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Lipschitz-like property of an implicit multifunction and its applications
Nonlinear Analysis 74 (2011) 6256–6264
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Lipschitz-like property of an implicit multifunction and its applications✩ Thai Doan Chuong Department of Mathematics & Applications, Saigon University, 273 An Duong Vuong Street, Ward 3, District 5, Ho Chi Minh City, Viet Nam
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Article history: Received 15 June 2010 Accepted 3 June 2011 Communicated by Enzo Mitidieri MSC: 49K40 49J52 90C29 90C26
The aim of this work is twofold. First, we use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of an implicit multifunction. More explicitly, new sufficient conditions in terms of the Fréchet coderivative and the normal/Mordukhovich coderivative of parametric multifunctions for this implicit multifunction to have the Lipschitz-like property at a given point are established. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map in parametric vector optimization problems by employing the above implicit multifunction results. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Implicit multifunction Parametric vector optimization Efficient solution map Lipschitz-like Coderivative
1. Introduction The paper mainly deals with the stability theory of implicit multifunctions and parametric vector optimization problems. We first give some notation and definitions. Let X , Y be Banach spaces and (P , d) be a metric space, and let F : P × X ⇒ Y be a parametric multifunction. By means of this parametric multifunction one can define an implicit multifunction G : P ⇒ X as follows: G(p) := {x ∈ X | 0 ∈ F (p, x)}.
(1.1)
Let K ⊂ Y be a pointed, closed and convex cone with an apex at the origin. Definition 1.1. We say that y ∈ A is an efficient point of a subset A ⊂ Y with respect to K if and only if (y − K ) ∩ A = {y}. The set of efficient points of A is denoted by EffK A. We stipulate that EffK ∅ = ∅. Given a vector function f : P × X → Y , we consider the following parametric vector optimization problem: EffK f (p, x) | x ∈ X ,
(1.2)
where x is the unknown (decision variable) and p ∈ P a parameter. For each p ∈ P, we put
F (p) := EffK {f (p, x) | x ∈ X }
(1.3)
✩ This work was supported in part by Joint research and training on Variational Analysis and Optimization Theory, with oriented applications in some technological areas (Viet Nam–USA). E-mail address: [email protected].
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.06.005
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and call F : P ⇒ Y the efficient point multifunction of (1.2). One writes x ∈ S (p) to indicate that x ∈ X is an efficient solution of (1.2) if f (p, x) ∈ F (p). The multifunction S : P ⇒ X assigns to p the set of all efficient solutions of (1.2), i.e.,
S (p) := x ∈ X | f (p, x) ∈ F (p) ,
(1.4)
is called the efficient solution map of (1.2). Stability analysis in implicit multifunctions/or vector optimization problems has been investigated intensively by many researchers. One of the main problems here is to find sufficient conditions for the implicit multifunction G/or the efficient solution map S to have a certain stability property such as lower (upper) semi-continuous, continuous properties, calmness, Aubin (Lipschitz-like, pseudo-Lipschitz) properties, Lipschitz properties, and Hölder continuities. We refer the reader to [1–17] and their references therein for more details and discussions. In this paper we first use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of the implicit multifunction G defined in (1.1). More precisely, new sufficient conditions in terms of the Fréchet coderivative and the normal/Mordukhovich coderivative [18] of parametric multifunctions for this implicit multifunction G to be Lipschitz-like at a given point are established. These results generalize some corresponding results in [10,12]. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map S defined in (1.4) in parametric vector optimization problems by exploiting the above implicit multifunction results. The main tools for the proofs of our main results involve the Ekeland variational principle [19], the nonsmooth version of Fermat’s rule (see e.g., [18]), the fuzzy sum rule for the Fréchet subdifferential (see e.g., [18]). The rest of the paper is organized as follows. In Section 2, we first provide the basic definitions and notation from variational analysis and set-valued analysis. Then we recall some known auxiliary results which will be useful hereafter. Section 3 is devoted to providing sufficient conditions for the implicit multifunction G to be Lipschitz-like at a given point. In the last section we derive the Lipschitz-like property of the efficient solution map S of (1.2) by means of exploiting the implicit multifunction results given in Section 3. 2. Preliminaries and auxiliary results Throughout the paper we use the standard notation of variational analysis and generalized differentiation; see, e.g., [18,20,21]. Unless otherwise stated, all spaces under consideration are Banach spaces whose norms are always denoted by ‖ · ‖. The canonical pairing between X and its topological dual X ∗ is denoted by ⟨· , ·⟩. In this setting, w ∗ denotes the weak∗ topology in X ∗ , and A∗ denotes the adjoint operator of a linear continuous operator A. The symbols BX and BX ∗ stand for the closed unit balls of X and its topological dual X ∗ , respectively. The closed ball with center x and radius ρ is denoted by Bρ (x). As usual, the distance from u ∈ X to Ω ⊂ X is denoted by dist(u, Ω ) := infx∈Ω ‖x − u‖. Given a set-valued mapping F : X ⇒ X ∗ between a Banach space X and its topological dual X ∗ , we denote by w Lim sup F (x) := x∗ ∈ X ∗ |∃ sequences xk → x¯ and x∗k − → x∗ with x∗k ∈ F (xk ) for all k ∈ N ∗
x→¯x
the sequential Painlevé–Kuratowski upper/outer limit with respect to the norm topology of X and the weak∗ topology of X ∗ , where N := {1, 2, . . .}. Given Ω ⊂ X and ε ≥ 0, define the collection of ε -normals to Ω at x¯ ∈ Ω by
⟨x∗ , x − x¯ ⟩ Nε (¯x; Ω ) := x∗ ∈ X ∗ | lim sup ≤ε , ‖x − x¯ ‖ Ω x− →x¯
(2.1)
Ω
where x − → x¯ means that x → x¯ with x ∈ Ω . When ε = 0, the set N (¯x; Ω ) := N0 (¯x; Ω ) in (2.1) is a cone called the prenormal cone or the Fréchet normal cone to Ω at x¯ . The Mordukhovich normal cone N (¯x; Ω ) is obtained from Nε (x; Ω ) by taking the sequential Painlevé–Kuratowski upper limit in the weak∗ topology of X ∗ as N (¯x; Ω ) := Lim sup Nε (x; Ω ), x
(2.2)
Ω
−−→x¯ ε↓0
where one can put ε = 0 when Ω is closed around x¯ and the space X is Asplund, i.e., a Banach space whose separable subspaces have separable duals. Here, Ω is said to be (locally) closed around x¯ if there is a neighborhood U of x¯ such that Ω ∩ U is closed. It is well known that the class of Asplund spaces is sufficiently large containing, in particular, all reflexive spaces and all spaces with separable duals. We refer the reader who is interested in this class, to the books [22,18,20] for numerous its characterizations and discussions on its applications. Let F : X ⇒ Y be a set-valued mapping between Banach spaces with the domain and the graph dom F := {x ∈ X | F (x) ̸= ∅},
gph F := (x, y) ∈ X × Y |y ∈ F (x) .
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The normal/Mordukhovich coderivative and the Fréchet coderivative of F at (¯x, y¯ ) ∈ gphF are defined, respectively, by D∗ F (¯x, y¯ )(y∗ ) := x∗ ∈ X ∗ |(x∗ , −y∗ ) ∈ N ((¯x, y¯ ); gph F )
D F (¯x, y¯ )(y ) := {x ∈ X | (x , −y ) ∈ N ((¯x, y¯ ); gph F )}
∗
∗
∗
∗
∗
∗
∀y∗ ∈ Y ∗ , ∀y ∈ Y . ∗
∗
(2.3) (2.4)
If F is a single-valued mapping, to simplify the notation, one writes D∗ F (¯x)(y∗ ) (resp., D∗ F (¯x)(y∗ )) instead of D∗ F (¯x, F (¯x))(y∗ ) (resp., D∗ F (¯x, F (¯x))(y∗ )). A single-valued mapping f : X → Y is said to be strictly differentiable at x¯ if there is a linear continuous operator ∇ f (¯x) : X → Y such that lim
x,u→¯x
f (x) − f (u) − ⟨∇ f (¯x), x − u⟩
‖ x − u‖
= 0.
It is well known that for such mappings one has
D∗ f (¯x)(y∗ ) = D∗ f (¯x)(y∗ ) = {(∇ f (¯x))∗ y∗ } ∀y∗ ∈ Y ∗ , i.e., the Fréchet coderivative/normal coderivative is a generalization of the adjoint operator to the classical Jacobian/strict derivative; see [18] for more details. For an extended real-valued function ϕ : X → R := [−∞, ∞], we define dom ϕ = {x ∈ X | ϕ(x) < ∞},
epi ϕ = {(x, µ) ∈ X × R | µ ≥ ϕ(x)}.
The limiting/basic subdifferential and the Fréchet subdifferential of ϕ at x¯ with |ϕ(¯x)| < ∞ are defined, respectively, by
∂ϕ(¯x) := {x∗ ∈ X ∗ | (x∗ , −1) ∈ N ((¯x, ϕ(¯x)); epiϕ)}, ∂ϕ(¯x) := {x∗ ∈ X ∗ | (x∗ , −1) ∈ N ((¯x, ϕ(¯x)); epiϕ)}. If |ϕ(¯x)| = ∞, then one puts ∂ϕ(¯x) = ∂ϕ(¯x) = ∅. The nonsmooth version of Fermat’s rule (see, e.g., [18, Proposition 1.114]) which is an important fact for many applications can be formulated as follows: If x¯ is a local minimizer for ϕ , then 0 ∈ ∂ϕ(¯x) ⊂ ∂ϕ(¯x).
(2.5)
The following fuzzy sum rule for the Fréchet subdifferential will be useful for our investigation in what follows. Lemma 2.1 (See [18, Lemma 2.32]). Let X be an Asplund space, let ϕi : X → R, i = 1, 2, be proper such that ϕ1 is Lipschitz continuous around x¯ ∈ domϕ1 ∩ domϕ2 , and ϕ2 is lower semi-continuous (l.s.c.) around this point. Assume that the sum ϕ1 + ϕ2 attains a local minimum at x¯ . Then for any η > 0 there are xi ∈ x¯ + ηBX with |ϕi (xi ) − ϕi (¯x)| ≤ η, i = 1, 2, such that 0 ∈ ∂ϕ1 (x1 ) + ∂ϕ2 (x2 ) + ηBX ∗ . In what follows we also use the so-called Ekeland variational principle (see [19]). Lemma 2.2 (Ekeland Variational Principle). Let (X , d) be a complete metric space and f : X → R be a proper l.s.c. function bounded from below. Let ϵ > 0 and x0 ∈ X be given such that f (x0 ) ≤ infx∈X f (x) + ϵ . Then for any λ > 0 there is x¯ ∈ X satisfying (i) f (¯x) ≤ f (x0 ), (ii) d(¯x, x0 ) ≤ λ, (iii) f (¯x) < f (x) + λϵ d(x, x¯ ) for all x ∈ X \ {¯x}. A multifunction F : P ⇒ Y from a metric space (P , d) into a Banach space Y is Lipschitz-like or pseudo-Lipschitz at a given point (¯p, y¯ ) of its graph (see, e.g., [23]) if there exist neighborhoods U of p¯ , V of y¯ and a real number ℓ > 0 such that F (p) ∩ V ⊂ F (p′ ) + ℓd(p, p′ )BY
∀p, p′ ∈ U .
With V := Y , Lipschitz-like property reduces to the (local) Lipschitz continuity of multifunctions which is stated as follows: F is said to be locally Lipschitz at p¯ if there exist U of p¯ and a real number ℓ > 0 such that F (p) ⊂ F (p′ ) + ℓd(p, p′ )BY
∀p, p′ ∈ U .
3. Lipschitz-like property of implicit multifunctions In this section we shall provide new sufficient conditions for the implicit multifunction G declared in (1.1) to have the Lipschitz-like property at a given point.
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For the sake of clarity, for each p ∈ P we use the notation Fp := F (p, ·), and for each (¯x, y¯ ) ∈ gphFp¯ we define
c [Fp¯ ](¯x, y¯ ) := lim inf ‖x∗ ‖ | x∗ ∈ D∗ Fp (x, y)(y∗ ), p ∈ Bδ (¯p), x ∈ Bδ (¯x), δ↓0
y¯ ̸∈ Fp (x), y ∈ Fp (x) ∩ Bδ (¯y), ‖y∗ ‖ = 1 .
(3.1)
Theorem 3.1. Let X and Y be Asplund spaces, P be a metric space and let F : P × X ⇒ Y be a multifunction. Consider (¯p, x¯ ) ∈ P × X such that 0 ∈ F (¯p, x¯ ). Suppose that there exist δ > 0 and ℓ > 0 such that (i) gphFp ∩ [(¯x, 0) + δ(BX × BY )] is closed for all p ∈ Bδ (¯p); (ii) F (p, x) ⊂ F (p′ , x) + ℓd(p, p′ )BY
∀x ∈ Bδ (¯x),
∀p, p′ ∈ Bδ (¯p);
(3.2)
(iii) c [Fp¯ ](¯x, 0) > 0. Then for any positive c < c [Fp¯ ](¯x, 0) the implicit multifunction G defined in (1.1) is Lipschitz-like at (¯p, x¯ ) with modulus ℓ/c. Proof. Let δ > 0 and ℓ > 0 be as stated in the assumptions of the theorem and let 0 < c < c ′ < c [Fp¯ ](¯x, 0). Based on definition (3.1), we assume without loss of generality that p ∈ Bδ (¯p), x ∈ Bδ (¯x), 0 ̸∈ Fp (x), y ∈ Fp (x) ∩ δ BY , ‖y∗ ‖ = 1, x∗ ∈ D∗ Fp (x, y)(y∗ )
⇒ ‖x∗ ‖ > c ′ .
(3.3)
Choose a positive ρ < min{1/2, c /(4ℓ), 1/(2ℓ)}δ . We are going to show that G(p) ∩ Bρ (¯x) ⊂ G(p′ ) +
ℓ c
d(p, p′ )BX
∀p, p′ ∈ intBρ (¯p).
Take any p, p′ ∈ P satisfying d(p, p¯ ) < ρ,
d(p′ , p¯ ) < ρ
(3.4)
and any z ∈ G(p) ∩ Bρ (¯x). Then p, p ∈ Bδ (¯p), z ∈ Bδ (¯x), 0 ∈ F (p, z ). To prove the assertion we need to find a point xˆ ∈ G(p′ ) such that ′
‖z − xˆ ‖ ≤ (ℓ/c )d(p, p′ ).
(3.5)
It follows from (3.2) that there exists v ∈ F (p′ , z ) such that
‖v‖ ≤ ℓd(p, p′ ).
(3.6)
We assert that the function w : X × Y → R ∪ {∞} defined by
‖y‖ w(x, y) := ∞
if y ∈ Fp′ (x) otherwise
is l.s.c. on (¯x, 0) + δ(BX × BY ). Indeed, let λ ∈ R be arbitrary. It suffices to show that Ω := {(x, y) ∈ (¯x, 0) + δ(BX × BY ) | w(x, y) ≤ λ} is closed. Assume an arbitrary sequence {(xn , yn )} ⊂ Ω converges to some (x0 , y0 ). We shall justify that (x0 , y0 ) ∈ Ω . By {(xn , yn )} ⊂ Ω ,
(xn , yn ) ∈ (¯x, 0) + δ(BX × BY ), w(xn , yn ) ≤ λ, ∀n ∈ N.
(3.7) (3.8)
From (3.8) implies that
w(xn , yn ) = ‖yn ‖,
∀n ∈ N
(3.9)
and thus, yn ∈ Fp′ (xn ) i.e., (xn , yn ) ∈ gphFp′ ,
∀n ∈ N .
(3.10)
Combining (3.7) with (3.10) gives us (xn , yn ) ∈ [(¯x, 0)+δ(BX ×BY )]∩gphFp′ for all n ∈ N. In addition, since (xn , yn ) → (x0 , y0 ) as n → ∞ and since [(¯x, 0) + δ(BX × BY )] ∩ gphFp′ is closed by virtue of hypothesis (i), one has (x0 , y0 ) ∈ [(¯x, 0) + δ(BX × BY )] ∩ gphFp′ . Hence, w(x0 , y0 ) = ‖y0 ‖. Besides, we get from (3.8) and (3.9) that ‖y0 ‖ = limn→∞ ‖yn ‖ ≤ λ. So, (x0 , y0 ) ∈ Ω and our assertion holds. Obviously, w(x, y) ≥ 0 for all (x, y) ∈ (¯x, 0) + δ(BX × BY ) and
w(z , v) = ‖v‖ ≤ ℓd(p, p′ ) ≤
inf
(x,y)∈(¯x,0)+δ(BX ×BY )
w(x, y) + ℓd(p, p′ ).
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We are going to apply the Ekeland variational principle (Lemma 2.2). The product space X × Y will be considered with the equivalent norm ‖(x, y)‖α := max{‖x‖, α‖y‖} depending on a small parameter α > 0 such that
α<
1 c
−
1 c′
·
(3.11)
By Lemma 2.2, there is some (ˆx, yˆ ) ∈ (¯x, 0) + δ(BX × BY ) such that
w(ˆx, yˆ ) ≤ w(z , v),
(3.12)
ℓ
‖(ˆx, yˆ ) − (z , v)‖α ≤ d(p, p′ ),
(3.13)
c
w(ˆx, yˆ ) ≤ w(x, y) + c ‖(x, y) − (ˆx, yˆ )‖α ,
∀(x, y) ∈ (¯x, 0) + δ(BX × BY ).
(3.14)
Recalling the definition of w , we have yˆ ∈ Fp′ (ˆx), and conditions (3.12) and (3.14) take the following form:
‖ˆy‖ ≤ ‖v‖,
(3.15)
‖ˆy‖ ≤ ‖y‖ + c ‖(x, y) − (ˆx, yˆ )‖α ,
∀(x, y) ∈ gphFp′ ∩ [(¯x, 0) + δ(BX × BY )].
(3.16)
Now we are going to show that xˆ is the sought point. Due to (3.13), it satisfies (3.5), and we only need to show that 0 ∈ Fp′ (ˆx). Suppose by contradiction that 0 ̸∈ Fp′ (ˆx). Then yˆ ̸= 0. Due to (3.4), (3.6), (3.13) and (3.15),
δ δ ℓ ‖ˆx − x¯ ‖ ≤ ‖ˆx − z ‖ + ‖z − x¯ ‖ ≤ 2ρ + ρ < + = δ, c
2
2
‖ˆy‖ ≤ ℓ2ρ < δ.
(3.17)
It follows from (3.16) that (ˆx, yˆ ) is a point of local minimizer for the function ψ1 + ψ2 + ψ3 , where
ψ1 (x, y) := ‖y‖, ψ2 (x, y) := c ‖(x, y) − (ˆx, yˆ )‖α , 0 if (x, y) ∈ gphFp′ , ψ3 (x, y) := ∞ otherwise. Thus, 0 ∈ ∂(ψ1 + ψ2 + ψ3 )(ˆx, yˆ ) by (2.5). Functions ψ1 and ψ2 are convex and Lipschitz continuous, while ψ3 is l.s.c. around (ˆx, yˆ ) by virtue of hypothesis (i). One can apply the fuzzy sum rule (Lemma 2.1). For any ε > 0 there exist points (x1 , y1 ), (x2 , y2 ) ∈ X × Y , (x3 , y3 ) ∈ gphFp′ and elements (x∗1 , y∗1 ) ∈ ∂ψ1 (x1 , y1 ), (x∗2 , y∗2 ) ∈ ∂ψ2 (x2 , y2 ), (x∗3 , y∗3 ) ∈ ∂ψ3 (x3 , y3 ) such that
‖xi − xˆ ‖ < ε,
‖yi − yˆ ‖ < ε,
‖x1 + x2 + x3 ‖ < ε, ∗
∗
∗
i = 1, 2, 3;
(3.18)
‖y1 + y2 + y3 ‖ < ε.
(3.19)
∗
∗
∗
It follows by definitions of functions ψ1 , and ψ3 that x∗1 = 0,
y∗1 ∈ ∂‖y1 ‖,
(x3 , y3 ) ∈ N ((x3 , y3 ); gphFp′ ). ∗
∗
(3.20) (3.21)
We now prove that
‖x∗2 ‖ +
1
α
‖y∗2 ‖ ≤ c .
(3.22)
According to the evaluating formula of subdifferential of a norm in a Banach space (see e.g., [24, Example 4, p. 198]), we have
∂‖(x, y) − (ˆx, yˆ )‖α ⊂ {(x∗ , y∗ ) ∈ X ∗ × Y ∗ | ‖(x∗ , y∗ )‖ ≤ 1} ∀(x, y) ∈ X × Y . Thus,
∂ψ2 (x2 , y2 ) = c ∂‖(x2 , y2 ) − (ˆx, yˆ )‖α ⊂ {c (x∗ , y∗ ) ∈ X ∗ × Y ∗ | ‖(x∗ , y∗ )‖ ≤ 1}. So, there is (x∗ , y∗ ) ∈ X ∗ × Y ∗ with ‖(x∗ , y∗ )‖ ≤ 1 such that (x∗2 , y∗2 ) = c (x∗ , y∗ ). This yields
‖(x∗2 , y∗2 )‖ ≤ c .
(3.23)
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Besides,
‖(x∗2 , y∗2 )‖ = max{|(x∗2 , y∗2 )(x, y)| | ‖(x, y)‖α ≤ 1} = max{|x∗2 (x) + y∗2 (y)| | max{‖x‖, α‖y‖} ≤ 1}
(3.24)
= max{|x2 (x)| | ‖x‖ ≤ 1} + max{|y2 (y)| | α‖y‖ ≤ 1}
(3.25)
∗
∗
= max{|x∗2 (x)| | ‖x‖ ≤ 1} + = max{|x∗2 (x)| | ‖x‖ ≤ 1} + = ‖x∗2 ‖ +
1
α
1
α 1
α
max{|y∗2 (α y)| | ‖α y‖ ≤ 1}
(3.26)
max{|y∗2 (z )| | ‖z ‖ ≤ 1}
(3.27)
‖y∗2 ‖
(3.28)
Equalities (3.24) and (3.26)–(3.28) are easy to see. Let us verify (3.25). Obviously, max{|x∗2 (x) + y∗2 (y)| | max{‖x‖, α‖y‖} ≤ 1} ≤ max{|x∗2 (x)| | ‖x‖ ≤ 1} + max{|y∗2 (y)| | α‖y‖ ≤ 1}.
(3.29)
Suppose that there exist x0 ∈ X , y0 ∈ Y with ‖x0 ‖ ≤ 1, α‖y0 ‖ ≤ 1 such that |x2 (x0 )| := max{|x2 (x)| | ‖x‖ ≤ 1}, and |y∗2 (y0 )| := max{|y∗2 (y)| | α‖y‖ ≤ 1}. Further, we may assume without loss of generality that x∗2 (x0 ) ≥ 0 (otherwise if x∗2 (x0 ) < 0, then we can replace x0 by x′0 = −x0 ), and that y∗2 (y0 ) ≥ 0. Then, ∗
∗
|x∗2 (x0 ) + y∗2 (y0 )| = |x∗2 (x0 )| + |y∗2 (y0 )| = max{|x∗2 (x)| | ‖x‖ ≤ 1} + max{|y∗2 (y)| | α‖y‖ ≤ 1}. This together with (3.29) gives (3.25). Combining equalities (3.24)–(3.28) with (3.23) we arrive at (3.22). If ε is small enough, then, due to (3.18), y1 ̸= 0. It follows from (3.20) that
‖y∗1 ‖ = 1,
⟨y∗1 , y1 ⟩ = ‖y1 ‖.
(3.30)
Taking a smaller ε if necessary, we can ensure that ε < 1 − c α and, by (3.17) and (3.18), that x3 ∈ Bδ (¯x), ‖y3 ‖ < δ . It follows from (3.19), (3.20), (3.22) and (3.30) that
‖x∗3 ‖ < c + ε,
| ‖y∗3 ‖ − 1| < c α + ε.
(3.31)
By (3.31), y∗3 ̸= 0 and we can define x∗ := x∗3 /‖y∗3 ‖, y∗ = −y∗3 /‖y∗3 ‖. Then, ‖y∗ ‖ = 1, and x∗ ∈ D∗ Fp′ (x3 , y3 )(y∗ ) by (3.21). From (3.31) we obtain
‖x ∗ ‖ <
c+ε 1 − cα − ε
.
(3.32)
By (3.11), 1−cc α = 1/c1−α < c ′ . Taking a smaller ε again, we can ensure that 1−cc+ε < c ′ . Then (3.32) implies that ‖x∗ ‖ < c ′ . α−ε This contradicts (3.23). Hence 0 ∈ Fp′ (ˆx). The proof is complete. Note that, when applying Theorem 3.1, it is important to be able to verify the regularity assumption (iii), i.e., compute constant (3.1). We consider now some special cases when the computation of this constant can be simplified. For simplicity we limit ourselves to the case y¯ = 0 which is needed in Theorem 3.1. For each p ∈ P we consider a projection multifunction Πp : X ⇒ Y defined by
Πp (x) := {y ∈ Fp (x) | ‖y‖ = dist(0, Fp (x))} ∀x ∈ X . The next assertion follows directly from the definitions. Proposition 3.2. If Πp (x) ̸= ∅ for all p near p¯ and all x near x¯ , then
c [Fp¯ ](¯x, 0) = lim inf ‖x∗ ‖ | x∗ ∈ D∗ Fp (x, y)(y∗ ), p ∈ Bδ (¯p), x ∈ Bδ (¯x), δ↓0
0 ̸∈ Fp (x), y ∈ Πp (x) ∩ δ BY , ‖y∗ ‖ = 1 .
(3.33)
The assumption in Proposition 3.2 is satisfied automatically if Fp is nonempty compact-valued near x¯ and for all p near p¯ (see [11, Lemma 3.3]). In particular, it happens if dim Y < ∞ and Fp is nonempty closed-valued near x¯ and all p near p¯ . Another condition which is much stronger in general ensures the fulfillment of the assumption in Proposition 3.2 is that Πp is inner semicompact [18] near x¯ for all p near p¯ (see condition (A2 ) in [12, Theorem 3.1]). Beside the computation of (3.1) or (3.33) where one needs to know Fréchet coderivatives of Fp at all points (x, y) with x ̸∈ Fp−1 (¯y) near (¯x, y¯ ), we wish in many situations to deal with limiting/Mordukhovich ones. More explicitly, one will
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compute normal/Mordukhovich coderivatives of Fp at the corresponding points. In this case we consider the following constant:
c1 [Fp¯ ](¯x, y¯ ) := lim inf ‖x∗ ‖ | x∗ ∈ D∗ Fp (x, y)(y∗ ), p ∈ Bδ (¯p), x ∈ Bδ (¯x), δ↓0
y¯ ̸∈ Fp (x), y ∈ Fp (x) ∩ Bδ (¯y), ‖y∗ ‖ = 1 .
(3.34)
There is a similar representation for constant c1 [Fp¯ ](¯x, 0) as the form given in Proposition 3.2. More precisely, if Πp (x) ̸= ∅ for all p near p¯ and all x near x¯ , then
c1 [Fp¯ ](¯x, 0) = lim inf ‖x∗ ‖ | x∗ ∈ D∗ Fp (x, y)(y∗ ), p ∈ Bδ (¯p), x ∈ Bδ (¯x), δ↓0
0 ̸∈ Fp (x), y ∈ Πp (x) ∩ δ BY , ‖y∗ ‖ = 1 . Obviously, c [Fp¯ ](¯x, y¯ ) ≥ c1 [Fp¯ ](¯x, y¯ ). So we obtain the following result. Theorem 3.3. The conclusion of Theorem 3.1 remains valid if its assumption (iii) is replaced by the following one: (iii′ ) c1 [Fp¯ ](¯x, 0) > 0. Remark 3.4. Theorem 3.3 generalizes some corresponding results in [10,12]. More exactly, condition (SC) given in [10, Theorem 3.5] and assumptions (A2 ), (A3 ) given in [12, Theorem 3.3] are not needed. 4. Lipschitz-like property of efficient solution maps We now give sufficient conditions for the efficient solution map S defined in (1.4) to be Lipschitz-like at the reference point. Observe first that this efficient solution map S can be rewritten in the implicit multifunction form
S (p) := x ∈ X | 0 ∈ H (p, x) ,
where H : P × X ⇒ Y is a multifunction which is constructed via the objective function f and the efficient point multifunction declared in (1.3), H (p, x) := −f (p, x) + F (p).
(4.1)
Once again, we use the notation Hp := H (p, ·). Theorem 4.1. Let X and Y be Asplund spaces, P be a metric space and let H be the multifunction defined in (4.1). Consider (¯p, x¯ ) ∈ gphS . Suppose that there exist δ > 0 and ℓ > 0 such that (i) gphHp ∩ [(¯x, 0) + δ(BX × BY )] is closed for all p ∈ Bδ (¯p); (ii) H (p, x) ⊂ H (p′ , x) + ℓd(p, p′ )BY
∀x ∈ Bδ (¯x), ∀p, p′ ∈ Bδ (¯p);
(4.2)
(iii) c [Hp¯ ](¯x, 0) > 0. Then for any positive c < c [Hp¯ ](¯x, 0) the efficient solution map S of (1.2) is Lipschitz-like at (¯p, x¯ ) with modulus ℓ/c. Proof. The proof is directly derived from Theorem 3.1 by considering the multifunction H instead of F .
Remark 4.2. It is easy to verify that the property of H in (4.2) is valid under the fulfillment of the following two conditions: (a) there exist neighborhoods U of x¯ , W of p¯ , and a real number l > 0 such that
‖f (p, x) − f (p′ , x)‖ ≤ ld(p, p′ ) ∀x ∈ U , ∀p, p′ ∈ W ;
(4.3)
(b) F in (1.3) is locally Lipschitz at p¯ . The following corollary provides sufficient conditions for the efficient solution map S in (1.4) to be Lipschitz-like at the reference point under the strict differentiability of the objective function f . Corollary 4.3. Let X and Y be Asplund spaces, P be a normed space and let H be the multifunction defined in (4.1). Consider (¯p, x¯ ) ∈ gphS and suppose that f is strictly differentiable at (¯p, x¯ ). The conclusion of Theorem 4.1 remains valid if its assumption (ii) is replaced by the following condition: (ii′ ) F defined in (1.3) is locally Lipschitz at p¯ . Proof. Since f is strictly differentiable at (¯p, x¯ ), condition (4.3) is fulfilled. This together with the local Lipschitz property of F implies by Remark 4.2 that H satisfies condition (4.2). To finish the proof it remains to apply Theorem 4.1.
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Remark 4.4. From Theorem 3.3 we assert that the conclusion of Theorem 4.1, and thus Corollary 4.3, remains valid if its assumption (iii) is replaced by the following one: (iii′ ) c1 [Hp¯ ](¯x, 0) > 0. We observe here that it is not difficult to show by examples that assumptions (i) and (iii) in Theorem 4.1 are essential. The next example illustrates the importance of assumption (ii), namely the Lipschitz-like property of S defined in (1.4) may be violated if this assumption is omitted. Example 4.5. Let P = X = Y = R, K = R+ := [0; +∞) and let f : P × X → Y be a function which is given as follows: f (p, x) =
|x − p√| + 2p | x − p| − p
if p < 0, x ∈ X , if p ≥ 0, x ∈ X .
We consider the problem (1.2) with the efficient solution map S defined in (1.4). Let us examine (¯p, x¯ ) ∈ gphS , where p¯ := x¯ := 0. By simple computation, one gets
−|x − √ p| Hp (x) = −|x − p|
if p < 0, x ∈ X , if p ≥ 0, x ∈ X .
So, the condition (i) of Theorem 4.1 is satisfied, while the condition (ii) is not. Consider two the following possibilities: Case 1: p < 0. Take any x ∈ X with 0 ̸∈ Hp (x). The last relation means that x ̸= p. Put y := Hp (x) = −|x − p|. Then for any y∗ ∈ Y ∗ = R we obtain
∗ y ∗ ∗ D Hp (x, y)(y ) = −y ∗
if x < p, if x > p.
Case 2: p ≥ 0. Take any x ∈ X with 0 ̸∈ Hp (x). The last relation means that x ̸= for any y∗ ∈ Y ∗ = R we have
∗ y D∗ Hp (x, y)(y∗ ) = −y ∗
√
p. Set y := Hp (x) = −|x −
√
p|. Then
√
if x < √p, if x > p.
Taking into account the possibilities shown above, we get from (3.1) that c [Hp¯ ](¯x, 0) = 1. This means that the condition (iii) of Theorem 4.1 is fulfilled. It is easy to see that the efficient solution map
p S (p) = √
p
if p < 0, if p ≥ 0,
is not Lipschitz-like at (¯p, x¯ ). Finally, we mention that an important issue when applying Theorem 4.1 or Corollary 4.3, is the computation of constants c [Hp¯ ](¯x, 0) or c1 [Hp¯ ](¯x, 0) (see Remark 4.4), which requires the knowledge of the family of Fréchet coderivatives/or normal coderivatives of Hp at nearby points. Moreover, from (4.1) one can see that for each p ∈ P , Hp is the sum of the function f (p, ·) and the constant multifunction Fp : X ⇒ Y defined by Fp (x) := F (p) for all x ∈ X . So, if f (p, ·) is Fréchet/or strictly differentiable at every x near x¯ and for all p near p¯ , then we can significantly simplify the computation by exploiting the coderivative sum rules in [18]. For example, suppose that f (p, ·) is strictly differentiable at every x near x¯ and all p near p¯ . Then by utilizing the coderivative sum rule given in [18, Theorem 1.62(ii)] and the scalarization of the normal coderivative given in [18, Theorem 3.28], we obtain for any x near x¯ , p near p¯ and y ∈ −f (p, x) + F (p), D∗ Hp (x, y)(y∗ ) = −(∇x f (p, x))∗ y∗ + D∗ Fp (x, y + f (p, x))(y∗ ) ∀y∗ ∈ Y ∗ , where ∇x f (p, x) denotes the Fréchet derivative of f (p, ·) at x. Acknowledgment The author would like to thank the anonymous referee for his/her valuable comments and suggestions. References [1] E.M. Bednarczuk, Some stability results for vector optimization problems in partially ordered topological vector, in: Proceedings of the First World Congress on World Congress of Nonlinear Analysts, vol. III Table of Contents, Tampa, Florida, United States, 1996, pp. 2371–2382. [2] E.M. Bednarczuk, Upper Hölder continuity of minimal points, J. Convex Anal. 9 (2) (2002) 327–338. [3] T.D. Chuong, N.Q. Huy, J.-C. Yao, Stability of semi-infinite vector optimization problems under functional purtubations, J. Global Optim. 45 (2009) 583–595.
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[4] T.D. Chuong, N.Q. Huy, J.-C. Yao, Pseudo-Lipschitz property of linear semi-infinite vector optimization problems, European J. Oper. Res. 200 (2010) 639–644. [5] T.D. Chuong, A. Y. Kruger, J.-C. Yao, Calmness of efficient solution maps in parametric vector optimization, J. Global Optim. (2011) doi:10.1007/s10898011-9651-z. [6] T.D. Chuong, J.-C. Yao, Sufficient conditions for pseudo-Lipschitz property in convex semi-infinite vector optimization problems, Nonlinear Anal. 71 (2009) 6312–6322. [7] T.D. Chuong, J.-C. Yao, N.D. Yen, Further results on the lower semicontinuity of efficient point multifunctions, Pac. J. Optim. 6 (2010) 405–422. [8] P.H. Dien, N.D. Yen, On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints, Appl. Math. Optim. 24 (1991) 35–54. [9] A. Gopfert, H. Riahi, C. Tammer, C. Zalinescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003. [10] N.Q. Huy, J.-C. Yao, Stability of implicit multifunctions in Asplund spaces, Taiwanese J. Math. 13 (2009) 47–65. [11] Y.S. Ledyaev, Q.J. Zhu, Implicit multifunctions theorems, Set-Valued Anal. 7 (1999) 209–238. [12] G.M. Lee, N.N. Tam, N.D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008) 11–22. [13] D.T. Luc, Theory of Vector Optimization, in: Lecture Notes in Economics and Mathematical Systems, vol. 319, Springer-Verlag, Berlin, 1989. [14] Y. Sawaragi, H. Nakayama, T. Tanino, Theory of Multiobjective Optimization, in: Mathematics in Science and Engineering, vol. 176, Academic Press, Inc., Orlando, FL, 1985. [15] S.W. Xiang, Y.H. Zhou, Continuity properties of solutions of vector optimization, Nonlinear Anal. 64 (2006) 2496–2506. [16] S.W. Xiang, W.S. Yin, Stability results for efficient solutions of vector optimization problems, J. Optim. Theory Appl. 134 (2007) 385–398. [17] N.D. Yen, Implicit function theorems for set-valued maps, Acta Math. Vietnam. 12 (2) (1987) 17–28. [18] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation. I: Basic Theory, Springer, Berlin, 2006. [19] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974) 324–353. [20] B.S. Mordukhovich, Variational Analysis and Generalized Differentiation. II: Applications, Springer, Berlin, 2006. [21] R.T. Rockafellar, R.J-B. Wets, Variational Analysis, Springer, Berlin, 1998. [22] R.R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd ed., in: Lecture Notes in Math., vol. 1364, Springer, Berlin, 1993. [23] J.-P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, 1990. [24] A.D. Ioffe, V.M. Tihomirov, Theory of Extremal Problems, North-Holland Publishing Co., Amsterdam, New York, 1979.
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Lipschitz-like property of an implicit multifunction and its applications✩ Thai Doan Chuong Department of Mathematics & Applications, Saigon University, 273 An Duong Vuong Street, Ward 3, District 5, Ho Chi Minh City, Viet Nam
article
abstract
info
Article history: Received 15 June 2010 Accepted 3 June 2011 Communicated by Enzo Mitidieri MSC: 49K40 49J52 90C29 90C26
The aim of this work is twofold. First, we use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of an implicit multifunction. More explicitly, new sufficient conditions in terms of the Fréchet coderivative and the normal/Mordukhovich coderivative of parametric multifunctions for this implicit multifunction to have the Lipschitz-like property at a given point are established. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map in parametric vector optimization problems by employing the above implicit multifunction results. © 2011 Elsevier Ltd. All rights reserved.
Keywords: Implicit multifunction Parametric vector optimization Efficient solution map Lipschitz-like Coderivative
1. Introduction The paper mainly deals with the stability theory of implicit multifunctions and parametric vector optimization problems. We first give some notation and definitions. Let X , Y be Banach spaces and (P , d) be a metric space, and let F : P × X ⇒ Y be a parametric multifunction. By means of this parametric multifunction one can define an implicit multifunction G : P ⇒ X as follows: G(p) := {x ∈ X | 0 ∈ F (p, x)}.
(1.1)
Let K ⊂ Y be a pointed, closed and convex cone with an apex at the origin. Definition 1.1. We say that y ∈ A is an efficient point of a subset A ⊂ Y with respect to K if and only if (y − K ) ∩ A = {y}. The set of efficient points of A is denoted by EffK A. We stipulate that EffK ∅ = ∅. Given a vector function f : P × X → Y , we consider the following parametric vector optimization problem: EffK f (p, x) | x ∈ X ,
(1.2)
where x is the unknown (decision variable) and p ∈ P a parameter. For each p ∈ P, we put
F (p) := EffK {f (p, x) | x ∈ X }
(1.3)
✩ This work was supported in part by Joint research and training on Variational Analysis and Optimization Theory, with oriented applications in some technological areas (Viet Nam–USA). E-mail address: [email protected].
0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.06.005
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and call F : P ⇒ Y the efficient point multifunction of (1.2). One writes x ∈ S (p) to indicate that x ∈ X is an efficient solution of (1.2) if f (p, x) ∈ F (p). The multifunction S : P ⇒ X assigns to p the set of all efficient solutions of (1.2), i.e.,
S (p) := x ∈ X | f (p, x) ∈ F (p) ,
(1.4)
is called the efficient solution map of (1.2). Stability analysis in implicit multifunctions/or vector optimization problems has been investigated intensively by many researchers. One of the main problems here is to find sufficient conditions for the implicit multifunction G/or the efficient solution map S to have a certain stability property such as lower (upper) semi-continuous, continuous properties, calmness, Aubin (Lipschitz-like, pseudo-Lipschitz) properties, Lipschitz properties, and Hölder continuities. We refer the reader to [1–17] and their references therein for more details and discussions. In this paper we first use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of the implicit multifunction G defined in (1.1). More precisely, new sufficient conditions in terms of the Fréchet coderivative and the normal/Mordukhovich coderivative [18] of parametric multifunctions for this implicit multifunction G to be Lipschitz-like at a given point are established. These results generalize some corresponding results in [10,12]. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map S defined in (1.4) in parametric vector optimization problems by exploiting the above implicit multifunction results. The main tools for the proofs of our main results involve the Ekeland variational principle [19], the nonsmooth version of Fermat’s rule (see e.g., [18]), the fuzzy sum rule for the Fréchet subdifferential (see e.g., [18]). The rest of the paper is organized as follows. In Section 2, we first provide the basic definitions and notation from variational analysis and set-valued analysis. Then we recall some known auxiliary results which will be useful hereafter. Section 3 is devoted to providing sufficient conditions for the implicit multifunction G to be Lipschitz-like at a given point. In the last section we derive the Lipschitz-like property of the efficient solution map S of (1.2) by means of exploiting the implicit multifunction results given in Section 3. 2. Preliminaries and auxiliary results Throughout the paper we use the standard notation of variational analysis and generalized differentiation; see, e.g., [18,20,21]. Unless otherwise stated, all spaces under consideration are Banach spaces whose norms are always denoted by ‖ · ‖. The canonical pairing between X and its topological dual X ∗ is denoted by ⟨· , ·⟩. In this setting, w ∗ denotes the weak∗ topology in X ∗ , and A∗ denotes the adjoint operator of a linear continuous operator A. The symbols BX and BX ∗ stand for the closed unit balls of X and its topological dual X ∗ , respectively. The closed ball with center x and radius ρ is denoted by Bρ (x). As usual, the distance from u ∈ X to Ω ⊂ X is denoted by dist(u, Ω ) := infx∈Ω ‖x − u‖. Given a set-valued mapping F : X ⇒ X ∗ between a Banach space X and its topological dual X ∗ , we denote by w Lim sup F (x) := x∗ ∈ X ∗ |∃ sequences xk → x¯ and x∗k − → x∗ with x∗k ∈ F (xk ) for all k ∈ N ∗
x→¯x
the sequential Painlevé–Kuratowski upper/outer limit with respect to the norm topology of X and the weak∗ topology of X ∗ , where N := {1, 2, . . .}. Given Ω ⊂ X and ε ≥ 0, define the collection of ε -normals to Ω at x¯ ∈ Ω by
⟨x∗ , x − x¯ ⟩ Nε (¯x; Ω ) := x∗ ∈ X ∗ | lim sup ≤ε , ‖x − x¯ ‖ Ω x− →x¯
(2.1)
Ω
where x − → x¯ means that x → x¯ with x ∈ Ω . When ε = 0, the set N (¯x; Ω ) := N0 (¯x; Ω ) in (2.1) is a cone called the prenormal cone or the Fréchet normal cone to Ω at x¯ . The Mordukhovich normal cone N (¯x; Ω ) is obtained from Nε (x; Ω ) by taking the sequential Painlevé–Kuratowski upper limit in the weak∗ topology of X ∗ as N (¯x; Ω ) := Lim sup Nε (x; Ω ), x
(2.2)
Ω
−−→x¯ ε↓0
where one can put ε = 0 when Ω is closed around x¯ and the space X is Asplund, i.e., a Banach space whose separable subspaces have separable duals. Here, Ω is said to be (locally) closed around x¯ if there is a neighborhood U of x¯ such that Ω ∩ U is closed. It is well known that the class of Asplund spaces is sufficiently large containing, in particular, all reflexive spaces and all spaces with separable duals. We refer the reader who is interested in this class, to the books [22,18,20] for numerous its characterizations and discussions on its applications. Let F : X ⇒ Y be a set-valued mapping between Banach spaces with the domain and the graph dom F := {x ∈ X | F (x) ̸= ∅},
gph F := (x, y) ∈ X × Y |y ∈ F (x) .
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The normal/Mordukhovich coderivative and the Fréchet coderivative of F at (¯x, y¯ ) ∈ gphF are defined, respectively, by D∗ F (¯x, y¯ )(y∗ ) := x∗ ∈ X ∗ |(x∗ , −y∗ ) ∈ N ((¯x, y¯ ); gph F )
D F (¯x, y¯ )(y ) := {x ∈ X | (x , −y ) ∈ N ((¯x, y¯ ); gph F )}
∗
∗
∗
∗
∗
∗
∀y∗ ∈ Y ∗ , ∀y ∈ Y . ∗
∗
(2.3) (2.4)
If F is a single-valued mapping, to simplify the notation, one writes D∗ F (¯x)(y∗ ) (resp., D∗ F (¯x)(y∗ )) instead of D∗ F (¯x, F (¯x))(y∗ ) (resp., D∗ F (¯x, F (¯x))(y∗ )). A single-valued mapping f : X → Y is said to be strictly differentiable at x¯ if there is a linear continuous operator ∇ f (¯x) : X → Y such that lim
x,u→¯x
f (x) − f (u) − ⟨∇ f (¯x), x − u⟩
‖ x − u‖
= 0.
It is well known that for such mappings one has
D∗ f (¯x)(y∗ ) = D∗ f (¯x)(y∗ ) = {(∇ f (¯x))∗ y∗ } ∀y∗ ∈ Y ∗ , i.e., the Fréchet coderivative/normal coderivative is a generalization of the adjoint operator to the classical Jacobian/strict derivative; see [18] for more details. For an extended real-valued function ϕ : X → R := [−∞, ∞], we define dom ϕ = {x ∈ X | ϕ(x) < ∞},
epi ϕ = {(x, µ) ∈ X × R | µ ≥ ϕ(x)}.
The limiting/basic subdifferential and the Fréchet subdifferential of ϕ at x¯ with |ϕ(¯x)| < ∞ are defined, respectively, by
∂ϕ(¯x) := {x∗ ∈ X ∗ | (x∗ , −1) ∈ N ((¯x, ϕ(¯x)); epiϕ)}, ∂ϕ(¯x) := {x∗ ∈ X ∗ | (x∗ , −1) ∈ N ((¯x, ϕ(¯x)); epiϕ)}. If |ϕ(¯x)| = ∞, then one puts ∂ϕ(¯x) = ∂ϕ(¯x) = ∅. The nonsmooth version of Fermat’s rule (see, e.g., [18, Proposition 1.114]) which is an important fact for many applications can be formulated as follows: If x¯ is a local minimizer for ϕ , then 0 ∈ ∂ϕ(¯x) ⊂ ∂ϕ(¯x).
(2.5)
The following fuzzy sum rule for the Fréchet subdifferential will be useful for our investigation in what follows. Lemma 2.1 (See [18, Lemma 2.32]). Let X be an Asplund space, let ϕi : X → R, i = 1, 2, be proper such that ϕ1 is Lipschitz continuous around x¯ ∈ domϕ1 ∩ domϕ2 , and ϕ2 is lower semi-continuous (l.s.c.) around this point. Assume that the sum ϕ1 + ϕ2 attains a local minimum at x¯ . Then for any η > 0 there are xi ∈ x¯ + ηBX with |ϕi (xi ) − ϕi (¯x)| ≤ η, i = 1, 2, such that 0 ∈ ∂ϕ1 (x1 ) + ∂ϕ2 (x2 ) + ηBX ∗ . In what follows we also use the so-called Ekeland variational principle (see [19]). Lemma 2.2 (Ekeland Variational Principle). Let (X , d) be a complete metric space and f : X → R be a proper l.s.c. function bounded from below. Let ϵ > 0 and x0 ∈ X be given such that f (x0 ) ≤ infx∈X f (x) + ϵ . Then for any λ > 0 there is x¯ ∈ X satisfying (i) f (¯x) ≤ f (x0 ), (ii) d(¯x, x0 ) ≤ λ, (iii) f (¯x) < f (x) + λϵ d(x, x¯ ) for all x ∈ X \ {¯x}. A multifunction F : P ⇒ Y from a metric space (P , d) into a Banach space Y is Lipschitz-like or pseudo-Lipschitz at a given point (¯p, y¯ ) of its graph (see, e.g., [23]) if there exist neighborhoods U of p¯ , V of y¯ and a real number ℓ > 0 such that F (p) ∩ V ⊂ F (p′ ) + ℓd(p, p′ )BY
∀p, p′ ∈ U .
With V := Y , Lipschitz-like property reduces to the (local) Lipschitz continuity of multifunctions which is stated as follows: F is said to be locally Lipschitz at p¯ if there exist U of p¯ and a real number ℓ > 0 such that F (p) ⊂ F (p′ ) + ℓd(p, p′ )BY
∀p, p′ ∈ U .
3. Lipschitz-like property of implicit multifunctions In this section we shall provide new sufficient conditions for the implicit multifunction G declared in (1.1) to have the Lipschitz-like property at a given point.
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For the sake of clarity, for each p ∈ P we use the notation Fp := F (p, ·), and for each (¯x, y¯ ) ∈ gphFp¯ we define
c [Fp¯ ](¯x, y¯ ) := lim inf ‖x∗ ‖ | x∗ ∈ D∗ Fp (x, y)(y∗ ), p ∈ Bδ (¯p), x ∈ Bδ (¯x), δ↓0
y¯ ̸∈ Fp (x), y ∈ Fp (x) ∩ Bδ (¯y), ‖y∗ ‖ = 1 .
(3.1)
Theorem 3.1. Let X and Y be Asplund spaces, P be a metric space and let F : P × X ⇒ Y be a multifunction. Consider (¯p, x¯ ) ∈ P × X such that 0 ∈ F (¯p, x¯ ). Suppose that there exist δ > 0 and ℓ > 0 such that (i) gphFp ∩ [(¯x, 0) + δ(BX × BY )] is closed for all p ∈ Bδ (¯p); (ii) F (p, x) ⊂ F (p′ , x) + ℓd(p, p′ )BY
∀x ∈ Bδ (¯x),
∀p, p′ ∈ Bδ (¯p);
(3.2)
(iii) c [Fp¯ ](¯x, 0) > 0. Then for any positive c < c [Fp¯ ](¯x, 0) the implicit multifunction G defined in (1.1) is Lipschitz-like at (¯p, x¯ ) with modulus ℓ/c. Proof. Let δ > 0 and ℓ > 0 be as stated in the assumptions of the theorem and let 0 < c < c ′ < c [Fp¯ ](¯x, 0). Based on definition (3.1), we assume without loss of generality that p ∈ Bδ (¯p), x ∈ Bδ (¯x), 0 ̸∈ Fp (x), y ∈ Fp (x) ∩ δ BY , ‖y∗ ‖ = 1, x∗ ∈ D∗ Fp (x, y)(y∗ )
⇒ ‖x∗ ‖ > c ′ .
(3.3)
Choose a positive ρ < min{1/2, c /(4ℓ), 1/(2ℓ)}δ . We are going to show that G(p) ∩ Bρ (¯x) ⊂ G(p′ ) +
ℓ c
d(p, p′ )BX
∀p, p′ ∈ intBρ (¯p).
Take any p, p′ ∈ P satisfying d(p, p¯ ) < ρ,
d(p′ , p¯ ) < ρ
(3.4)
and any z ∈ G(p) ∩ Bρ (¯x). Then p, p ∈ Bδ (¯p), z ∈ Bδ (¯x), 0 ∈ F (p, z ). To prove the assertion we need to find a point xˆ ∈ G(p′ ) such that ′
‖z − xˆ ‖ ≤ (ℓ/c )d(p, p′ ).
(3.5)
It follows from (3.2) that there exists v ∈ F (p′ , z ) such that
‖v‖ ≤ ℓd(p, p′ ).
(3.6)
We assert that the function w : X × Y → R ∪ {∞} defined by
‖y‖ w(x, y) := ∞
if y ∈ Fp′ (x) otherwise
is l.s.c. on (¯x, 0) + δ(BX × BY ). Indeed, let λ ∈ R be arbitrary. It suffices to show that Ω := {(x, y) ∈ (¯x, 0) + δ(BX × BY ) | w(x, y) ≤ λ} is closed. Assume an arbitrary sequence {(xn , yn )} ⊂ Ω converges to some (x0 , y0 ). We shall justify that (x0 , y0 ) ∈ Ω . By {(xn , yn )} ⊂ Ω ,
(xn , yn ) ∈ (¯x, 0) + δ(BX × BY ), w(xn , yn ) ≤ λ, ∀n ∈ N.
(3.7) (3.8)
From (3.8) implies that
w(xn , yn ) = ‖yn ‖,
∀n ∈ N
(3.9)
and thus, yn ∈ Fp′ (xn ) i.e., (xn , yn ) ∈ gphFp′ ,
∀n ∈ N .
(3.10)
Combining (3.7) with (3.10) gives us (xn , yn ) ∈ [(¯x, 0)+δ(BX ×BY )]∩gphFp′ for all n ∈ N. In addition, since (xn , yn ) → (x0 , y0 ) as n → ∞ and since [(¯x, 0) + δ(BX × BY )] ∩ gphFp′ is closed by virtue of hypothesis (i), one has (x0 , y0 ) ∈ [(¯x, 0) + δ(BX × BY )] ∩ gphFp′ . Hence, w(x0 , y0 ) = ‖y0 ‖. Besides, we get from (3.8) and (3.9) that ‖y0 ‖ = limn→∞ ‖yn ‖ ≤ λ. So, (x0 , y0 ) ∈ Ω and our assertion holds. Obviously, w(x, y) ≥ 0 for all (x, y) ∈ (¯x, 0) + δ(BX × BY ) and
w(z , v) = ‖v‖ ≤ ℓd(p, p′ ) ≤
inf
(x,y)∈(¯x,0)+δ(BX ×BY )
w(x, y) + ℓd(p, p′ ).
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We are going to apply the Ekeland variational principle (Lemma 2.2). The product space X × Y will be considered with the equivalent norm ‖(x, y)‖α := max{‖x‖, α‖y‖} depending on a small parameter α > 0 such that
α<
1 c
−
1 c′
·
(3.11)
By Lemma 2.2, there is some (ˆx, yˆ ) ∈ (¯x, 0) + δ(BX × BY ) such that
w(ˆx, yˆ ) ≤ w(z , v),
(3.12)
ℓ
‖(ˆx, yˆ ) − (z , v)‖α ≤ d(p, p′ ),
(3.13)
c
w(ˆx, yˆ ) ≤ w(x, y) + c ‖(x, y) − (ˆx, yˆ )‖α ,
∀(x, y) ∈ (¯x, 0) + δ(BX × BY ).
(3.14)
Recalling the definition of w , we have yˆ ∈ Fp′ (ˆx), and conditions (3.12) and (3.14) take the following form:
‖ˆy‖ ≤ ‖v‖,
(3.15)
‖ˆy‖ ≤ ‖y‖ + c ‖(x, y) − (ˆx, yˆ )‖α ,
∀(x, y) ∈ gphFp′ ∩ [(¯x, 0) + δ(BX × BY )].
(3.16)
Now we are going to show that xˆ is the sought point. Due to (3.13), it satisfies (3.5), and we only need to show that 0 ∈ Fp′ (ˆx). Suppose by contradiction that 0 ̸∈ Fp′ (ˆx). Then yˆ ̸= 0. Due to (3.4), (3.6), (3.13) and (3.15),
δ δ ℓ ‖ˆx − x¯ ‖ ≤ ‖ˆx − z ‖ + ‖z − x¯ ‖ ≤ 2ρ + ρ < + = δ, c
2
2
‖ˆy‖ ≤ ℓ2ρ < δ.
(3.17)
It follows from (3.16) that (ˆx, yˆ ) is a point of local minimizer for the function ψ1 + ψ2 + ψ3 , where
ψ1 (x, y) := ‖y‖, ψ2 (x, y) := c ‖(x, y) − (ˆx, yˆ )‖α , 0 if (x, y) ∈ gphFp′ , ψ3 (x, y) := ∞ otherwise. Thus, 0 ∈ ∂(ψ1 + ψ2 + ψ3 )(ˆx, yˆ ) by (2.5). Functions ψ1 and ψ2 are convex and Lipschitz continuous, while ψ3 is l.s.c. around (ˆx, yˆ ) by virtue of hypothesis (i). One can apply the fuzzy sum rule (Lemma 2.1). For any ε > 0 there exist points (x1 , y1 ), (x2 , y2 ) ∈ X × Y , (x3 , y3 ) ∈ gphFp′ and elements (x∗1 , y∗1 ) ∈ ∂ψ1 (x1 , y1 ), (x∗2 , y∗2 ) ∈ ∂ψ2 (x2 , y2 ), (x∗3 , y∗3 ) ∈ ∂ψ3 (x3 , y3 ) such that
‖xi − xˆ ‖ < ε,
‖yi − yˆ ‖ < ε,
‖x1 + x2 + x3 ‖ < ε, ∗
∗
∗
i = 1, 2, 3;
(3.18)
‖y1 + y2 + y3 ‖ < ε.
(3.19)
∗
∗
∗
It follows by definitions of functions ψ1 , and ψ3 that x∗1 = 0,
y∗1 ∈ ∂‖y1 ‖,
(x3 , y3 ) ∈ N ((x3 , y3 ); gphFp′ ). ∗
∗
(3.20) (3.21)
We now prove that
‖x∗2 ‖ +
1
α
‖y∗2 ‖ ≤ c .
(3.22)
According to the evaluating formula of subdifferential of a norm in a Banach space (see e.g., [24, Example 4, p. 198]), we have
∂‖(x, y) − (ˆx, yˆ )‖α ⊂ {(x∗ , y∗ ) ∈ X ∗ × Y ∗ | ‖(x∗ , y∗ )‖ ≤ 1} ∀(x, y) ∈ X × Y . Thus,
∂ψ2 (x2 , y2 ) = c ∂‖(x2 , y2 ) − (ˆx, yˆ )‖α ⊂ {c (x∗ , y∗ ) ∈ X ∗ × Y ∗ | ‖(x∗ , y∗ )‖ ≤ 1}. So, there is (x∗ , y∗ ) ∈ X ∗ × Y ∗ with ‖(x∗ , y∗ )‖ ≤ 1 such that (x∗2 , y∗2 ) = c (x∗ , y∗ ). This yields
‖(x∗2 , y∗2 )‖ ≤ c .
(3.23)
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Besides,
‖(x∗2 , y∗2 )‖ = max{|(x∗2 , y∗2 )(x, y)| | ‖(x, y)‖α ≤ 1} = max{|x∗2 (x) + y∗2 (y)| | max{‖x‖, α‖y‖} ≤ 1}
(3.24)
= max{|x2 (x)| | ‖x‖ ≤ 1} + max{|y2 (y)| | α‖y‖ ≤ 1}
(3.25)
∗
∗
= max{|x∗2 (x)| | ‖x‖ ≤ 1} + = max{|x∗2 (x)| | ‖x‖ ≤ 1} + = ‖x∗2 ‖ +
1
α
1
α 1
α
max{|y∗2 (α y)| | ‖α y‖ ≤ 1}
(3.26)
max{|y∗2 (z )| | ‖z ‖ ≤ 1}
(3.27)
‖y∗2 ‖
(3.28)
Equalities (3.24) and (3.26)–(3.28) are easy to see. Let us verify (3.25). Obviously, max{|x∗2 (x) + y∗2 (y)| | max{‖x‖, α‖y‖} ≤ 1} ≤ max{|x∗2 (x)| | ‖x‖ ≤ 1} + max{|y∗2 (y)| | α‖y‖ ≤ 1}.
(3.29)
Suppose that there exist x0 ∈ X , y0 ∈ Y with ‖x0 ‖ ≤ 1, α‖y0 ‖ ≤ 1 such that |x2 (x0 )| := max{|x2 (x)| | ‖x‖ ≤ 1}, and |y∗2 (y0 )| := max{|y∗2 (y)| | α‖y‖ ≤ 1}. Further, we may assume without loss of generality that x∗2 (x0 ) ≥ 0 (otherwise if x∗2 (x0 ) < 0, then we can replace x0 by x′0 = −x0 ), and that y∗2 (y0 ) ≥ 0. Then, ∗
∗
|x∗2 (x0 ) + y∗2 (y0 )| = |x∗2 (x0 )| + |y∗2 (y0 )| = max{|x∗2 (x)| | ‖x‖ ≤ 1} + max{|y∗2 (y)| | α‖y‖ ≤ 1}. This together with (3.29) gives (3.25). Combining equalities (3.24)–(3.28) with (3.23) we arrive at (3.22). If ε is small enough, then, due to (3.18), y1 ̸= 0. It follows from (3.20) that
‖y∗1 ‖ = 1,
⟨y∗1 , y1 ⟩ = ‖y1 ‖.
(3.30)
Taking a smaller ε if necessary, we can ensure that ε < 1 − c α and, by (3.17) and (3.18), that x3 ∈ Bδ (¯x), ‖y3 ‖ < δ . It follows from (3.19), (3.20), (3.22) and (3.30) that
‖x∗3 ‖ < c + ε,
| ‖y∗3 ‖ − 1| < c α + ε.
(3.31)
By (3.31), y∗3 ̸= 0 and we can define x∗ := x∗3 /‖y∗3 ‖, y∗ = −y∗3 /‖y∗3 ‖. Then, ‖y∗ ‖ = 1, and x∗ ∈ D∗ Fp′ (x3 , y3 )(y∗ ) by (3.21). From (3.31) we obtain
‖x ∗ ‖ <
c+ε 1 − cα − ε
.
(3.32)
By (3.11), 1−cc α = 1/c1−α < c ′ . Taking a smaller ε again, we can ensure that 1−cc+ε < c ′ . Then (3.32) implies that ‖x∗ ‖ < c ′ . α−ε This contradicts (3.23). Hence 0 ∈ Fp′ (ˆx). The proof is complete. Note that, when applying Theorem 3.1, it is important to be able to verify the regularity assumption (iii), i.e., compute constant (3.1). We consider now some special cases when the computation of this constant can be simplified. For simplicity we limit ourselves to the case y¯ = 0 which is needed in Theorem 3.1. For each p ∈ P we consider a projection multifunction Πp : X ⇒ Y defined by
Πp (x) := {y ∈ Fp (x) | ‖y‖ = dist(0, Fp (x))} ∀x ∈ X . The next assertion follows directly from the definitions. Proposition 3.2. If Πp (x) ̸= ∅ for all p near p¯ and all x near x¯ , then
c [Fp¯ ](¯x, 0) = lim inf ‖x∗ ‖ | x∗ ∈ D∗ Fp (x, y)(y∗ ), p ∈ Bδ (¯p), x ∈ Bδ (¯x), δ↓0
0 ̸∈ Fp (x), y ∈ Πp (x) ∩ δ BY , ‖y∗ ‖ = 1 .
(3.33)
The assumption in Proposition 3.2 is satisfied automatically if Fp is nonempty compact-valued near x¯ and for all p near p¯ (see [11, Lemma 3.3]). In particular, it happens if dim Y < ∞ and Fp is nonempty closed-valued near x¯ and all p near p¯ . Another condition which is much stronger in general ensures the fulfillment of the assumption in Proposition 3.2 is that Πp is inner semicompact [18] near x¯ for all p near p¯ (see condition (A2 ) in [12, Theorem 3.1]). Beside the computation of (3.1) or (3.33) where one needs to know Fréchet coderivatives of Fp at all points (x, y) with x ̸∈ Fp−1 (¯y) near (¯x, y¯ ), we wish in many situations to deal with limiting/Mordukhovich ones. More explicitly, one will
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compute normal/Mordukhovich coderivatives of Fp at the corresponding points. In this case we consider the following constant:
c1 [Fp¯ ](¯x, y¯ ) := lim inf ‖x∗ ‖ | x∗ ∈ D∗ Fp (x, y)(y∗ ), p ∈ Bδ (¯p), x ∈ Bδ (¯x), δ↓0
y¯ ̸∈ Fp (x), y ∈ Fp (x) ∩ Bδ (¯y), ‖y∗ ‖ = 1 .
(3.34)
There is a similar representation for constant c1 [Fp¯ ](¯x, 0) as the form given in Proposition 3.2. More precisely, if Πp (x) ̸= ∅ for all p near p¯ and all x near x¯ , then
c1 [Fp¯ ](¯x, 0) = lim inf ‖x∗ ‖ | x∗ ∈ D∗ Fp (x, y)(y∗ ), p ∈ Bδ (¯p), x ∈ Bδ (¯x), δ↓0
0 ̸∈ Fp (x), y ∈ Πp (x) ∩ δ BY , ‖y∗ ‖ = 1 . Obviously, c [Fp¯ ](¯x, y¯ ) ≥ c1 [Fp¯ ](¯x, y¯ ). So we obtain the following result. Theorem 3.3. The conclusion of Theorem 3.1 remains valid if its assumption (iii) is replaced by the following one: (iii′ ) c1 [Fp¯ ](¯x, 0) > 0. Remark 3.4. Theorem 3.3 generalizes some corresponding results in [10,12]. More exactly, condition (SC) given in [10, Theorem 3.5] and assumptions (A2 ), (A3 ) given in [12, Theorem 3.3] are not needed. 4. Lipschitz-like property of efficient solution maps We now give sufficient conditions for the efficient solution map S defined in (1.4) to be Lipschitz-like at the reference point. Observe first that this efficient solution map S can be rewritten in the implicit multifunction form
S (p) := x ∈ X | 0 ∈ H (p, x) ,
where H : P × X ⇒ Y is a multifunction which is constructed via the objective function f and the efficient point multifunction declared in (1.3), H (p, x) := −f (p, x) + F (p).
(4.1)
Once again, we use the notation Hp := H (p, ·). Theorem 4.1. Let X and Y be Asplund spaces, P be a metric space and let H be the multifunction defined in (4.1). Consider (¯p, x¯ ) ∈ gphS . Suppose that there exist δ > 0 and ℓ > 0 such that (i) gphHp ∩ [(¯x, 0) + δ(BX × BY )] is closed for all p ∈ Bδ (¯p); (ii) H (p, x) ⊂ H (p′ , x) + ℓd(p, p′ )BY
∀x ∈ Bδ (¯x), ∀p, p′ ∈ Bδ (¯p);
(4.2)
(iii) c [Hp¯ ](¯x, 0) > 0. Then for any positive c < c [Hp¯ ](¯x, 0) the efficient solution map S of (1.2) is Lipschitz-like at (¯p, x¯ ) with modulus ℓ/c. Proof. The proof is directly derived from Theorem 3.1 by considering the multifunction H instead of F .
Remark 4.2. It is easy to verify that the property of H in (4.2) is valid under the fulfillment of the following two conditions: (a) there exist neighborhoods U of x¯ , W of p¯ , and a real number l > 0 such that
‖f (p, x) − f (p′ , x)‖ ≤ ld(p, p′ ) ∀x ∈ U , ∀p, p′ ∈ W ;
(4.3)
(b) F in (1.3) is locally Lipschitz at p¯ . The following corollary provides sufficient conditions for the efficient solution map S in (1.4) to be Lipschitz-like at the reference point under the strict differentiability of the objective function f . Corollary 4.3. Let X and Y be Asplund spaces, P be a normed space and let H be the multifunction defined in (4.1). Consider (¯p, x¯ ) ∈ gphS and suppose that f is strictly differentiable at (¯p, x¯ ). The conclusion of Theorem 4.1 remains valid if its assumption (ii) is replaced by the following condition: (ii′ ) F defined in (1.3) is locally Lipschitz at p¯ . Proof. Since f is strictly differentiable at (¯p, x¯ ), condition (4.3) is fulfilled. This together with the local Lipschitz property of F implies by Remark 4.2 that H satisfies condition (4.2). To finish the proof it remains to apply Theorem 4.1.
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Remark 4.4. From Theorem 3.3 we assert that the conclusion of Theorem 4.1, and thus Corollary 4.3, remains valid if its assumption (iii) is replaced by the following one: (iii′ ) c1 [Hp¯ ](¯x, 0) > 0. We observe here that it is not difficult to show by examples that assumptions (i) and (iii) in Theorem 4.1 are essential. The next example illustrates the importance of assumption (ii), namely the Lipschitz-like property of S defined in (1.4) may be violated if this assumption is omitted. Example 4.5. Let P = X = Y = R, K = R+ := [0; +∞) and let f : P × X → Y be a function which is given as follows: f (p, x) =
|x − p√| + 2p | x − p| − p
if p < 0, x ∈ X , if p ≥ 0, x ∈ X .
We consider the problem (1.2) with the efficient solution map S defined in (1.4). Let us examine (¯p, x¯ ) ∈ gphS , where p¯ := x¯ := 0. By simple computation, one gets
−|x − √ p| Hp (x) = −|x − p|
if p < 0, x ∈ X , if p ≥ 0, x ∈ X .
So, the condition (i) of Theorem 4.1 is satisfied, while the condition (ii) is not. Consider two the following possibilities: Case 1: p < 0. Take any x ∈ X with 0 ̸∈ Hp (x). The last relation means that x ̸= p. Put y := Hp (x) = −|x − p|. Then for any y∗ ∈ Y ∗ = R we obtain
∗ y ∗ ∗ D Hp (x, y)(y ) = −y ∗
if x < p, if x > p.
Case 2: p ≥ 0. Take any x ∈ X with 0 ̸∈ Hp (x). The last relation means that x ̸= for any y∗ ∈ Y ∗ = R we have
∗ y D∗ Hp (x, y)(y∗ ) = −y ∗
√
p. Set y := Hp (x) = −|x −
√
p|. Then
√
if x < √p, if x > p.
Taking into account the possibilities shown above, we get from (3.1) that c [Hp¯ ](¯x, 0) = 1. This means that the condition (iii) of Theorem 4.1 is fulfilled. It is easy to see that the efficient solution map
p S (p) = √
p
if p < 0, if p ≥ 0,
is not Lipschitz-like at (¯p, x¯ ). Finally, we mention that an important issue when applying Theorem 4.1 or Corollary 4.3, is the computation of constants c [Hp¯ ](¯x, 0) or c1 [Hp¯ ](¯x, 0) (see Remark 4.4), which requires the knowledge of the family of Fréchet coderivatives/or normal coderivatives of Hp at nearby points. Moreover, from (4.1) one can see that for each p ∈ P , Hp is the sum of the function f (p, ·) and the constant multifunction Fp : X ⇒ Y defined by Fp (x) := F (p) for all x ∈ X . So, if f (p, ·) is Fréchet/or strictly differentiable at every x near x¯ and for all p near p¯ , then we can significantly simplify the computation by exploiting the coderivative sum rules in [18]. For example, suppose that f (p, ·) is strictly differentiable at every x near x¯ and all p near p¯ . Then by utilizing the coderivative sum rule given in [18, Theorem 1.62(ii)] and the scalarization of the normal coderivative given in [18, Theorem 3.28], we obtain for any x near x¯ , p near p¯ and y ∈ −f (p, x) + F (p), D∗ Hp (x, y)(y∗ ) = −(∇x f (p, x))∗ y∗ + D∗ Fp (x, y + f (p, x))(y∗ ) ∀y∗ ∈ Y ∗ , where ∇x f (p, x) denotes the Fréchet derivative of f (p, ·) at x. Acknowledgment The author would like to thank the anonymous referee for his/her valuable comments and suggestions. References [1] E.M. Bednarczuk, Some stability results for vector optimization problems in partially ordered topological vector, in: Proceedings of the First World Congress on World Congress of Nonlinear Analysts, vol. III Table of Contents, Tampa, Florida, United States, 1996, pp. 2371–2382. [2] E.M. Bednarczuk, Upper Hölder continuity of minimal points, J. Convex Anal. 9 (2) (2002) 327–338. [3] T.D. Chuong, N.Q. Huy, J.-C. Yao, Stability of semi-infinite vector optimization problems under functional purtubations, J. Global Optim. 45 (2009) 583–595.
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