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

Abstract

For a set-valued mapping 𝑀 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 𝒯_𝑀,Ω(𝑥,𝑦)∶=inf{𝑡≥0∶𝑀(𝑥)∩(𝑦+𝑡Ω)≠∅}. We prove and extend various properties on directional derivatives and subdifferentials of 𝒯_𝑀,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 𝒯_𝑀,Ω to the calmness property of perturbed optimization problems in Hausdorff topological vector spaces is given in the last section of the paper.

1. Introduction

Let 𝐸 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 𝒯_𝑀,Ω(𝑥)∩,∀𝑥,𝑦∶=inf𝑡≥0∶𝑀𝑦+𝑡Ω≠∅𝑥,𝑦∈𝐸×𝐹. (1)

The case 𝑀(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 𝑇_𝑆,Ω(𝑢)∶=inf{𝑡≥0∶𝑆∩(𝑢+𝑡Ω)≠∅},∀𝑢∈𝐹. (2)

This function 𝑇_𝑆,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 𝒯_𝑀,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, 𝒯_𝑀,Ω𝑥₀,𝑦₀=(⋅)𝑦(̇𝜏)[](0)inf𝑡>0∶∃𝑦s.t.∈Ω,a.e.0,𝑡,𝑦=𝑦₀,𝑦(𝑡)∈𝑀𝑥₀,∀𝑦₀∉𝑀𝑥₀,0,∀𝑦₀∈𝑀𝑥₀. (3)

We notice that the minimal time function 𝒯_𝑀,Ω covers many crucial functions in variational analysis: the distance function to images Δ_𝑀, the indicator function, the usual distance function, and the Minkowski function, by taking some particular cases of 𝑀 and Ω as follows.(1)If 𝑀(𝑥)={0}, then 𝒯_𝑀,Ω(𝑥,𝑦)=𝑇_{0},Ω(−𝑦)=𝜌_Ω(𝑦) which coincides with the Minkowski (gauge) function associated with Ω; that is, 𝜌_Ω𝑦=inf𝑡≥0∶𝑦∈𝑡Ω,∀𝑦∈𝐹. (4)(2)If Ω={0}, then 𝒯_𝑀,Ω(𝑥,𝑦)=𝑇_{𝑀(𝑥),{0}}(𝑦)=𝜓_𝑀(𝑥)(𝑦)=𝜓_{𝑔𝑝ℎ𝑀}(𝑥,𝑦), which coincides with the indicator function associated with 𝑔𝑝ℎ𝑀. Here, 𝜓_𝑆(𝑢)=0 if 𝑢∈𝑆 and 𝜓_𝑆(𝑢)=+∞, otherwise.(3)If 𝐹 is normed, 𝑀(𝑥)={0}, and Ω=𝔹_𝐹 is the closed unit ball in 𝐹, then 𝒯_𝑀,Ω(𝑥,𝑦)=𝑇_{0},𝔹𝐹(𝑦)=‖𝑦‖.(4)If 𝐹 is normed and Ω=𝔹_𝐹, then 𝒯_𝑀,Ω𝑥,𝑦=𝒯_𝑀,𝔹(𝑥)=inf_{𝑠∈𝑀(𝑥)}𝑦−𝑠=Δ_𝑀𝑥,𝑦 (5) 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)If 𝐹 is normed, 𝑀(𝑥)=𝑆, and Ω=𝔹_𝐹, then 𝒯_𝑀,Ω𝑥,𝑦=𝑇_𝑆,𝔹𝑦=inf_𝑠∈𝑆𝑦−𝑠=𝑑_𝑆𝑦, (6) 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 function 𝒯_𝑀,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 set 𝑆 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 𝒯_𝑀,Ω is [16], in which the author studied the Fréchet subdifferential of 𝒯_𝑀,Ω in Banach spaces. We mention that there are no results on the directional derivatives and subdifferentials of 𝒯_𝑀,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 𝒯_𝑀,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 𝒯_𝑀,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 that 𝐸 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^∗.

Let 𝑆 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}.

Let 𝑓 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 𝑓^′(𝑢;𝑣)∶=lim_𝑡↓0𝑡⁻¹𝑓(𝑢+𝑡𝑣)−𝑓(𝑢(),∀𝑣∈𝐸.resp.,𝜕𝑓𝑥)∶=𝑢^∗∈𝐸^∗∶𝑢^∗,𝑣≤𝑓^′()=𝑢𝑢;𝑣,∀𝑣∈𝐸_∗∈𝐸^∗∶𝑢^∗,𝑢−𝑢≤𝑓(𝑢)−𝑓(𝑢),∀𝑢∈𝐸. (7)

Note that for l.s.c convex functions we have 𝑓^′(𝑢;ℎ)=𝜎_{𝜕𝑓(𝑢)}(ℎ), whenever 𝑢∈core(dom𝑓) and 𝐸 is a Hausdorff locally convex topological vector space. Here, 𝜎_𝐴 denotes the support function associated with a closed subset 𝐴⊂𝐸^∗ defined by 𝜎_𝐴(ℎ)∶=sup{⟨𝑥^∗,ℎ⟩∶𝑥^∗∈𝐴} and core(𝑆)∶={𝑥∈𝐸∶forallℎ∈𝐸,∃𝑡_𝑥,ℎ>0,suchthat𝑥+[0,𝑡_𝑥,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 mappings 𝑀∶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. Let 𝐸 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)If 𝒯_𝑀,Ω(𝑥,𝑦)=0, then 𝑦∈cl(𝑀(𝑥)).(2)Conversely, for any 𝑦∈𝑀(𝑥), one has 𝑇_𝑀,Ω(𝑥,𝑦)=0.(3)If in addition 0∈intΩ, then for any 𝑦∈cl(𝑀(𝑥)) one has 𝑇_𝑀,Ω(𝑥,𝑦)=0.

Proof. (1) Let 𝑥∈𝐸 with 𝒯_𝑀,Ω(𝑥,𝑦)=0, then for all 𝑛≥1, ∃𝑡_𝑛≥0 such that 𝒯_𝑀,Ω(𝑥,𝑦)+1/𝑛>𝑡_𝑛≥𝒯_𝑀,Ω(𝑥,𝑦) with 𝑀(𝑥)∩(𝑦+𝑡_𝑛Ω)≠∅; that is, there exists 𝑢_𝑛∈Ω such that 𝑦+𝑡_𝑛𝑢_𝑛∈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 𝑤_𝑛∈𝑊 such that 𝑢_𝑛=𝛼𝑤_𝑛. Clearly, for any 𝑛>𝛼, we have 𝑡_𝑛𝛼∈(0,1), and so by the fact that 𝑊 is balanced we obtain 𝑡_𝑛𝑢_𝑛=𝑡_𝑛𝛼𝑤_𝑛∈𝑡_𝑛𝛼𝑊⊂𝑊⊂𝑉. (8)
Since 𝑉 is arbitrary, we get the convergence of the sequence 𝑡_𝑛𝑢_𝑛 to zero, that is, the convergence of the sequence 𝑦+𝑡_𝑛𝑢_𝑛 to 𝑦, and since 𝑦+𝑡_𝑛𝑢_𝑛∈𝑀(𝑥), we get 𝑦∈cl(𝑀(𝑥)).
(2) Let 𝑦∈𝑀(𝑥) and let 𝑤∈Ω. Clearly, we have 𝑦=𝑦+0⋅𝑤∈(𝑦+0⋅Ω)∩𝑀(𝑥). This ensures by definition that 𝒯_𝑀,Ω(𝑥,𝑦)=0.
(3) Assume now that 0∈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≤𝒯_𝑀,Ω(𝑥,𝑦)≤𝜖 which ensures that 𝒯_𝑀,Ω(𝑥,𝑦)=0.

Note that the assumption 0∈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 𝒯_𝑀,Ω(𝑥,𝑦)=𝜓_𝑀(𝑥)(𝑦)=+∞.

As a corollary of parts (1) and (2) in Lemma 1 we have the following.

Corollary 2. Let 𝐸 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 𝒯_𝑀,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. Let 𝐸 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, 𝒯_𝑀,Ω(⋅,⋅) is convex on its domain if and only if the graph 𝑔𝑝ℎ𝑀 is convex.

Proof. Assume that 𝒯_𝑀,Ω(⋅,⋅) is convex on its domain; that is, 𝒯_𝑀,Ω𝜆𝑥₁,𝑦₁+(1−𝜆)𝑥₂,𝑦₂≤𝜆𝒯_𝑀,Ω𝑥₁,𝑦₁+(1−𝜆)𝒯_𝑀,Ω𝑥₂,𝑦₂, (9) for all (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈dom𝒯_𝑀,Ω, and all 𝜆∈[0,1]. Therefore, by Corollary 2, for any (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝑔𝑝ℎ𝑀, we have 𝒯_𝑀,Ω(𝑥₁,𝑦₁)=𝒯_𝑀,Ω(𝑥₂,𝑦₂)=0, and so by the previous inequality, we obtain 𝒯_𝑀,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 (𝑥_𝑖,𝑦_𝑖)∈dom𝒯_𝑀,Ω (𝑖=1,2), and let any 𝜆∈[0,1]. Fix any 𝜖>0. Since 𝒯_𝑀,Ω(𝑥₁,𝑦₁)<∞ and 𝒯_𝑀,Ω(𝑥₂,𝑦₂)<∞, we can find that 𝑤_𝑖∈Ω, (𝑖=1,2) and 𝑡_𝑖≥0 (𝑖=1,2) such that 𝑡_𝑖<𝒯_𝑀,Ω(𝑥_𝑖,𝑦_𝑖)+𝜖 and 𝑦_𝑖+𝑡_𝑖𝑤_𝑖∈𝑀(𝑥_𝑖) (𝑖=1,2). So the convexity of 𝑔𝑝ℎ𝑀 ensures that 𝜆(𝑥₁,𝑦₁+𝑡₁𝑤₁)+(1−𝜆)(𝑥₂,𝑦₂+𝑡₂𝑤₂)∈𝑔𝑝ℎ𝑀. Hence, 𝜆𝑦₁+(1−𝜆)𝑦₂+𝜆𝑡₁+(1−𝜆)𝑡₂𝑤∈𝑀𝜆𝑥₁+(1−𝜆)𝑥₂, (10) 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 𝑤=𝜆𝑡₁𝜆𝑡₁+(1−𝜆)𝑡₂𝑤₁+(1−𝜆)𝑡₂𝜆𝑡₁+(1−𝜆)𝑡₂𝑤₂∈Ω, (11) which ensures that 𝜆𝑦₁+(1−𝜆)𝑦₂+𝜆𝑡₁+(1−𝜆)𝑡₂Ω∩𝑀𝜆𝑥₁+(1−𝜆)𝑥₂≠∅. (12)
This ensures that 𝒯_𝑀,Ω𝜆𝑥₁,𝑦₁+(1−𝜆)𝑥₂,𝑦₂=𝒯_𝑀,Ω𝜆𝑥₁+(1−𝜆)𝑥₂,𝜆𝑦₁+(1−𝜆)𝑦₂≤𝜆𝑡₁+(1−𝜆)𝑡₂<𝜆𝒯_𝑀,Ω𝑥₁,𝑦₁+(1−𝜆)𝒯_𝑀,Ω𝑥₂,𝑦₂+𝜖, (13) for any 𝜖>0. Thus, taking 𝜖→0 completes the proof.

Note that the convexity of Ω, in the proof of Lemma 3, is needed only in one direction (reverse implication); that is, the convexity of 𝒯_𝑀,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. Let 𝐸 and 𝐹 be two Hausdorff topological vector spaces. Assume that Ω is compact in 𝐹 and 𝑔𝑝ℎ𝑀 is closed in 𝐸×𝐹. Then 𝒯_𝑀,Ω is lower semicontinuous at any (𝑥,𝑦)∈dom𝒯_𝑀,Ω.

Proof. Let (𝑥,𝑦)∈dom𝒯_𝑀,Ω. We have to prove that 𝒯_𝑀,Ω𝑥,𝑦≤liminf_{(𝑥,𝑦)→(𝑥,𝑦)}𝒯_𝑀,Ω𝑥,𝑦. (14)
The case liminf_{(𝑥,𝑦)→(𝑥,𝑦)}𝒯_𝑀,Ω(𝑥,𝑦)=∞ is obvious, so we assume that 𝛼∶=liminf_{→𝑥,𝑦𝑥,𝑦}𝒯_𝑀,Ω𝑥,𝑦<∞. (15)
Let (𝑥_𝑗,𝑦_𝑗)_𝑗∈𝐽 be a net satisfying the above liminf; that is, (𝑥_𝑗,𝑦_𝑗)→(𝑥,𝑦) and 𝛼=lim_𝑗𝒯_𝑀,Ω(𝑥_𝑗,𝑦_𝑗). Let (𝑡_𝑗)_𝑗∈𝐽 be a net of positive real numbers satisfying 𝑡_𝑗→0. By the definition of the minimal time function, we can find for any 𝑗∈𝐽 a real number 𝛿_𝑗 such that 0≤𝒯_𝑀,Ω𝑥_𝑗,𝑦_𝑗≤𝛿_𝑗<𝒯_𝑀,Ω𝑥_𝑗,𝑦_𝑗+𝑡_𝑗,𝑦_𝑗+𝛿_𝑗Ω∩𝑀𝑥_𝑗≠∅,∀𝑗∈𝐽. (16) Hence, for any 𝑗∈𝐽, there exists 𝑤_𝑗∈Ω such that (𝑥_𝑗,𝑦_𝑗+𝛿_𝑗𝑤_𝑗)∈𝑔𝑝ℎ𝑀. Using the compactness of Ω, we get the convergence of a subnet of (𝑤_𝑗)_𝑗 to some point 𝑤∈Ω. Without loss of generality the subnet is still denoted by (𝑤_𝑗)_𝑗. Therefore, (𝑥_𝑗,𝑦_𝑗+𝛿_𝑗𝑤_𝑗)→(𝑥,𝑦+𝛼𝑤)∈𝑔𝑝ℎ𝑀 (by the closedness of 𝑔𝑝ℎ𝑀). This shows that (𝑦+𝛼Ω)∩𝑀(𝑥)≠∅, which ensures that 𝒯_𝑀,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 that Ω is weakly compact and 𝑔𝑝ℎ𝑀 is weakly closed. Consequently, we have the two following corollaries.

Corollary 5. Let 𝐸 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, 𝒯_𝑀,Ω is lower semicontinuous at any (𝑥,𝑦)∈dom𝒯_𝑀,Ω.

Corollary 6. Assume that 𝐹 is a reflexive Banach space, Ω is closed convex bounded set in 𝐹, and 𝑔𝑝ℎ𝑀 is closed convex set in 𝐸×𝐹. Then 𝒯_𝑀,Ω is lower semicontinuous at any (𝑥,𝑦)∈dom𝒯_𝑆,Ω.

The following lemma is technical and is needed in some forthcoming proofs. Its proof is straightforward.

Lemma 7. Let 𝐸 and 𝐹 be two Hausdorff topological vector spaces, 𝑀∶𝐸⇉𝐹 a set-valued mapping, and Ω a nonempty set in 𝐹. Then, 𝑡⁻¹𝒯_𝑀,Ω𝑥+𝑡ℎ,𝑦+𝑡𝑘=𝑇_{𝑡−1[𝑀(𝑥+𝑡ℎ)−𝑦],Ω}(𝑘),∀𝑥,𝑦,(ℎ,𝑘)∈𝐸×𝐹,∀𝑡>0. (17)

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 of 𝒯_𝑀,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. Let 𝐸 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)𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)=lim_𝑡→0+𝑡⁻¹𝒯_𝑀,Ω𝑥+𝑡ℎ,𝑦+𝑡𝑘=lim_𝑡→0+𝑇_{𝑡−1[𝑀(𝑥+𝑡ℎ)−𝑦],Ω}(𝑘),∀(ℎ,𝑘)∈𝐸×𝐹; (18)(2)𝜕𝒯_𝑀,Ω𝑥,𝑦=𝑁𝑔𝑝ℎ𝑀;𝑥,𝑦∩𝑥^∗,𝑦^∗∈𝐸^∗×𝐹^∗∶𝜎_Ω−𝑦^∗≤1; (19)(3)if, in addition, (𝑥,𝑦)∈𝑐𝑜𝑟𝑒(dom𝒯_𝑀,Ω), 𝐸×𝐹 is locally convex, then 𝒯^′_𝑀,Ω𝑥,𝑦;(𝑇ℎ,𝑘)=𝒯_𝑀,Ω(),∀()ℎ,𝑘ℎ,𝑘∈𝐸×𝐹; (20)(4)𝑇𝑔𝑝ℎ𝑀;𝑥,𝑘=(ℎ,𝑘)∈𝐸×𝐹∶𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)=0; (21)(5)and 𝑁𝑔𝑝ℎ𝑀;𝑥,𝑦=cl_𝑤∗ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦. (22)

Proof. (1) By Lemma 3, we have the convexity of 𝒯_𝑀,Ω, and so 𝒯^′_𝑀,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 𝜎_Ω(−𝑦^∗)≤1. Then, by definition of the convex normal cone and by Corollary 2, we have 𝑥^∗,𝑦^∗,𝑥−𝑥,𝑦−𝑦≤0=𝒯_𝑀,Ω𝑥,𝑦−𝒯_𝑀,Ω𝑥,𝑦,∀𝑥,𝑦∈𝑔𝑝ℎ𝑀. (23)
Assume now that (𝑥,𝑦)∉𝑔𝑝ℎ𝑀, and let 𝜖>0. If 𝒯_𝑀,Ω(𝑥,𝑦)=∞, then the previous inequality holds obviously. Assume that 𝒯_𝑀,Ω(𝑥,𝑦)<∞. By definition of minimal time function, there exists 𝛿>0 and 𝑤∈Ω such that 0<𝒯_𝑀,Ω𝑥,𝑦≤𝛿<𝒯_𝑀,Ω𝑥,𝑦+𝜖with𝑦+𝛿𝑤∈𝑀(𝑥). (24)
Hence, we get by the definition of convex normal cones that 𝑥^∗,𝑦^∗,𝑥,𝑦+𝛿𝑤−𝑥,𝑦≤0. (25)
Therefore, combining this inequality with the assumption 𝜎_Ω(−𝑦^∗)≤1, we obtain that 𝑥^∗,𝑦^∗,𝑥−𝑥,𝑦−𝑦=𝑥^∗,𝑦^∗,𝑥−𝑥,𝑦+𝛿𝑤−𝑦+𝛿−𝑦^∗,𝑤≤𝛿𝜎_Ω−𝑦^∗≤𝛿<𝒯_𝑀,Ω𝑥,𝑦+𝜖, (26) which ensures, by taking 𝜖→0, that 𝑥^∗,𝑦^∗,𝑥−𝑥,𝑦−𝑦≤𝒯_𝑀,Ω𝑥,𝑦−𝒯_𝑀,Ω𝑥,𝑦,∀𝑥,𝑦∉𝑔𝑝ℎ𝑀. (27)
Thus, this inequality holds for any (𝑥,𝑦)∈𝐸×𝐹 which means that (𝑥^∗,𝑦^∗)∈𝜕𝒯_𝑀,Ω(𝑥,𝑦).
Conversely, let (𝑥^∗,𝑦^∗)∈𝜕𝒯_𝑀,Ω(𝑥,𝑦). Then, by definition, we have ⟨𝑥^∗,𝑦^∗,𝑥,𝑦−𝑥,𝑦⟩≤𝒯_𝑀,Ω𝑥,𝑦−𝒯_𝑀,Ω𝑥,𝑦,∀𝑥,𝑦∈𝐸×𝐹. (28)
Using Corollary 2, we have 𝑥^∗,𝑦^∗,𝑥,𝑦−𝑥,𝑦≤0∀𝑥,𝑦∈𝑔𝑝ℎ𝑀. (29)
This means that (𝑥^∗,𝑦^∗)∈𝑁(𝑔𝑝ℎ𝑀;(𝑥,𝑦)). We have to prove that 𝜎_Ω(−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 𝒯_𝑀,Ω(𝑥,𝑦−𝛿𝑤)≤𝛿. Therefore, by (28), we have −𝑦^∗,𝛿𝑤=𝑥^∗,𝑦^∗,𝑥,𝑦−𝛿𝑤−𝑥,𝑦≤𝒯_𝑀,Ω𝑥,𝑦−𝛿𝑤≤𝛿, (30) which ensures that −𝑦^∗,𝑤≤1,∀𝑤∈Ω. (31)
This gives 𝜎_Ω(−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^∗)∈𝜕𝒯_𝑀,Ω(𝑥,𝑦), and so 𝜎_{𝜕𝒯𝑀,Ω(𝑥,𝑦)}(ℎ,𝑘)≤0. Now we use the assumption (𝑥,𝑦)∈core(dom𝒯_𝑀,Ω) to write 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))=𝜎_{𝜕𝒯𝑀,Ω(𝑥,𝑦)}(ℎ,𝑘) and so 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))≤0. Hence by Part (1) we obtain the needed equality 𝒯^′_𝑀,Ω𝑥,𝑦;(𝑇ℎ,𝑘)=0=𝒯_𝑀,Ω(ℎ,𝑘). (32) Assume now that (ℎ,𝑘)∉𝑇(𝑔𝑝ℎ𝑀;(𝑥,𝑦)). Without loss of generality, we assume that 𝒯𝑇_𝑀,Ω(ℎ,𝑘)<∞. Put 𝑇𝛼∶=𝒯_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^∗)∈𝜕𝒯_𝑀,Ω(𝑥,𝑦), we have 𝜎_Ω(−𝑦^∗)≤1 and (𝑥^∗,𝑦^∗)∈𝑁(𝑔𝑝ℎ𝑀;(𝑥,𝑦)), and hence, we get the following 𝑥^∗,𝑦^∗,(=𝑥ℎ,𝑘)_∗,𝑦^∗,(ℎ,𝑘+𝛿𝑤)+𝛿−𝑦^∗,𝑤≤𝛿𝜎_Ω−𝑦^∗≤𝛿,∀𝑥^∗,𝑦^∗∈𝜕𝒯_𝑀,Ω𝑥,𝑦. (33)
This ensures that 𝒯^′_𝑀,Ω𝑥,𝑦,(ℎ,𝑘)=𝜎_{𝜕𝒯𝑀,Ω(𝑥,𝑦)}(ℎ,𝑘)≤𝛿, (34) and since 𝛿 is taken arbitrarily, we take the infimum to obtain that 𝒯^′_𝑀,Ω𝑥,𝑦,(()∩𝑇ℎ,𝑘)≤inf𝛿>0∶suchthat𝑘+𝛿Ω_𝑀(ℎ)𝑇≠∅=𝒯_𝑀,Ω(ℎ,𝑘). (35)
Let us prove the reverse inequality 𝒯𝑇_𝑀,Ω(ℎ,𝑘)≤𝒯^′_𝑀,Ω𝑥,𝑦,(ℎ,𝑘). (36)
Obviously, we assume that 𝒯^′_𝑀,Ω((𝑥,𝑦),(ℎ,𝑘))<∞. Put 𝛼∶=𝒯^′_𝑀,Ω((𝑥,𝑦),(ℎ,𝑘)). Let (𝑡_𝑗)_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=lim_𝑗𝑇_𝑡𝑗⁻¹_{[𝑀(𝑥+𝑡𝑗ℎ)−𝑦],Ω}(𝑘). Let 𝛿_𝑗>0 and 𝑤_𝑗∈Ω such that 𝑇_𝑡𝑗⁻¹_{[𝑀(𝑥+𝑡𝑗ℎ)−𝑦],Ω}(𝑘)≤𝛿_𝑗<𝑇_𝑡𝑗⁻¹_{[𝑀(𝑥+𝑡𝑗ℎ)−𝑦],Ω}(𝑘)+𝑡_𝑗,𝑘+𝛿_𝑗𝑤_𝑗∈𝑡_𝑗⁻¹𝑀𝑥+𝑡_𝑗ℎ−𝑦;thatis,ℎ,𝑘+𝛿_𝑗𝑤_𝑗∈𝑡_𝑗⁻¹𝑔𝑝ℎ𝑀−𝑥,𝑦. (37)
Since 𝑔𝑝ℎ𝑀 is closed convex, we have 𝑇(𝑔𝑝ℎ𝑀;(𝑥,𝑦))=cl(ℝ₊[𝑔𝑝ℎ𝑀−(𝑥,𝑦)]), and so (ℎ,𝑘+𝛿_𝑗𝑤_𝑗)∈𝑇(𝑔𝑝ℎ𝑀;(𝑥,𝑦)), for all 𝑗; that is, 𝑘+𝛿_𝑗𝑤_𝑗∈𝑇_𝑀(ℎ), for all 𝑗.
On the other hand we clearly have 𝒯𝑇_𝑀,Ω(ℎ,𝑘)≤𝜌_Ω(𝑠−𝑘), for all 𝑠∈𝑇_𝑀(ℎ). Consequently, we obtain for all 𝑗 that 𝒯𝑇_𝑀,Ω(ℎ,𝑘)≤𝜌_Ω𝑘+𝛿_𝑗𝑤_𝑗−𝑘=𝛿_𝑗𝜌_Ω𝑤_𝑗≤𝛿_𝑗<𝑇_𝑡𝑗⁻¹_{[𝑀(𝑥+𝑡𝑗ℎ)−𝑦],Ω}(𝑘)+𝑡_𝑗,∀𝑗. (38) Taking the limit on this inequality, we obtain that 𝒯𝑇_𝑀,Ω(ℎ,𝑘)≤𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘), (39) thus completing the proof of (3).
(4) The inclusion 𝑇(𝑔𝑝ℎ𝑀,(𝑥,𝑦))⊂{(ℎ,𝑘)∈𝐸×𝐹∶𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))=0} follows from the assertion (3) and Corollary 2. Conversely, let (ℎ,𝑘)∈𝐸×𝐹 with 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))=0. Let (𝑡_𝑗) be a net of positive real numbers converging to zero and satisfying the limit 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))=lim_𝑡𝑗→0𝒯_𝑡𝑗⁻¹_{[𝑀(𝑥+𝑡𝑗ℎ)−𝑦],Ω}(𝑘). For any 𝑗∈𝐽, we can find that 𝛿_𝑗>0 and 𝑤_𝑗∈Ω such that 𝒯_𝑡𝑗⁻¹_{[𝑀(𝑥+𝑡𝑗ℎ)−𝑦],Ω}(𝑘)≤𝛿_𝑗<𝒯_𝑡𝑗⁻¹_{[𝑀(𝑥+𝑡𝑗ℎ)−𝑦],Ω}(𝑘)+𝑡_𝑗,𝑘+𝛿_𝑗𝑤_𝑗∈𝑡_𝑗⁻¹𝑀𝑥+𝑡_𝑗ℎ−𝑦;thatis,ℎ,𝑘+𝛿_𝑗𝑤_𝑗∈𝑡_𝑗⁻¹𝑔𝑝ℎ𝑀−𝑥,𝑦. (40)
Since the graph of 𝑀 is closed convex, we obtain that ℎ,𝑘+𝛿_𝑗𝑤_𝑗∈𝑡_𝑗⁻¹𝑔𝑝ℎ𝑀−𝑥,𝑦⊂𝑇𝑔𝑝ℎ𝑀;𝑥,𝑦. (41)
On the other hand, we have lim_𝑗𝛿_𝑗=lim_𝑡𝑗→0𝒯_𝑡𝑗⁻¹_{[𝑀(𝑥+𝑡𝑗ℎ)−𝑦],Ω}(𝑘)=0, and hence, by the boundedness of the set Ω, we get the convergence of 𝛿_𝑗𝑤_𝑗 to zero. This ensures the convergence of 𝑘+𝛿_𝑗𝑤_𝑗→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 inclusion cl_𝑤∗(ℝ₊𝜕𝒯_𝑀,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 𝜕𝒯_𝑀,Ω𝑥,𝑦⁻⊂𝑇𝑔𝑝ℎ𝑀;𝑥,𝑦. (42)
Fix any (ℎ,𝑘)∈(𝜕𝒯_𝑀,Ω((𝑥,𝑦)))⁻. Then 𝑥^∗,𝑦^∗;(ℎ,𝑘)≤0,∀𝑥^∗,𝑦^∗∈𝜕𝒯_𝑀,Ω𝑥,𝑦. (43)
Since 𝜕𝒯_𝑀,Ω((𝑥,𝑦))≠∅ (because 0∈𝜕𝒯_𝑀,Ω((𝑥,𝑦))) and (𝑥,𝑦)∈core(dom𝒯_𝑀,Ω), then 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))=𝜎_{𝜕𝒯𝑀,Ω((𝑥,𝑦))}(ℎ,𝑘). Thus 𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)=𝜎_{𝜕𝒯𝑀,Ω((𝑥,𝑦))}(ℎ,𝑘)≤0. (44) Since the reverse inequality 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))≥0 follows directly from part (1), then we get the equality form 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))=0, and hence, part (4) ensures that (ℎ,𝑘)∈𝑇(𝑔𝑝ℎ𝑀;(𝑥,𝑦)). Therefore, (𝜕𝒯_𝑀,Ω((𝑥,𝑦)))⁻⊂𝑇(𝑔𝑝ℎ𝑀;(𝑥,𝑦)). Since clearly we have (𝜕𝒯_𝑀,Ω((𝑥,𝑦)))⁻=(cl_𝑤∗(ℝ₊𝜕𝒯_𝑀,Ω((𝑥,𝑦))))⁻, then cl_𝑤∗ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦⁻⊂𝑇𝑔𝑝ℎ𝑀;𝑥,𝑦, (45) which ensures that 𝑁𝑔𝑝ℎ𝑀;𝑥,𝑦=𝑇𝑔𝑝ℎ𝑀;𝑥,𝑦⁻⊂cl_𝑤∗ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦⁻⁻=cl_𝑤∗ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦. (46)
The last equality follows from the fact that 𝐸×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 that 𝐹 is a normed vector space, 𝑆 is a closed convex subset in 𝐹, and 𝑦∈𝑆. Then, one has (1)𝑑^′_𝑆𝑦;𝑘=lim_𝑡→0+𝑡⁻¹𝑑_𝑆𝑦+𝑡𝑘=lim_𝑡→0+𝑑_{𝑡−1[𝑆−𝑦]}(𝑘)(2)∀𝑘∈𝐹;𝜕𝑑_𝑆𝑦=𝑁𝑆;𝑦∩𝑦^∗∈𝐹^∗∶𝑦^∗;(3)𝑑≤1_′𝑆𝑦;𝑘=𝑑_{𝑇(𝑆;𝑦)}(𝑘)(4)𝑇,∀𝑘∈𝐹;𝑆;𝑦=𝑘∈𝐹∶𝑑^′_𝑆𝑦;𝑘=0. (47)

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

Now, we consider those points outside the graph of set-valued mappings 𝑀 with nonempty closed values. Let (𝑥,𝑦)∉𝑔𝑝ℎ𝑀. Clearly, by Corollary 2, we have 𝒯_𝑀,Ω(𝑥,𝑦)>0. Denote 𝑟∶=𝒯_𝑀,Ω(𝑥,𝑦)>0, and define the enlargement set-valued mapping 𝑀_𝑟(𝑥)∶=𝑦∈𝐹∶𝒯_𝑀,Ω𝑥,𝑦≤𝑟. (48) Clearly, the graph of 𝑀_𝑟 is defined as 𝑔𝑝ℎ𝑀_𝑟∶={(𝑥,𝑦)∈𝐸×𝐹∶𝒯_𝑀,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∈𝑀_𝑟(𝑥) and 𝑀(𝑥)⊂𝑀_𝑟(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 𝑀_𝑟 whenever Ω is bounded convex. However, the l.s.c. of 𝒯_𝑀,Ω ensures the closedness of the graph of 𝑔𝑝ℎ𝑀_𝑟. It follows then, from Lemma 3, the convexity of the minimal time function 𝒯_𝑀𝑟,Ω whenever 𝒯_𝑀,Ω is lower semicontinuous and Ω is bounded convex.

Lemma 10. Let 𝐸 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 𝒯_𝑀,Ω𝑥,𝑦≤𝒯_𝑀𝑟,Ω𝑥,𝑦+𝑟 (49) and for any (𝑥,𝑦)∉𝑔𝑝ℎ𝑀_𝑟 we have 𝒯_𝑀,Ω𝑥,𝑦=𝒯_𝑀𝑟,Ω𝑥,𝑦+𝑟. (50)

Proof. First we prove the inequality 𝒯_𝑀,Ω𝑥,𝑦≤𝒯_𝑀𝑟,Ω𝑥,𝑦+𝑟,forany𝑦∈𝐹. (51) Assume that (𝑥,𝑦)∈𝐸×𝐹 with 𝒯_𝑀𝑟,Ω(𝑥,𝑦)<∞. The case 𝒯_𝑀𝑟,Ω(𝑥,𝑦)=∞ is obvious. Fix any 𝛿≥0 satisfying [𝑦+𝛿Ω]∩𝑀_𝑟(𝑥)≠∅. Then, there exists 𝑤∈Ω such that 𝑦+𝛿𝑤∈𝑀_𝑟(𝑥); that is, 𝒯_𝑀,Ω(𝑥,𝑦+𝛿𝑤)≤𝑟. Then, for any 𝜖>0, we can choose 𝜆>0 and 𝑧∈Ω such that 𝒯_𝑀,Ω𝑥,𝑦+𝛿𝑤≤𝜆<𝒯_𝑀,Ω𝑥,𝑦+𝛿𝑤+𝜖,𝑦+𝛿𝑤+𝜆𝑧∈𝑀(𝑥). (52) By convexity of Ω, we have 𝛿𝑤+𝜆𝑧∈(𝛿+𝜆)Ω, and hence, 𝑦+𝛿𝑤+𝜆𝑧∈𝑀(𝑥)∩𝑦+(𝛿+𝜆)Ω≠∅. (53) Therefore, 𝒯_𝑀,Ω𝑥,𝑦≤𝛿+𝜆<𝛿+𝒯_𝑀,Ω𝑥,𝑦+𝛿𝑤+𝜖<𝛿+𝑟+𝜖. (54) Since 𝛿 is taken arbitrarily satisfying (𝑦+𝛿Ω)∩𝑀_𝑟(𝑥)≠∅, we obtain by taking the infimum over all 𝛿 the inequality 𝒯_𝑀,Ω𝑥,𝑦≤𝒯_𝑀𝑟,Ω𝑥,𝑦+𝑟+𝜖,∀𝜖>0, (55) and by taking 𝜖→0 we get the first desired inequality 𝒯_𝑀,Ω𝑥,𝑦≤𝒯_𝑀𝑟,Ω𝑥,𝑦+𝑟. (56) Assume now that 𝑦∉𝑀_𝑟(𝑥), and let us prove the reverse inequality; that is, 𝒯_𝑀𝑟,Ω(𝑥,𝑦)+𝑟≤𝒯_𝑀,Ω(𝑥,𝑦). The case 𝒯_𝑀,Ω(𝑥,𝑦)=∞ is obvious so we suppose that 𝒯_𝑀,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, 𝒯_𝑀,Ω(𝑥,𝑦)≤𝛿, and since 𝑦∉𝑀_𝑟(𝑥), we have 𝛿>𝑟. Therefore, 𝑦+(𝛿−𝑟)𝑤+𝑟𝑤∈𝑀(𝑥)∩𝑦+(𝛿−𝑟)𝑤+𝑟Ω≠∅, (57) and so 𝒯_𝑀,Ω()𝑤()𝑥,𝑦+𝛿−𝑟≤𝑟,thatis,𝑦+𝛿−𝑟𝑤∈𝑀_𝑟(𝑥). (58)
Thus, 𝑦+(𝛿−𝑟)𝑤∈𝑀_𝑟(𝑥)∩𝑦+(𝛿−𝑟)Ω≠∅. (59)
Then, 𝒯_𝑀𝑟,Ω𝑥,𝑦≤𝛿−𝑟. (60)
Since 𝛿 is taken arbitrarily with [𝑦+𝛿Ω]∩𝑀(𝑥)≠∅, we can take the infimum over all those 𝛿, and hence, we obtain that 𝒯_𝑀𝑟,Ω𝑥,𝑦+𝑟≤𝒯_𝑀,Ω𝑥,𝑦. (61)
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 Δ_𝑀 to the case of minimal time function 𝒯_𝑀,Ω.

The first consequence of Lemma 10 is the following proposition in which we establish a relationship between the directional derivatives of 𝒯_𝑀,Ω and 𝒯_𝑀𝑟,Ω.

Proposition 11. Let 𝐸 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∶=𝒯_𝑀,Ω(𝑥,𝑦)>0. Then, for any (ℎ,𝑘)∈𝐸×𝐹, one has 𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)≤𝒯^′_𝑀𝑟,Ω𝑥,𝑦;(ℎ,𝑘). (62)

Proof. It follows, directly from the first part of the previous lemma, the fact that 𝒯_𝑀𝑟,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 mapping 𝑀_𝑟 as the set of all directions (ℎ,𝑘) in 𝐸×𝐹 for which the directional derivative of 𝑇_𝑀,Ω is nonnegative.

Theorem 12. Let 𝐸 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 𝒯_𝑀,Ω is l.s.c. at (𝑥,𝑦). Then, one has 𝑇𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦=(ℎ,𝑘)∈𝐸×𝐹∶𝒯^′_𝑀𝑟,Ω𝑥,𝑦;()=)ℎ,𝑘=0(ℎ,𝑘∈𝐸×𝐹∶𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)≤0. (63)

Proof. The first equality follows directly from part (4) in Theorem 8. The direct inclusion of the second equality follows directly from the previous proposition. So it remains to prove the inclusion (ℎ,𝑘)∈𝐸×𝐹∶𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)≤0⊂𝑇𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦. (64)
Let (ℎ,𝑘)∈𝐸×𝐹 with 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))≤0. Let (𝑡_𝑗)_𝑗 be a net in (0,∞) converging to 0 and satisfying the limit 𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)=lim_𝑗𝑡_𝑗⁻¹𝒯_𝑀,Ω𝑥+𝑡_𝑗ℎ,𝑦+𝑡_𝑗𝑘−𝒯_𝑀,Ω𝑥,𝑦. (65)
Use the lower semicontinuity of 𝒯_𝑀,Ω at (𝑥,𝑦) and the assumption 𝑦∉𝑀(𝑥) to find some 𝑗₀∈𝐽 such that 𝒯_𝑀,Ω𝑥+𝑡_𝑗ℎ,𝑦+𝑡_𝑗𝑘>2𝑡²_𝑗>0,forany𝑗≥𝑗₀. (66)
Using now the assumption 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))≤0 ensures the existence of some 𝑗₁≥𝑗₀ such that 𝒯_𝑀,Ω𝑥+𝑡_𝑗ℎ,𝑦+𝑡_𝑗𝑘≤𝑟+𝑡²_𝑗,forany𝑗≥𝑗₁. (67)
Thus, combining (66) and (67), we obtain 0<2𝑡²_𝑗<𝒯_𝑀,Ω𝑥+𝑡_𝑗ℎ,𝑦+𝑡_𝑗𝑘≤𝑟+𝑡²_𝑗,forany𝑗≥𝑗₁. (68)
Use now the definition of the minimal time function to choose for any 𝑗≥𝑗₁ some 𝑤_𝑗∈Ω and 𝛿_𝑗≥0 such that 𝒯_𝑀,Ω𝑥+𝑡_𝑗ℎ,𝑦+𝑡_𝑗𝑘≤𝛿_𝑗<𝒯_𝑀,Ω𝑥+𝑡_𝑗ℎ,𝑦+𝑡_𝑗𝑘+𝑡²_𝑗,𝑦+𝑡_𝑗𝑘+𝛿_𝑗𝑤_𝑗∈𝑀𝑥+𝑡_𝑗ℎ. (69)
Hence, 0≤𝛿_𝑗−2𝑡²_𝑗<𝑟,∀𝑗≥𝑗₁. (70)
Let 𝑧_𝑗∶=𝑦+𝑡_𝑗𝑘+2𝑡²_𝑗𝑤_𝑗 and 𝑘_𝑗∶=𝑘+2𝑡_𝑗𝑤_𝑗. Then, 𝑧_𝑗+𝛿_𝑗−2𝑡²_𝑗𝑤_𝑗=𝑦+𝑡_𝑗𝑘+𝛿_𝑗𝑤_𝑗∈𝑀𝑥+𝑡_𝑗ℎ∩𝑧_𝑗+𝛿_𝑗−2𝑡²_𝑗Ω≠∅. (71)
So 𝒯_𝑀,Ω(𝑥+𝑡_𝑗ℎ,𝑧_𝑗)≤𝛿_𝑗−2𝑡²_𝑗≤𝑟; that is, 𝑧_𝑗∈𝑀_𝑟(𝑥+𝑡_𝑗ℎ).
Therefore, 𝑦+𝑡_𝑗𝑘_𝑗=𝑦+𝑡_𝑗𝑘+2𝑡²_𝑗𝑤_𝑗∈𝑀_𝑟𝑥+𝑡_𝑗ℎ, (72) and so ℎ,𝑘_𝑗∈𝑡_𝑗⁻¹𝑔𝑝ℎ𝑀_𝑟−𝑥,𝑦⊂𝑇𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦. (73)
Now, the boundedness of Ω ensures the convergence of 2𝑡_𝑗𝑤_𝑗 to zero, which gives the convergence of 𝑘_𝑗 to 𝑘. Thus, by the closedness of the convex tangent cone, we get (ℎ,𝑘)∈𝑇(𝑔𝑝ℎ𝑀_𝑟,(𝑥,𝑦)), and hence, the proof is finished.

Various corollaries can be deduced from the above theorem by taking special cases of 𝑀 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, 𝑀_𝑟(𝑥) coincides with the enlargement set 𝑆(𝑟) defined by 𝑆(𝑟)={𝑦∈𝐹∶𝑑_𝑆(𝑦)≤𝑟}.

Corollary 13. Let 𝐹 be a normed vector space, 𝑆 a closed convex subset in 𝐹, and 𝑦∈𝑆. Then, one has 𝑑^′_𝑆𝑦;𝑘≤𝑑^′_𝑆(𝑟)𝑦;𝑘≤𝑑_{𝑇(𝑆(𝑟);𝑦)}𝑘,∀𝑇𝑆(𝑟);𝑘∈𝐹,𝑦=𝑘∈𝐹∶𝑑^′_𝑆(𝑟)𝑦;𝑘==0𝑘∈𝐹∶𝑑^′_𝑆𝑦;𝑘≤0. (74)

Now, we turn to establish some relationships between the convex normal cone to 𝑔𝑝ℎ𝑀_𝑟 and the convex subdifferential of 𝒯_𝑀,Ω at (𝑥,𝑦)∉𝑔𝑝ℎ𝑀.

Theorem 14. Let 𝐸 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 𝒯_𝑀,Ω is l.s.c. at (𝑥,𝑦), 𝐸 and 𝐹 are locally convex, and 𝜕𝒯_𝑀,Ω((𝑥,𝑦))≠∅. Then, 𝑁𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦=cl_𝑤∗ℝ₊𝜕𝒯_𝑀𝑟,Ω𝑥,𝑦=cl_𝑤∗ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦. (75)

Proof. The first equality follows directly from Theorem 8. We will prove the equality 𝑁(𝑔𝑝ℎ𝑀_𝑟;(𝑥,𝑦))=cl_𝑤∗(ℝ₊𝜕𝒯_𝑀,Ω((𝑥,𝑦))). Assume that 𝜕𝒯_𝑀,Ω(𝑥,𝑦)≠∅. First, we prove that 𝜕𝒯_𝑀,Ω𝑥,𝑦⊂𝑁𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦. (76)
Since 𝜕^𝐶𝒯_𝑀,Ω(𝑥,𝑦)≠∅, we take some (𝑥^∗,𝑦^∗)∈𝐸^∗×𝐹^∗ such that 𝑥^∗,𝑦^∗,(ℎ,𝑘)≤𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘),∀(ℎ,𝑘)∈𝐸×𝐹. (77)
Hence, for any (ℎ,𝑘)∈𝑇(𝑔𝑝ℎ𝑀_𝑟;(𝑥,𝑦)), we have 𝒯^′_𝑀,Ω((𝑥,𝑦);(ℎ,𝑘))≤0 (by Theorem 12), and so 𝑥^∗,𝑦^∗,(ℎ,𝑘)≤𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)≤0; (78) that is, (𝑥^∗,𝑦^∗)∈(𝑇(𝑔𝑝ℎ𝑀_𝑟;(𝑥,𝑦)))⁻=𝑁(𝑔𝑝ℎ𝑀_𝑟;(𝑥,𝑦)). Since 𝑁(𝑔𝑝ℎ𝑀_𝑟;(𝑥,𝑦)) is a closed convex cone in 𝐸^∗×𝐹^∗, we obtain that cl_𝑤∗ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦⊂𝑁𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦. (79)
Conversely, we have to prove that 𝑁𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦⊂cl_𝑤∗ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦. (80)
We will prove that the negative polar of cl_𝑤∗(ℝ₊𝜕𝒯_𝑀,Ω(𝑥,𝑦)) is included in 𝑇(𝑔𝑝ℎ𝑀_𝑟;(𝑥,𝑦))=(𝑁(𝑔𝑝ℎ𝑀_𝑟;(𝑥,𝑦)))⁻; that is, (by Theorem 12) cl_𝑤∗(ℝ₊𝜕𝒯_𝑀,Ω(𝑥,𝑦))⁻⊂(ℎ,𝑘)∈𝐸×𝐹∶𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)≤0. (81)
Note first that cl_𝑤∗ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦⁻=𝜕𝒯_𝑀,Ω𝑥,𝑦⁻. (82)
So consider any (ℎ,𝑘)∈(𝜕𝒯_𝑀,Ω(𝑥,𝑦))⁻. Then, we have 𝑥^∗,𝑦^∗,(ℎ,𝑘)≤0,∀𝑥^∗,𝑦^∗∈𝜕𝒯_𝑀,Ω𝑥,𝑦. (83)
Hence, (since 𝜕𝒯_𝑀,Ω(𝑥,𝑦))≠∅ and (𝑥,𝑦)∈core(dom𝑇_𝑀,Ω)) 𝒯^′_𝑀,Ω𝑥,𝑦;(ℎ,𝑘)=𝜎_{𝜕𝒯𝑀,Ω(𝑥,𝑦)}(ℎ,𝑘)=sup_{(𝑥∗,𝑦∗)∈𝜕𝒯𝑀,Ω(𝑥,𝑦)}⟨𝑥^∗,𝑦^∗,(ℎ,𝑘)⟩≤0, (84) which ensures by Theorem 12 that (ℎ,𝑘)∈𝑇(𝑔𝑝ℎ𝑀_𝑟;(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 𝒯_𝑀,Ω instead of its l.s.c. and the assumption 0∉𝜕𝒯_𝑀,Ω(𝑥,𝑦) instead of the assumption 𝜕𝒯_𝑀,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 𝑓(𝑢)−𝑓(𝑢)≤𝜌_𝑉(𝑢−𝑢),∀𝑢∈𝑈. (85)

Corollary 15. Let 𝐸 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 𝒯_𝑀,Ω is calm at (𝑥,𝑦) with 0∉𝜕𝒯_𝑀,Ω(𝑥,𝑦), then one has 𝑁𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦=ℝ₊𝜕𝒯_𝑀,Ω𝑥,𝑦. (86)

Proof. Using the calmness of 𝒯_𝑀,Ω at (𝑥,𝑦) and Banach-Alaoglu theorem (see Theorem  3.5 in [21]), we get the weak star compactness of 𝜕^𝐶𝒯_𝑀,Ω(𝑥,𝑦) in 𝐸^∗×𝐹^∗. Therefore, the assumption 0∉𝜕^𝐶𝒯_𝑀,Ω(𝑥,𝑦) with the weak star compactness of 𝜕^𝐶𝒯_𝑀,Ω(𝑥,𝑦) ensures the weak star closedness of the cone generated by 𝜕^𝐶𝒯_𝑀,Ω(𝑥,𝑦); that is, cl_𝑤∗(ℝ₊𝜕^𝐶𝒯_𝑀,Ω(𝑥,𝑦))=ℝ₊𝜕^𝐶𝒯_𝑀,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 function 𝒯_𝑀,Ω defined as follows 𝜋_{𝑀(𝑥),Ω}(𝑢)∶=𝑦∈𝑀(𝑥)∶𝒯_𝑀,Ω𝑥,𝑦=𝜌_Ω𝑦−𝑢. (87) We begin with the following lemma.

Lemma 16. Let 𝐸 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 𝒯_𝑀,Ω(𝑥,𝑦)<∞ and 𝜋_{𝑀(𝑥),Ω}(𝑦)≠∅. For every 𝑦∈𝜋_{𝑀(𝑥),Ω}(𝑦) and for every 𝑡∈(0,1], one has 𝒯_𝑀,Ω𝑥,𝑡𝑦+(1−𝑡)𝑦=(1−𝑡)𝒯_𝑀,Ω𝑥,𝑦. (88)
This ensures that 𝑦∈𝜋_{𝑀(𝑥),Ω}(𝑡𝑦+(1−𝑡)𝑦), for any 𝑡∈(0,1].

Proof. Let 𝑦∈𝜋_{𝑀(𝑥),Ω}(𝑦). Then 𝑦∈𝑀(𝑥) with 𝒯_𝑀,Ω(𝑥,𝑦)=𝜌_Ω(𝑦−𝑦)<∞. First, we prove the inequality 𝒯_𝑀,Ω𝑥,𝑡𝑦+(1−𝑡)𝑦≤(1−𝑡)𝒯_𝑀,Ω𝑥,𝑦,∀𝑡∈(0,1]. (89)
Put 𝛼∶=𝜌_Ω(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 𝑦=𝑡𝑦+(1−𝑡)𝑦+𝛼₀(1−𝑡)𝑦−𝑦𝛼₀∈𝑀(𝑥)∩𝑡𝑦+(1−𝑡)𝑦+𝛼₀(1−𝑡)Ω≠∅, (90) which ensures that 𝒯_𝑀,Ω𝑥,𝑡𝑦+(1−𝑡)𝑦≤𝛼₀(1−𝑡)<(1−𝑡)𝜌_Ω𝑦−𝑦≤()𝒯+𝜖1−𝑡_𝑀,Ω𝑥,𝑦+𝜖,∀𝜖>0. (91) Taking 𝜖→0⁺ finishes the proof of the first inequality.
By the first part, we have 𝒯_𝑀,Ω(𝑥,𝑡𝑦+(1−𝑡)𝑦)<∞ whenever 𝒯_𝑀,Ω(𝑥,𝑦)<∞ and so for any 𝜖>0, we can find some 𝑤∈Ω, some 𝛿>0, and some 𝛿₀>0 such that 𝒯_𝑀,Ω𝑥,𝑡𝑦+(1−𝑡)𝑦≤𝛿<𝒯_𝑀,Ω𝑥,𝑡𝑦+(1−𝑡)𝑦()+𝜖,𝑡𝑦+1−𝑡𝑦+𝛿𝑤∈𝑀(𝑥),𝑦−𝑦∈𝛿₀Ωwith0<𝜌_Ω𝑦−𝑦≤𝛿₀<𝜌_Ω𝑦−𝑦+𝜖. (92)
Then, by convexity of Ω, we have 𝛿₀𝑡((𝑦−𝑦)/𝛿₀)+𝛿𝑤∈(𝛿₀𝑡+𝛿)Ω, and so 𝑦+𝑡𝑦−𝑦+𝛿𝑤=𝑦+𝛿₀𝑡𝑦−𝑦𝛿₀+𝛿𝑤∈𝑀(𝑥)∩𝑦+𝛿₀𝑡+𝛿Ω≠∅. (93)
Hence, 𝒯_𝑀,Ω𝑥,𝑦≤𝛿₀𝜌𝑡+𝛿<𝑡_Ω𝑦−𝑦+𝜖+𝒯_𝑀,Ω𝑥,𝑡𝑦+(1−𝑡)𝑦+𝜖<𝑡𝒯_𝑀,Ω𝑥,𝑦+𝒯_𝑀,Ω𝑥,𝑡𝑦+(1−𝑡)𝑦+2𝜖, (94) and so by taking 𝜖→0⁺, we obtain that (1−𝑡)𝒯_𝑀,Ω𝑥,𝑦≤𝒯_𝑀,Ω𝑥,𝑡𝑦+(1−𝑡)𝑦. (95)
This completes the proof of the lemma.

We use this lemma to prove the following proposition on directional derivatives and convex subdifferentials of 𝒯_𝑀,Ω at points outside the graph.

Proposition 17. Assume that the assumptions of Lemma 16 are fulfilled. Then, for every (𝑥,𝑦)∉𝑔𝑝ℎ𝑀 and for every 𝑦∈𝜋_{𝑀(𝑥),Ω}(𝑦), one has 𝒯^′_𝑀,Ω𝑥,𝑦;0,𝑦−𝑦=−𝒯_𝑀,Ω𝑥,𝑦,𝜕𝒯_𝑀,Ω𝑥,𝑦⊂𝑥^∗,𝑦^∗∈𝐸^∗×𝐹^∗∶𝜎_Ω−𝑦^∗=1. (96)

Proof. By Lemma 16, we have 𝑇_𝑀,Ω𝑥,𝑦;0,𝑦−𝑦=lim_𝑡→0+𝑡⁻¹𝒯_𝑀,Ω𝑥,𝑦+𝑡𝑦−𝑦−𝒯_𝑀,Ω𝑥,𝑦=−𝒯_𝑀,Ω𝑥,𝑦. (97)
We turn now to show the inclusion 𝜕𝒯_𝑀,Ω𝑥,𝑦⊂𝑥^∗,𝑦^∗∈𝐸^∗×𝐹^∗∶𝜎_Ω−𝑦^∗=1. (98) Fix any (𝑥^∗,𝑦^∗)∈𝜕𝒯_𝑀,Ω(𝑥,𝑦). Then 𝑦^∗,𝑘≤𝒯^′_𝑀,Ω𝑥,𝑦;(0,𝑘),∀𝑘∈𝐹. (99)
By assumption, we have 𝜋_{𝑀(𝑥),Ω}(𝑦)≠∅. Then, there exists 𝑦₀∈𝜋_{𝑀(𝑥),Ω}(𝑦); that is, 𝑦₀∈𝑀(𝑥) with 𝒯_𝑀,Ω(𝑥,𝑦)=𝜌_Ω(𝑦₀−𝑦)≠0. By the first part, we obtain that 𝑦^∗,𝑦₀−𝑦≤𝒯^′_𝑀,Ω𝑥,𝑦;0,𝑦₀−𝑦=−𝒯_𝑀,Ω𝑥,𝑦=−𝜌_Ω𝑦₀−𝑦, (100) and so −𝑦^∗,𝑦₀−𝑦𝜌_Ω𝑦₀−𝑦≥1, (101) since 𝜌_Ω(𝑦₀−𝑦)≠0, because 𝑦∉𝑀(𝑥). Thus, for any 𝜖>0, there exist 𝑤∈Ω and 𝛼₀>0 such that 0<𝜌_Ω𝑦₀−𝑦≤𝛼₀<𝜌_Ω𝑦₀−𝑦+𝜖 (102) and 𝑦₀−𝑦∈𝛼₀Ω. Thus, 𝜎_Ω−𝑦^∗≥−𝑦^∗,𝑦₀−𝑦𝛼₀=𝜌_Ω𝑦₀−𝑦𝛼₀−𝑦^∗,𝑦₀−𝑦𝜌_Ω𝑦₀−𝑦≥𝜌_Ω𝑦₀−𝑦𝛼₀≥𝜌_Ω𝑦₀−𝑦𝜌_Ω𝑦₀−𝑦,+𝜖 (103) and hence, 𝜎_Ω−𝑦^∗𝜖≥1−𝜌_Ω𝑦₀−𝑦+𝜖,∀𝜖>0. (104)
Taking 𝜖→0⁺ yields 𝜎_Ω−𝑦^∗≥1. (105)
Let us prove the reverse inequality 𝜎_Ω−𝑦^∗≤1. (106)
Let (𝑡_𝑗)_𝑗 be a net in (0,∞) converging to 0 and satisfying 𝒯^′_𝑀,Ω𝑥,𝑦;(0,𝑘)=lim_𝑗𝑡_𝑗⁻¹𝒯_𝑀,Ω𝑥,𝑦+𝑡_𝑗𝑘−𝒯_𝑀,Ω𝑥,𝑦. (107)
By Lemma 10, we have, for 𝑟∶=𝒯_𝑀,Ω(𝑥,𝑦), the following inequality 𝒯_𝑀,Ω𝑥,𝑦−𝒯_𝑀,Ω𝑥,𝑦≤𝒯_𝑀𝑟,Ω𝑥,𝑦−𝒯_𝑀𝑟,Ω𝑥,𝑦,∀𝑥,𝑦∈𝐸×𝐹, (108) which ensures that 𝑡_𝑗⁻¹𝒯_𝑀,Ω𝑥,𝑦+𝑡_𝑗𝑘−𝒯_𝑀,Ω𝑥,𝑦≤𝑡_𝑗⁻¹𝒯_𝑀𝑟,Ω𝑥,𝑦+𝑡_𝑗𝑘=𝑡_𝑗⁻¹inf_{𝑢∈𝑀𝑟(𝑥)}𝜌_Ω𝑢−𝑦+𝑡_𝑗𝑘≤𝑡_𝑗⁻¹𝜌_Ω𝑦−𝑦+𝑡_𝑗𝑘=𝜌_Ω(−𝑘), (109) and so 𝑡_𝑗⁻¹𝒯_𝑀,Ω𝑥,𝑦+𝑡_𝑗𝑘−𝒯_𝑀,Ω𝑥,𝑦≤𝜌_Ω(−𝑘). (110)
Taking the limit on 𝑗, we get that 𝒯^′_𝑀,Ω𝑥,𝑦;(0,𝑘)=lim_𝑗𝑡_𝑗⁻¹𝒯_𝑀,Ω𝑥,𝑦+𝑡_𝑗𝑘−𝒯_𝑀,Ω𝑥,𝑦≤𝜌_Ω(−𝑘). (111)
Using the fact that (𝑥^∗,𝑦^∗)∈𝜕𝒯_𝑀,Ω(𝑥,𝑦), we write −𝑦^∗,−𝑘=𝑥^∗,𝑦^∗;(0,𝑘)≤𝒯_𝑀,Ω𝑥,𝑦;(0,𝑘)≤𝜌_Ω(−𝑘),∀𝑘∈𝐹. (112)
Fix any 𝑣∈Ω, and take 𝑘∶=−𝑣 in the previous inequality we get −𝑦^∗,𝑣=−𝑦^∗,−𝑘≤𝜌_Ω(𝑣)≤1. (113) This ensures that 𝜎_Ω(−𝑦^∗)≤1, and hence, the proof is finished.

The following theorem establishes another relationship between the convex subdifferential of 𝒯_𝑀,Ω and the convex normal cone of 𝑀_𝑟 at points outside the graph 𝑔𝑝ℎ𝑀.

Theorem 18. Let 𝐸 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 𝜋_{𝑀(𝑥),Ω}(𝑦)≠∅. Then, one has 𝜕𝒯_𝑀,Ω𝑥,𝑦⊂𝑁𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦∩𝑥^∗,𝑦^∗∈𝐸^∗×𝐹^∗∶𝜎_Ω−𝑦^∗=1. (114)
If, in addition, 𝒯_𝑀,Ω is calm at (𝑥,𝑦), then one has 𝜕𝒯_𝑀,Ω𝑥,𝑦=𝑁𝑔𝑝ℎ𝑀_𝑟;𝑥,𝑦∩𝑥^∗,𝑦^∗∈𝐸^∗×𝐹^∗∶𝜎_Ω−𝑦^∗=1. (115)

Proof. The first inclusion follows directly from the proof of Theorem 14 and Proposition 17. Let us prove the reverse inclusion. Fix any (𝑥^∗,𝑦^∗)∈𝑁(𝑔𝑝ℎ𝑀_𝑟;(𝑥,𝑦)) with 𝜎_Ω(−𝑦^∗)=1. First, let us check that 0∉𝜕𝒯_𝑀,Ω(𝑥,𝑦). Indeed, by the assumption 𝜋_{𝑀(𝑥),Ω}(𝑦)≠∅ and Proposition 17, there exists 𝑧₀∈𝜋_{𝑀(𝑥),Ω}(𝑦) such that 𝒯^′_𝑀,Ω𝑥,𝑦;0,𝑧₀−𝑦=−𝒯_𝑀,Ω𝑥,𝑦<0, (116) which ensures that 0∉𝜕𝒯_𝑀,Ω(𝑥,𝑦). Therefore, by Corollary 15, there exists some 𝜆>0 and (𝑢^∗,𝑣^∗)∈𝜕𝒯_𝑀,Ω(𝑥,𝑦), such that (𝑥^∗,𝑦^∗)=𝜆(𝑢^∗,𝑣^∗). Using the first part of the proof, we obtain that 𝜎_Ω(−𝑣^∗)=1, and so by combining this equality with the assumption 𝜎_Ω(−𝑦^∗)=1, we get 𝜆=1. Thus completing the proof.

5. Application of 𝒯_𝑀,Ω: Calmness and Exact Penalization

The primary goal in the present section is to make clear that the scalar function 𝒯_𝑀,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), (𝒫(𝑥)(𝑥),)minimize𝑓subjectto0∈𝑀 (117) 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. Let 𝑓, 𝑀, 𝐸, and 𝐹 be as in the statement of (𝒫), and consider the following perturbed problem 𝒫_𝑦(𝑥)(𝑥).minimize𝑓subjectto𝑦∈𝑀 (118)
Let 𝑆∶𝐹⇉𝐸 be the feasible set-valued mapping associated with (𝒫_𝑦); that is, 𝑆𝑦∶=𝑥∈dom𝑓∶𝑦∈𝑀(𝑥). (119)
Let (𝑥,𝑦)∈𝑔𝑝ℎ𝑆 and let Ω be a bounded set in 𝐹. One will say that the problem (𝒫_𝑦) is calm at 𝑥 with respect to Ω if there exist a constant 𝛼≥0 and 𝑋∈𝒩_𝐸(𝑥) such that for every 𝑥∈𝑋 and any 𝑦∈𝑀(𝑥) one has 𝑓(𝑥)≤𝑓(𝑥)+𝛼𝜌_Ω𝑦−𝑦. (120)
The constant 𝛼 and 𝑋 are called the modulus of calmness and neighborhood of calmness for (𝒫_𝑦) at 𝑥, respectively.

Remark 20. When 𝐹 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 (𝒫_𝑦) is calm at 𝑥 with respect to a bounded set Ω, then 𝑥 is necessarily a local solution to (𝒫_𝑦).

For any problem (𝒫_𝑦), any real number 𝛼≥0, and any bounded set Ω, we will associate the function 𝑃_𝛼,𝑦,Ω defined by 𝑃_𝛼,𝑦,Ω(𝑥)∶=𝑓(𝑥)+𝛼𝒯_𝑀,Ω𝑥,𝑦. (121)

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 (𝒫_𝑦) 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 (𝒫_𝑦) is calm at 𝑥 with respect to Ω with modulus 𝛼 and neighborhood 𝑋∈𝒩_𝐸(𝑥), then 𝑥 is a minimum over 𝑋 of the function 𝑃_𝛼,𝑦,Ω, for all 𝛼≥𝛼. If, in addition, Ω is convex and 0∈Ω, the converse holds.

Proof. Let 𝜖>0. Given any 𝑥∈𝑋∩dom𝑀≠∅. Then there exist 𝑡₀≥0 and 𝑤∈Ω such that 𝑦+𝑡₀𝑤∈𝑀(𝑥) and 𝒯_𝑀,Ω(𝑥,𝑦)≤𝑡₀<𝒯_𝑀,Ω(𝑥,𝑦)+𝜖. Thus, if 𝛼≥𝛼, we obtain from the calmness hypothesis that 𝑓(𝑥)≤𝑓(𝑥)+𝛼𝜌_Ω𝑦−𝑦,∀𝑥∈𝑋,∀𝑦∈𝑀(𝑥). (122)
Fix any 𝑥∈𝑋. Then by taking 𝑦∶=𝑦+𝑡₀𝑤 in the previous inequality and observing that 𝜌_Ω(𝑤)≤1 we obtain that 𝑓(𝑥)≤𝑓(𝑥)+𝛼𝜌_Ω𝑡₀𝑤≤𝑓(𝑥)+𝑡₀𝛼𝜌_Ω(𝑤)≤𝑓(𝑥)+𝑡₀𝛼≤𝑓(𝑥)+𝛼𝒯_𝑀,Ω𝑥,𝑦+𝛼𝜖. (123)
Hence 𝑃_𝛼,𝑦,Ω(𝑥)=𝑓(𝑥)≤𝑃_𝛼,𝑦,Ω(𝑥)+𝛼𝜖,∀𝑥∈𝑋. (124)
Since 𝜖>0 was taken arbitrarily, the direct implication is established by taking 𝜖→0.
Conversely, we assume that Ω is convex and 0∈Ω. Fix any 𝑥∈𝑋 and any 𝑦∈𝑀(𝑥). We claim first the inequality 𝒯_𝑀,Ω(𝑥,𝑦)≤𝜌_Ω(𝑦−𝑦). Clearly the case 𝜌_Ω(𝑦−𝑦)=∞ is obvious. So we assume that 𝜌_Ω(𝑦−𝑦)<∞. Since 𝑦∈𝑀(𝑥), we have 𝒯_𝑀,Ω(𝑥,𝑦)=0. Then by the finiteness of 𝒯_𝑀,Ω(𝑥,𝑦) we can find for any 𝜖>0 some 𝜆₀≥0 such that 0=𝒯_𝑀,Ω𝑥,𝑦≤𝜆₀<𝒯_𝑀,Ω𝑥,𝑦+𝜖=𝜖,𝑦+𝜆₀Ω∩𝑀(𝑥)≠∅. (125)
Hence, there exists 𝑤∈Ω such that 𝑦+𝜆₀𝑤∈𝑀(𝑥). Therefore, for any 𝑣∈Ω and any 𝛽>0, we have 𝑦+𝜆₀𝑤=𝑦−𝛽𝑣+𝜆₀𝜆𝑤+𝛽𝑣=𝑦−𝛽𝑣+₀𝜆+𝛽⁰𝜆₀𝛽+𝛽𝑤+𝜆₀𝑣(𝑥).+𝛽∈𝑀 (126)
Since Ω 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 𝒯_𝑀,Ω𝑥,𝑦−𝛽𝑣≤𝜆₀+𝛽<𝜖+𝛽,∀𝜖>0. (127)
Since 𝜌_Ω(𝑦−𝑦)<∞, we let 𝛽∶=𝜌_Ω(𝑦−𝑦)+𝜖<∞ and 𝑣∶=(𝑦−𝑦)/𝛽. Since Ω is convex and 0∈Ω, we have {ℎ∈𝐸∶𝜌_Ω(ℎ)<1}⊂Ω, and hence 𝑣∈Ω. Thus, (127) ensures that 𝒯_𝑀,Ω𝑥,𝑦=𝒯_𝑀,Ω𝑥,𝑦−𝛽𝑣<𝜖+𝛽=𝜌_Ω𝑦−𝑦+2𝜖,∀𝜖>0. (128)
After taking 𝜖→0, this inequality ensures the desired inequality; that is, 𝒯_𝑀,Ω(𝑥,𝑦)≤𝜌_Ω(𝑦−𝑦). Therefore, for any 𝑥∈𝑋 and any 𝑦∈𝑀(𝑥), we have 𝑓(𝑥)=𝑃_𝛼,𝑦,Ω(𝑥)≤𝑃_𝛼,𝑦,Ω(𝑥)=𝑓(𝑥)+𝛼𝒯_𝑀,Ω𝑥,𝑦≤𝑓(𝑥)+𝛼𝜌_Ω𝑦−𝑦. (129)
Hence, (𝒫_𝑦) is calm at 𝑥 with respect to Ω, and the proof of the theorem is complete.

Acknowledgment

The author extends his appreciations to the Deanship of Scientific Research at King Saud University for funding the work through the research group project no. RGP-VPP-024.