∈N(gphMr;(x-,y-)) with σΩ(-y*)=1. First, let us check that 0∉∂𝒯M,Ω(x-,y-). Indeed, by the assumption πM(x-),Ω(y-)≠∅ and Proposition 17, there exists z0∈πM(x-),Ω(y-) such that
(116)𝒯M,Ω′((x-,y-);(0,z0-y-))=-𝒯M,Ω(x-,y-)<0,
which ensures that 0∉∂𝒯M,Ω(x-,y-). Therefore, by Corollary 15, there exists some λ>0 and (u*,v*)∈∂𝒯M,Ω(x-,y-), such that (x*,y*)=λ(u*,v*). Using the first part of the proof, we obtain that σΩ(-v*)=1, and so by combining this equality with the assumption σΩ(-y*)=1, we get λ=1. Thus completing the proof.


## 5. Application of𝒯M,Ω: Calmness and Exact Penalization

The primary goal in the present section is to make clear that the scalar function𝒯M,Ω can also be a powerful tool in the study of the calmness property of optimization problems in Hausdorff topological vector spaces. Here, we are interested in the concept of calmness of perturbed optimization problems with a constraint defined by a set used and studied by Burke [22, 23] in normed vector spaces. We will adapt his definition for a general perturbed problem with a constraint defined by a set-valued mapping in Hausdorff topological vector spaces, and we will prove that it is equivalent to the existence of an exact penalization in terms of the minimal time function associated with the set-valued mapping defining the constraint of the problem.Consider the problem(𝒫), which consists in minimizing the function f over all x∈E satisfying 0∈M(x),
(117)(𝒫){minimizef(x)subjectto0∈M(x),
where M:E⇉F is a closed set-valued mapping between two normed vector spaces E and F and f:E→ℝ∪{+∞} is an extended real-valued function. We begin with the definition of calmness.Definition 19.
Letf, M, E, and F be as in the statement of (𝒫), and consider the following perturbed problem
(118)(𝒫y){minimizef(x)subjecttoy∈M(x).
LetS:F⇉E be the feasible set-valued mappingassociated with (𝒫y); that is,
(119)S(y)∶={x∈domf:y∈M(x)}.
Let(x-,y-)∈gphS and let Ω be a bounded set in F. One will say that the problem (𝒫y-) is calmat x- with respect to Ω if there exist a constant α-≥0 and X∈𝒩E(x-) such that for every x∈X and any y∈M(x) one has
(120)f(x-)≤f(x)+α-ρΩ(y-y-).
The constantα- and X are called the modulus of calmnessand neighborhood of calmnessfor (𝒫y-) at x-, respectively.Remark 20.
WhenF is assumed to be normed, the above definition coincides with the definition used in [22, 23] by taking Ω to be the closed unit ball in F. Observe that if (𝒫y-) is calm at x- with respect to a bounded set Ω, then x- is necessarily a local solution to (𝒫y-).For any problem(𝒫y), any real number α≥0, and any bounded set Ω, we will associate the function Pα,y,Ω defined by
(121)Pα,y,Ω(x)∶=f(x)+α𝒯M,Ω(x,y).In the following theorem we state our main result in this section. It establishes a relationship between the calmness property and the existence of an exact penalization of the general perturbed problem(𝒫y-) in terms of the minimal time function to images associated with the set-valued mapping M defining the constraint of the problem.Theorem 21.
Let(x-,y-)∈gphS. If (𝒫y-) is calm at x- with respect to Ω with modulus α- and neighborhood X∈𝒩E(x-), then x- is a minimum over X of the function Pα,y-,Ω, for all α≥α-. If, in addition, Ω is convex and 0∈Ω, the converse holds.Proof.
Letϵ>0. Given any x∈X∩domM≠∅. Then there exist t0≥0 and w∈Ω such that y-+t0w∈M(x) and 𝒯M,Ω(x-,y-)≤t0<𝒯M,Ω(x-,y-)+ϵ. Thus, if α≥α-, we obtain from the calmness hypothesis that
(122)f(x-)≤f(x)+αρΩ(y-y-),∀x∈X,∀y∈M(x).
Fix anyx∈X. Then by taking y:=y-+t0w in the previous inequality and observing that ρΩ(w)≤1 we obtain that
(123)f(x-)≤f(x)+αρΩ(t0w)≤f(x)+t0αρΩ(w)f(x-)≤f(x)+t0α≤f(x)+α𝒯M,Ω(x-,y-)+αϵ.
Hence(124)Pα,y-,Ω(x-)=f(x-)≤Pα,y-,Ω(x)+αϵ,∀x∈X.
Sinceϵ>0 was taken arbitrarily, the direct implication is established by taking ϵ→0.
Conversely, we assume thatΩ is convex and 0∈Ω. Fix any x∈X and any y∈M(x). We claim first the inequality 𝒯M,Ω(x,y-)≤ρΩ(y-y-). Clearly the case ρΩ(y-y-)=∞ is obvious. So we assume that ρΩ(y-y-)<∞. Since y∈M(x), we have 𝒯M,Ω(x,y)=0. Then by the finiteness of 𝒯M,Ω(x,y) we can find for any ϵ>0 some λ0≥0 such that
(125)0=𝒯M,Ω(x,y)0≤λ0<𝒯M,Ω(x,y)+ϵ=ϵ,(y+λ0Ω)∩M(x)≠∅.
Hence, there existsw∈Ω such that y+λ0w∈M(x). Therefore, for any v∈Ω and any β>0, we have
(126)y+λ0w=y-βv+(λ0w+βv)=y-βv+(λ0+β)(λ0λ0+βw+βλ0+βv)∈M(x).
SinceΩ is convex, we have (λ0/(λ0+β))w+(β/(λ+β))v∈Ω, and so (y-βv+(λ0+β)Ω)∩M(x)≠∅, which ensures by the definition of the minimal time function that
(127)𝒯M,Ω(x,y-βv)≤λ0+β<ϵ+β,∀ϵ>0.
SinceρΩ(y-y-)<∞, we let β:=ρΩ(y-y-)+ϵ<∞ and v∶=(y-y-)/β. Since Ω is convex and 0∈Ω, we have {h∈E:ρΩ(h)<1}⊂Ω, and hence v∈Ω. Thus, (127) ensures that
(128)𝒯M,Ω(x,y-)=𝒯M,Ω(x,y-βv)<ϵ+β𝒯M,Ω(x,y-)=ρΩ(y-y-)+2ϵ,∀ϵ>0.
After takingϵ→0, this inequality ensures the desired inequality; that is, 𝒯M,Ω(x,y-)≤ρΩ(y-y-). Therefore, for any x∈X and any y∈M(x), we have
(129)f(x-)=Pα-,y-,Ω(x-)≤Pα-,y-,Ω(x)=f(x)+α-𝒯M,Ω(x,y-)f(x-)≤f(x)+α-ρΩ(y-y-).
Hence,(𝒫y-) is calm at x- with respect to Ω, and the proof of the theorem is complete.



---

*Source: 707603-2013-11-27.xml*