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We establish formulas for computingestimating the Fr´echet and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces. These formulas are applied to obtain conditions for solution stability of parametric variational systems over perturbed smoothboundary constraint sets.
Coderivatives of Implicit Multifunctions and Stability of Variational Systems∗ Nguyen Thanh Qui† June 22, 2015 Abstract We establish formulas for computing/estimating the Fr´echet and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces These formulas are applied to obtain conditions for solution stability of parametric variational systems over perturbed smooth-boundary constraint sets Key words Coderivative, generalized equation, implicit multifunction, local Lipschitz-like property, normal cone operator, variational system AMS subject classification 49J53, 49J52, 49J40 Introduction It is well-known that the classical implicit function theorem (see, e.g., [4, Theorem 2.1] or [22, Theorem 9.28]) is one of important theorems in mathematical analysis It gives a formula for computing the Fr´echet derivative of an implicit function at a given point in its effective domain Using the obtained formula one can find the adjoint of the Fr´echet derivative of the implicit function, which is called the coderivative (a kind of generalized derivatives) of the implicit function It can be seen in [2] that implicit function theorems are used extensively in numerical analysis and solution stability theory In accordance with developing of variational analysis, nonsmooth analysis, and set-valued analysis, implicit multifunctions appear naturally as extensions of implicit functions Generalized differentiation properties of implicit multifunctions allow one to obtain interesting results on solution sensitivity/stability of (parametric) variational systems (see the references cited below) In variational analysis, the Fr´echet coderivative and the normal coderivative (also called the Mordukhovich coderivative) of multifunctions are well-known notions of generalized differentiation; see [10, 11, 21] Implicit multifunctions defined by an inclusion involving a single-valued function and a multifunction (generalized equation in the terminology of Robinson [20]) have been considered by many researchers In [8], Levy and Mordukhovich obtained an upper estimate for the values of the normal coderivative of the implicit multifunctions in finite dimensional spaces Later, Lee and Yen [6] provided us with formulas for computing/estimating coderivatives of the implicit multifunctions Furthermore, Mordukhovich [9] established an upper estimate for the values of the normal coderivative of the implicit multifunctions, in a Banach space ∗ This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.56 † College of Information and Communication Technology, Can Tho University, Campus II, 3/2 Street, Can Tho, Vietnam; Email: ntqui@cit.ctu.edu.vn; ntqui.vn@gmail.com setting, under the assumption that the multifunction in the inclusion has the SNC property (see the definition in the next section) Related to the topic of coderivatives of implicit multifunctions in Banach spaces, a lower estimate for the values of the Fr´echet coderivative of implicit multifunctions under a select of assumptions can be found in [5] The results of Lee and Yen in [6] on coderivatives of implicit multifunctions are applied to establish verifiable necessary and sufficient conditions for the local Lipschitz-like property and the local metric regularity in Robinson’s sense of the solution maps of parametric variational inequalities over unperturbed/perturbed polyhedral convex sets or smooth-boundary sets in finite dimensional spaces; see e.g., [6, 7, 14, 15, 16, 18, 17, 19] Concerning solution stability of variational inequalities/systems using coderivative criteria, we refer the reader to [1, 3, 12, 23, 24, 25] for more details Note that, in the paper [25], the smooth-boundary constraint sets of variational systems are fixed In this paper, we focus on implicit multifunctions defined by inclusions involving a singlevalued function and a multifunction Analyzing and developing furthermore the aforemention upper estimate for the values of the Mordukhovich coderivative of the implicit multifunctions in [9], we obtain formulas for computing/estimating the Fr´echet coderivative and the Mordukhovich coderivative of the implicit multifunctions in a class of infinite dimensional spaces; called Asplund spaces An important assumption for our investigation is that the multifunctions in the inclusions defined the implicit multifunctions have the SNC property On the basis of the explicit formulas of coderivatives of the normal cone operator to parametric smooth-boundary sets in Asplund spaces given in [13], the coderivative formulas of implicit multifunctions are used to derive solution stability criteria of parametric variational systems given by inclusions involving a C -function and the normal cone operator to a smooth-boundary constraint set in finite dimensional spaces When the constraint sets are convex, the parametric variational systems are termed parametric variational inequalities over perturbed smooth-boundary sets We will show that the normal cone operator to parametric smooth-boundary sets may not be SNC in infinite dimensional Asplund spaces (such as the Hilbert space ); see Lemma 4.1 Hence, it is important to stress that we cannot establish solution stability criteria for the above parametric variational systems via the coderivative formulas of the normal cone operator in infinite dimensional spaces because of the surprising property of the operator The rest of this paper is organized as follows Section recalls some basic concepts and facts from variational analysis Section provides exact formulas for computing the Fr´echet and Mordukhovich coderivatives of the implicit multifunctions in Asplund spaces Section establishes criteria for stability of parametric variational systems Several examples are also given in this section Preliminaries Let F : X ⇒ Y be a multifunction between Banach spaces The graph and kernel of F are gphF := {(x, y) ∈ X × Y | y ∈ F (x)} and ker F := {x ∈ X| ∈ F (x)} respectively A set Ω ⊂ X is locally closed around x ∈ Ω if there exists a closed ball in X centered at x with ¯ ε), such that Ω ∩ B(x, ¯ ε) is closed One says that F is locally radius ε > 0, denoted by B(x, closed around (¯ x, y¯) ∈ gphF if gphF is locally closed around (¯ x, y¯) in the product space X × Y Unless otherwise stated, every norm in question of a product normed space is the sum norm The sequential Painlev´e-Kuratowski upper/outer limit of F as x → x¯ is defined by Limsup F (x) = y ∈ Y ∃xk → x¯ and yk → y with yk ∈ F (xk ), ∀k ∈ IN , x→¯ x (2.1) where IN := {1, 2, } and the limits are understood in the sequential sense Let Ω ⊂ X with x¯ ∈ Ω The set T (¯ x; Ω) ⊂ X defined by T (¯ x; Ω) := Limsup t↓0 Ω − x¯ , t (2.2) where “Limsup” is taken with respect to the norm topology of X, is the Bouligand-Severi tangent/contingent cone to Ω at x¯ From (2.2) and (2.1) we have T (¯ x; Ω) = z ∈ X| ∃tk ↓ 0, ∃zk → z with x¯ + tk zk ∈ Ω, ∀k ∈ IN (2.3) The set of ε-normals to Ω at x¯ ∈ Ω is given by Nε (¯ x; Ω) := x∗ ∈ X ∗ limsup Ω x→¯ x x∗ , x − x¯ ≤ε , x − x¯ (2.4) Ω where x → x¯ means x → x¯ with x ∈ Ω When ε = 0, the set in (2.4) is called the Fr´echet normal cone to Ω at x¯ and denoted by N (¯ x; Ω) If x¯ ∈ Ω, one puts Nε (¯ x; Ω) = ∅ for all ε ≥ by convention The limiting/Mordukhovich normal cone to Ω at x¯ ∈ Ω is the set N (¯ x; Ω) := Limsup Nε (x; Ω), (2.5) x→¯ x ε↓0 where “Limsup” is taken with respect to the norm topology of X and the weak* topology of X ∗ By convention, one puts N (¯ x; Ω) = ∅ when x¯ ∈ Ω A Banach space X is Asplund if every convex continuous function ϕ : U → IR defined on an open convex subset U of X is Fr´echet differentiable on a dense subset of U The class of Asplund spaces is large For instance, it contains the class of reflexive Banach spaces The calculus of normal cones in Asplund spaces is simpler than that in general Banach spaces Namely, when X is an Asplund space and Ω ⊂ X is locally closed around x¯ ∈ Ω, the limiting normal cone to Ω at x¯ defined by (2.5) is computed by the formula N (¯ x, Ω) = Limsup N (x; Ω) (2.6) x→¯ x Given (¯ x, y¯) ∈ gphF , we call the multifunction Dε∗ F (¯ x, y¯) : Y ∗ ⇒ X ∗ defined by Dε∗ F (¯ x, y¯)(y ∗ ) = x∗ ∈ X ∗ (x∗ , −y ∗ ) ∈ Nε ((¯ x, y¯); gphF ) , ∀y ∗ ∈ Y ∗ , (2.7) the ε-coderivative of F at (¯ x, y¯) When ε = 0, Dε∗ F (¯ x, y¯) is called the Fr´echet coderivative ∗ of F at (¯ x, y¯) and it is denoted by D F (¯ x, y¯) The multifunction D∗ F (¯ x, y¯) : Y ∗ ⇒ X ∗ , D∗ F (¯ x, y¯)(y ∗ ) = x∗ ∈ X ∗ (x∗ , −y ∗ ) ∈ N ((¯ x, y¯); gphF ) , ∀y ∗ ∈ Y ∗ , (2.8) is said to be the Mordukhovich coderivative of F at (¯ x, y¯) From (2.6), (2.7), and (2.8) we see that if X and Y are Asplund spaces and gphF is locally closed around (¯ x, y¯), then D∗ F (¯ x, y¯)(y ∗ ) = Limsup D∗ F (x, y)(z ∗ ), ∀y ∗ ∈ Y ∗ (2.9) (x,y)→(¯ x,¯ y) w∗ z ∗ →y ∗ The multifunction F is graphically regular at (¯ x, y¯) if D∗ F (¯ x, y¯)(y ∗ ) = D∗ F (¯ x, y¯)(y ∗ ) for all y ∗ ∈ Y ∗ The last condition is equivalent to that N ((¯ x, y¯); gphF ) = N ((¯ x, y¯); gphF ) If F is a single-valued mapping and y¯ = F (¯ x), we write D∗ F (¯ x) and D∗ F (¯ x) for D∗ F (¯ x, y¯) and D∗ F (¯ x, y¯) respectively One says that F is locally Lipschitz-like (or, in the terminology of [2], F has the Aubin property) around (¯ x, y¯) ∈ gphF if there exist > and neighborhoods U of x¯ and V of y¯ such that ¯Y , ∀x, u ∈ U, F (x) ∩ V ⊂ F (u) + x − u B ¯Y is the closed unit ball of Y where B A multifunction F : X ⇒ Y is sequentially normally compact (SNC) at (¯ x, y¯) ∈ gphF if for any sequence (εk , xk , yk , x∗k , yk∗ ) ∈ [0, ∞) × (gphF ) × X ∗ × Y ∗ satisfying εk ↓ 0, (xk , yk ) → (¯ x, y¯), x∗k ∈ Dε∗k F (xk , yk )(yk∗ ) one has (2.10) w∗ (x∗k , yk∗ ) → (0, 0) =⇒ (x∗k , yk∗ ) → as k → ∞ The multifunction F is partially sequentially normally compact (PSNC) at (¯ x, y¯) ∈ gphF if ∗ ∗ ∗ ∗ for any sequence (εk , xk , yk , xk , yk ) ∈ [0, ∞) × (gphF ) × X × Y satisfying (2.10) one has w∗ x∗k → and yk∗ → =⇒ x∗k → as k → ∞ When both X and Y are Asplund and gphF is locally around (¯ x, y¯), we can put εk = in the above properties Coderivatives of implicit multifunctions In this section we establish formulas for computing/estimating the Fr´echet and Mordukhovich coderivatives of implicit multifunctions defined by generalized equations in Asplund spaces Let us consider the implicit multifunction S : X ⇒ Y defined by a generalized equation as follows: S(x) := {y ∈ Y | ∈ f (x, y) + Q(x, y)}, (3.1) where f : X × Y → Z is a single-valued mapping and Q : X × Y ⇒ Z is a multifunction between Banach spaces The theorem below gives an upper estimate for the values of the Mordukhovich coderivative of S(·) defined by (3.1) Theorem 3.1 (See [9, Theorem 4.1]) Consider S(·) defined by (3.1) and (¯ x, y¯) ∈ gphS, where X, Y , Z are Asplund Suppose that f : X × Y → Z is continuous around (¯ x, y¯), gphQ is locally closed around (¯ x, y¯, z¯) with z¯ := −f (¯ x, y¯) If Q is SNC at (¯ x, y¯, z¯) and (x∗ , y ∗ ) ∈ D∗ f (¯ x, y¯)(z ∗ ) ∩ − D∗ Q(¯ x, y¯, z¯)(z ∗ ) =⇒ (x∗ , y ∗ , z ∗ ) = (0, 0, 0), (3.2) then one has the upper estimate D∗ S(¯ x, y¯)(y ∗ ) ⊂ x∗ ∈ X ∗ ∃z ∗ ∈ Z ∗ with (x∗ , −y ∗ ) ∈ D∗ f (¯ x, y¯)(z ∗ ) + D∗ Q(¯ x, y¯, z¯)(z ∗ ) (3.3) From now on, we suppose that X, Y , Z are Asplund spaces We first consider the implicit multifunction (3.1) in the case that f ≡ 0, i.e., the implicit multifunction S : X ⇒ Y defined by S(x) = {y ∈ Y | ∈ Q(x, y)} (3.4) For every ω = (x, y) ∈ gphS, we define the sets Λ(Q, ω)(y ∗ ) := x∗ ∈ X ∗ (x∗ , −y ∗ ) ∈ D∗ Q(x, y, 0)(z ∗ ) , (3.5) x∗ ∈ X ∗ (x∗ , −y ∗ ) ∈ D∗ Q(x, y, 0)(z ∗ ) (3.6) z ∗ ∈Z ∗ and Λ(Q, ω)(y ∗ ) := z ∗ ∈Z ∗ Coderivative formulas of S(·) defined by (3.4) are established in the following theorem Theorem 3.2 Consider the implicit multifunction (3.4) and let ω ¯ := (¯ x, y¯) ∈ gphS Then the following estimates hold Λ(Q, ω ¯ )(y ∗ ) ⊂ D∗ S(¯ ω )(y ∗ ) ⊂ D∗ S(¯ ω )(y ∗ ), ∀y ∗ ∈ Y ∗ (3.7) If Q is locally closed around (¯ x, y¯, 0), SNC at this point, and ker D∗ Q(¯ x, y¯, 0) = {0}, (3.8) Λ(Q, ω ¯ )(y ∗ ) ⊂ D∗ S(¯ ω )(y ∗ ) ⊂ D∗ S(¯ ω )(y ∗ ) ⊂ Λ(Q, ω ¯ )(y ∗ ), ∀y ∗ ∈ Y ∗ (3.9) then the estimates below hold If in addition Q is graphically regular at (¯ x, y¯, 0), then Λ(Q, ω ¯ )(y ∗ ) = D∗ S(¯ ω )(y ∗ ) = D∗ S(¯ ω )(y ∗ ) = Λ(Q, ω ¯ )(y ∗ ), ∀y ∗ ∈ Y ∗ (3.10) Proof The last inclusion in (3.7) is obvious due to the definitions of coderivatives We now verify the first inclusion in (3.7) For any x∗ ∈ Λ(Q, ω ¯ )(y ∗ ), one can find z ∗ ∈ Z ∗ satisfying (x∗ , −y ∗ ) ∈ D∗ Q(¯ x, y¯, 0)(z ∗ ) We show that x∗ ∈ D∗ S(¯ x, y¯)(y ∗ ) The last inclusion means that (x∗ , −y ∗ ) ∈ N ((¯ x, y¯); gphS), i.e., limsup gphS (x,y) −→ (¯ x,¯ y) x∗ , x − x¯ − y ∗ , y − y¯ ≤ x − x¯ + y − y¯ (3.11) Since (x∗ , −y ∗ ) ∈ D∗ Q(¯ x, y¯, 0)(z ∗ ), we have (x∗ , −y ∗ , −z ∗ ) ∈ N ((¯ x, y¯, 0); gphQ) The last inclusion means limsup gphQ (x,y,z) −→ (¯ x,¯ y ,0) gphS x∗ , x − x¯ − y ∗ , y − y¯ − z ∗ , z ≤ x − x¯ + y − y¯ + z (3.12) gphQ For any (x, y) −→ (¯ x, y¯), we have (x, y, 0) −→ (¯ x, y¯, 0) Hence, from (3.12) we obtain (3.11) ∗ ∗ ∗ which yields x ∈ D S(¯ x, y¯)(y ) Therefore, (3.7) has been proved We now suppose that Q is locally closed around (¯ x, y¯, 0), SNC at this point, and the condi∗ tion (3.8) holds Letting f ≡ 0, we have D f (¯ x, y¯)(z ∗ ) = {(0, 0)} Hence, the condition (3.8) yields (3.2) By Theorem 3.1, the estimate D∗ S(¯ x, y¯)(y ∗ ) ⊂ Λ(Q, ω ¯ )(y ∗ ) holds, and thus we obtain (3.9) If Q is graphically regular at (¯ x, y¯, 0), then D∗ Q(¯ x, y¯, 0)(z ∗ ) = D∗ Q(¯ x, y¯, 0)(z ∗ ) for all z ∗ ∈ Z ∗ From (3.9), (3.5), and (3.6) we get (3.10) ✷ We now consider the implicit multifunction S(·) defined by (3.1) For ω = (x, y) ∈ gphS and (x, y, z) ∈ gphQ with z := −f (x, y), when f : X × Y → Z is continuously differentiable around ω, we define the sets Λ(Q, f, ω)(y ∗ ) := x∗ ∈ X ∗ (x∗ , −y ∗ ) ∈ ∇f (x, y)∗ z ∗ + D∗ Q(x, y, z)(z ∗ ) , z ∗ ∈Z ∗ (3.13) and Λ(Q, f, ω)(y ∗ ) := x∗ ∈ X ∗ (x∗ , −y ∗ ) ∈ ∇f (x, y)∗ z ∗ + D∗ Q(x, y, z)(z ∗ ) (3.14) z ∗ ∈Z ∗ We derive formulas for estimating/computing the Fr´echet and Mordukhovich coderivatives of S(·) defined by (3.1) in the forthcoming theorem Theorem 3.3 Consider the implicit multifunction (3.1) and let ω ¯ := (¯ x, y¯) ∈ gphS and (¯ x, y¯, z¯) ∈ gphQ with z¯ := −f (¯ x, y¯) Suppose that f : X × Y → Z is continuously differentiable around ω ¯ Then the following estimates hold Λ(Q, f, ω ¯ )(y ∗ ) ⊂ D∗ S(¯ ω )(y ∗ ) ⊂ D∗ S(¯ ω )(y ∗ ), ∀y ∗ ∈ Y ∗ (3.15) If Q is locally closed around (¯ x, y¯, z¯) and SNC at this point, and the constraint qualification ∈ ∇f (¯ x, y¯)∗ z ∗ + D∗ Q(¯ x, y¯, z¯)(z ∗ ) =⇒ z ∗ = (3.16) is satisfied, then the estimates below hold Λ(Q, f, ω ¯ )(y ∗ ) ⊂ D∗ S(¯ ω )(y ∗ ) ⊂ D∗ S(¯ ω )(y ∗ ) ⊂ Λ(Q, f, ω ¯ )(y ∗ ), ∀y ∗ ∈ Y ∗ (3.17) If, in addition, Q is graphically regular at (¯ x, y¯, z¯), then Λ(Q, f, ω ¯ )(y ∗ ) = D∗ S(¯ ω )(y ∗ ) = D∗ S(¯ ω )(y ∗ ) = Λ(Q, f, ω ¯ )(y ∗ ), ∀y ∗ ∈ Y ∗ (3.18) Proof Define Q(x, y) = f (x, y) + Q(x, y) By [10, Theorem 1.62], we have the equalities D∗ Q(¯ x, y¯, 0)(z ∗ ) = ∇f (¯ x, y¯)∗ z ∗ + D∗ Q(¯ x, y¯, −f (¯ x, y¯))(z ∗ ), ∀z ∗ ∈ Z ∗ , and D∗ Q(¯ x, y¯, 0)(z ∗ ) = ∇f (¯ x, y¯)∗ z ∗ + D∗ Q(¯ x, y¯, −f (¯ x, y¯))(z ∗ ), ∀z ∗ ∈ Z ∗ Observe that Q is locally closed around (¯ x, y¯, 0) Indeed, since Q is locally closed around ¯ (¯ x, y¯, z¯), there exists ε > such that B (¯ x, y¯, z¯), ε ∩ gphQ is closed From the continuity ¯ x, y¯), δ) of f one can find δ ∈ (0, ε/2] such that f (x, y) − f (¯ x, y¯) ≤ ε/2 for all (x, y) ∈ B((¯ ¯ (¯ We show that G := B x, y¯, 0), δ ∩ gphQ is closed Suppose that (xk , yk , zk ) ∈ G and ¯ (¯ (xk , yk , zk ) → (x, y, z) Then we have (x, y, z) ∈ B x, y¯, 0), δ In addition, using the sum norm in the product spaces we have (xk , yk ) − (¯ x, y¯) ≤ (xk , yk , zk ) − (¯ x, y¯, 0) ≤ δ Hence, f (xk , yk ) − f (¯ x, y¯) ≤ ε/2 Since z¯ = −f (¯ x, y¯), we have (xk , yk , zk − f (xk , yk )) − (¯ x, y¯, z¯) ≤ (xk , yk , zk ) − (¯ x, y¯, 0) + f (xk , yk ) − f (¯ x, y¯) ≤ δ + ε/2 ≤ ε ¯ (¯ Thus, we have (xk , yk , zk − f (xk , yk )) ∈ B x, y¯, z¯), ε ∩ gphQ and (xk , yk , zk − f (xk , yk )) → (x, y, z − f (x, y)) as k → ∞ ¯ (¯ ¯ (¯ Since B x, y¯, z¯), ε ∩gphQ is closed, (x, y, z−f (x, y)) ∈ B x, y¯, z¯), ε ∩gphQ The inclusion (x, y, z − f (x, y)) ∈ gphQ implies that (x, y, z) ∈ gphQ We have shown that Q is locally closed around (¯ x, y¯, 0) We see that Q is SNC at (¯ x, y¯, 0) Indeed, taking any sequences (xk , yk , zk , x∗k , yk∗ , zk∗ ) from gphQ × X ∗ × Y ∗ × Z ∗ with (xk , yk , zk ) → (¯ x, y¯, 0), (x∗k , yk∗ ) ∈ D∗ Q(xk , yk , zk )(zk∗ ), w∗ (x∗k , yk∗ , zk∗ ) → (0, 0, 0), (3.19) it holds that (xk , yk , zk − f (xk , yk )) ∈ gphQ, (xk , yk , zk − f (xk , yk )) → (¯ x, y¯, z¯), and (x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ ∈ D∗ Q(xk , yk , zk − f (xk , yk ))(zk∗ ) The last inclusion is due to (x∗k , yk∗ ) ∈ D∗ Q(xk , yk , zk )(zk∗ ) and [10, Theorem 1.62] From w∗ (3.19) we have ((x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ , zk∗ ) → (0, 0, 0) Since Q is SNC at (¯ x, y¯, z¯), using the sum norm in the product space we have ((x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ , zk∗ ) = (x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ + zk∗ → as k → ∞ Hence, it holds as k → ∞ that (x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ → 0, zk∗ → 0, ∇f (xk , yk )∗ zk∗ → This implies that (x∗k , yk∗ ) → as k → ∞ Consequently, (x∗k , yk∗ , zk∗ ) → as k → ∞, which verifies the SNC property of Q at (¯ x, y¯, 0) x, y¯, 0) = {0} The constraint qualification (3.16) is equivalent to ker D∗ Q(¯ Applying Theorem 3.2 to the implicit multifunction S(x) = {y ∈ Y | ∈ Q(x, y)} we get all the desired assertions of the theorem ✷ Furthermore, we consider the parametric implicit multifunction S1 : X × Z ⇒ Y defined by parametric generalized equation: S1 (x, q) = {y ∈ Y | q ∈ f (x, y) + Q(x, y)}, (3.20) where f : X × Y → Z and Q : X × Y ⇒ Z For ω = (x, q, y) ∈ gphS1 and (x, y, z) ∈ gphQ with z := q − f (x, y), when f : X × Y → Z is continuously differentiable around ω, we put (x∗ , q ∗ ) ∈ X ∗ × Z ∗ (x∗ , −y ∗ , q ∗ ) ∈ ∇f (x, y)∗ z ∗ × {−z ∗ } Λ1 (Q, f, ω)(y ∗ ) := z ∗ ∈Z ∗ +D∗ Q(x, y, z)(z ∗ ) × {0} , (3.21) and Λ1 (Q, f, ω)(y ∗ ) := (x∗ , q ∗ ) ∈ X ∗ × Z ∗ (x∗ , −y ∗ , q ∗ ) ∈ ∇f (x, y)∗ z ∗ × {−z ∗ } z ∗ ∈Z ∗ +D∗ Q(x, y, z)(z ∗ ) × {0} (3.22) We will see that in coderivative formulas of S1 (x, q) obtained via the sets (3.21) and (3.22), constraint qualifications as (3.16) can be removed The next theorem provides formulas for estimating/computing the Fr´echet and Mordukhovich coderivatives of S1 (x, q) without using any constraint qualification as (3.16) Theorem 3.4 Let ω ¯ := (¯ x, q¯, y¯) ∈ gphS1 with S1 (·) given by (3.20) and let (¯ x, y¯, z¯) ∈ gphQ with z¯ := q¯ − f (¯ x, y¯) Suppose that f : X × Y → Z is continuously differentiable around ω ¯ ∗ ∗ Then for every y ∈ Y one has Λ1 (Q, f, ω ¯ )(y ∗ ) ⊂ D∗ S1 (¯ ω )(y ∗ ) ⊂ D∗ S1 (¯ ω )(y ∗ ) (3.23) If Q is locally closed around (¯ x, y¯, z¯), SNC at this point, then the estimates below hold Λ1 (Q, f, ω ¯ )(y ∗ ) ⊂ D∗ S1 (¯ ω )(y ∗ ) ⊂ D∗ S1 (¯ ω )(y ∗ ) ⊂ Λ1 (Q, f, ω ¯ )(y ∗ ) (3.24) If, in addition, Q is graphically regular at (¯ x, y¯, z¯), then Λ1 (Q, f, ω ¯ )(y ∗ ) = D∗ S1 (¯ ω )(y ∗ ) = D∗ S1 (¯ ω )(y ∗ ) = Λ1 (Q, f, ω ¯ )(y ∗ ) (3.25) Proof Define Q(x, y, q) = f (x, y) − q + Q(x, y) By [10, Theorem 1.62], for every z ∗ ∈ Z ∗ one has the following equalities D∗ Q(¯ x, y¯, q¯, 0)(z ∗ ) = ∇f (¯ x, y¯)∗ z ∗ × {0} − {0} × {0} × {z ∗ } + D∗ Q(¯ x, y¯, z¯)(z ∗ ) × {0} = ∇f (¯ x, y¯)∗ z ∗ × {−z ∗ } + D∗ Q(¯ x, y¯, z¯)(z ∗ ) × {0}, and D∗ Q(¯ x, y¯, q¯, 0)(z ∗ ) = ∇f (¯ x, y¯)∗ z ∗ × {0} − {0} × {0} × {z ∗ } + D∗ Q(¯ x, y¯, z¯)(z ∗ ) × {0} = ∇f (¯ x, y¯)∗ z ∗ × {−z ∗ } + D∗ Q(¯ x, y¯, z¯)(z ∗ ) × {0} (3.26) ¯ Since Q is locally closed around (¯ x, y¯, z¯), the set B (¯ x, y¯, z¯), ε ∩ gphQ is closed for some ε > Note that there exists δ ∈ (0, ε/2] such that f (x, y) − f (¯ x, y¯) ≤ ε/2 for every ¯ ¯ (x, y) ∈ B((¯ x, y¯), δ) The set G := B (¯ x, y¯, q¯, 0), δ ∩ gphQ is closed Indeed, suppose that ¯ (¯ (xk , yk , qk , zk ) ∈ G and (xk , yk , qk , zk ) → (x, y, p, z) Then, (x, y, q, z) ∈ B x, y¯, q¯, 0), δ as ¯ B (¯ x, y¯, q¯, 0), δ is closed Since (xk , yk ) − (¯ x, y¯) ≤ (xk , yk , qk , zk ) − (¯ x, y¯, q¯, 0) ≤ δ, one has f (xk , yk ) − f (¯ x, y¯) ≤ ε/2 We have z¯ = q¯ − f (¯ x, y¯), thus (xk , yk , zk + qk − f (xk , yk )) − (¯ x, y¯, z¯) ≤ (xk , yk , zk ) − (¯ x, y¯, 0) + qk − q¯ + f (xk , yk ) − f (¯ x, y¯) = (xk , yk , qk , zk ) − (¯ x, y¯, q¯, 0) + f (xk , yk ) − f (¯ x, y¯) ≤ δ + ε/2 ≤ ε ¯ (¯ ¯ (¯ This means that (xk , yk , zk + qk − f (xk , yk )) ∈ B x, y¯, z¯), ε ∩ gphQ Since B x, y¯, z¯), ε ∩ gphQ is closed and (xk , yk , zk + qk − f (xk , yk )) → (x, y, z + q − f (x, y)) as k → ∞, one has ¯ (¯ (x, y, z + q − f (x, y)) ∈ B x, y¯, z¯), ε ∩ gphQ The inclusion (x, y, z + q − f (x, y)) ∈ gphQ yields (x, y, q, z) ∈ gphQ which verifies the local closedness of Q around (¯ x, y¯, q¯, 0) We see that Q is SNC at (¯ x, y¯, q¯, 0) Indeed, taking any (xk , yk , qk , zk , x∗k , yk∗ , qk∗ , zk∗ ) in w∗ gphQ × X ∗ × Y ∗ × Z ∗ × Z ∗ with (xk , yk , qk , zk ) → (¯ x, y¯, q¯, 0), (x∗k , yk∗ , qk∗ , zk∗ ) → (0, 0, 0, 0), and (x∗k , yk∗ , qk∗ ) ∈ D∗ Q(xk , yk , qk , zk )(zk∗ ), (3.27) one has (xk , yk , zk + qk − f (xk , yk )) ∈ gphQ, (xk , yk , zk + qk − f (xk , yk )) → (¯ x, y¯, z¯), and (x∗k , yk∗ , qk∗ ) ∈ (∇f (xk , yk )∗ zk∗ , −zk∗ ) + D∗ Q(xk , yk , zk + qk − f (xk , yk ))(zk∗ ) × {0} (3.28) The last inclusion is due to (3.27) and [10, Theorem 1.62] The inclusion (3.28) is equivalent to (x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ ∈ D∗ Q(xk , yk , zk + qk − f (xk , yk ))(zk∗ ) qk∗ = −zk∗ w∗ w∗ By (x∗k , yk∗ , qk∗ , zk∗ ) → (0, 0, 0, 0), we get ((x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ , zk∗ ) → (0, 0, 0) Since Q is SNC at (¯ x, y¯, z¯), using the sum norm in the product space we have ((x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ , zk∗ ) = (x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ + zk∗ → as k → ∞ Therefore, letting k → ∞, we obtain (x∗k , yk∗ ) − ∇f (xk , yk )∗ zk∗ → 0, qk∗ = zk∗ → 0, ∇f (xk , yk )∗ zk∗ → Consequently, (x∗k , yk∗ ) → as k → ∞ It follows that (x∗k , yk∗ , qk∗ , zk∗ ) → as k → ∞ We have shown that Q is SNC at (¯ x, y¯, q¯, 0) We verify that ker D∗ Q(¯ x, y¯, q¯, 0) = {0} For any z ∗ ∈ Z ∗ with ∈ D∗ Q(¯ x, y¯, q¯, 0)(z ∗ ), by (3.26) we infer that z ∗ = Hence, ker D∗ Q(¯ x, y¯, q¯, 0) = {0} Applying Theorem 3.2 to the implicit multifunction S1 (x, q) = {y ∈ Y | ∈ Q(x, y, q)} we obtain all the desired assertions ✷ Solution stability of variational systems Lipschitzian stability of (parametric) variational systems are investigated in this section The coderivative formulas of implicit multifunctions are applied to construct criteria for the local Lipschitz-like property of the solution maps of the variational systems Let us consider a C -smooth function φ : X × Z → X ∗ and a normal cone operator F(x, p) = N (x; C(p)) with N (x; C(p)) being the normal cone in the sense of Mordukhovich to C(p) at x Herein, C(p) is the constraint set given by C(p) := {x ∈ X| ψ(x, p) ≤ 0}, (4.1) where ψ : X × P → IR is a C -smooth function with P being an Asplund space For every p ∈ P , the boundary of C(p) is the set ∂C(p) := {x ∈ X| ψ(x, p) = 0} Following [13], the values of the normal cone operator F(·) are computed by the formula: if ψ(x, p) < {0}, (4.2) F(x, p) = µ∇x ψ(x, p) µ ≥ , if ψ(x, p) = and ∇x ψ(x, p) = ∅, if ψ(x, p) > for every (x, p) ∈ X × P For each (¯ x, p¯, v¯∗ ) ∈ X × P × X ∗ with ∇x ψ(¯ x, p¯) = 0, we define the sets Ω1 (¯ x, p¯, v¯∗ )(v ∗∗ ) := (x∗ , p∗ ) ∈ X ∗ × P ∗ x∗ = γ∇x ψ(¯ x, p¯) + µ∇2xx ψ(¯ x, p¯)∗ v ∗∗ , p∗ = γ∇p ψ(¯ x, p¯) + µ∇2xp ψ(¯ x, p¯)∗ v ∗∗ , γ ∈ IR , where µ := v¯∗ · ∇x ψ(¯ x, p¯) −1 (4.3) , Ω2 (¯ x, p¯, v¯∗ )(v ∗∗ ) := IR+ ∇ψ(¯ x, p¯) = γ ∇x ψ(¯ x, p¯), ∇p ψ(¯ x, p¯) γ ∈ IR+ , (4.4) Ω3 (¯ x, p¯, v¯∗ )(v ∗∗ ) := IR∇ψ(¯ x, p¯) = γ ∇x ψ(¯ x, p¯), ∇p ψ(¯ x, p¯) γ ∈ IR , (4.5) and for every v ∗∗ ∈ X ∗∗ Exact formulas for computing the Fr´echet and Mordukhovich coderivatives of F(·) are as follows Theorem 4.1 (See [13, Theorems 3.2 and 3.3]) For any (¯ x, p¯, v¯∗ ) ∈ gphF, the following assertions are valid: (i) If ψ(¯ x, p¯) < 0, then v¯∗ = and D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = {(0X ∗ , 0P ∗ )}, ∀v ∗∗ ∈ X ∗∗ (4.6) (ii) If ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, and v¯∗ = 0, then v¯∗ = µ∇x ψ(¯ x, p¯) for some µ > In this case, for every v ∗∗ ∈ X ∗∗ , one has D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = Ω1 (¯ x, p¯, v¯∗ )(v ∗∗ ), for v ∗∗ , ∇x ψ(¯ x, p¯) = ∗∗ ∅, for v , ∇x ψ(¯ x, p¯) = 0, (4.7) where Ω1 (¯ x, p¯, v¯∗ )(v ∗∗ ) is defined by (4.3) (iii) If ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, and v¯∗ = 0, then for every v ∗∗ ∈ X ∗∗ one has Ω2 (¯ x, p¯, v¯∗ )(v ∗∗ ), for v ∗∗ , ∇x ψ(¯ x, p¯) ≥ ∅, for v ∗∗ , ∇x ψ(¯ x, p¯) < 0, (4.8) for v ∗∗ , ∇x ψ(¯ x, p¯) < {(0X ∗ , 0P ∗ )}, ∗ ∗ ∗∗ D F(¯ x, p¯, v¯ )(v ) = Ω2 (¯ x, p¯, v¯∗ )(v ∗∗ ), for v ∗∗ , ∇x ψ(¯ x, p¯) > ∗ ∗∗ ∗∗ Ω3 (¯ x, p¯, v¯ )(v ), for v , ∇x ψ(¯ x, p¯) = 0, (4.9) D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = and where Ω2 (¯ x, p¯, v¯∗ )(v ∗∗ ) and Ω3 (¯ x, p¯, v¯∗ )(v ∗∗ ) are respectively given by (4.4) and (4.5) The following lemma provides us with important information on the SNC property of the normal cone operator F(·) Lemma 4.1 The normal cone operator F(·) may be not SNC in some infinite dimensional Asplund spaces Proof For X = , we have X ∗∗ = X ∗ = X = Let (¯ x, p¯, v¯∗ ) ∈ gphF We consider two cases of the point (¯ x, p¯, v¯∗ ) below: In the case ψ(¯ x, p¯) < 0: F(·) is not SNC at the point (¯ x, p¯, v¯∗ ) Indeed, for every k ∈ IN , choose (xk , pk , vk∗ , x∗k , p∗k , vk∗∗ ) ∈ gphF × X ∗ × P ∗ × X ∗∗ as follows (xk , pk , vk∗ ) = (¯ x, p¯, v¯∗ ), (x∗k , p∗k ) = (0, 0), vk∗∗ = (0, , 0, 1, 0, ) Then, we have (x∗k , p∗k ) ∈ D∗ F(xk , pk , vk∗ )(vk∗∗ ) for all k ∈ IN Letting k → ∞, we also have w∗ (xk , pk , vk∗ ) → (¯ x, p¯, v¯∗ ) and (x∗k , p∗k , vk∗∗ ) → (0, 0, 0) However, when k → ∞, we obtain (x∗k , p∗k , vk∗∗ ) = x∗k + p∗k + vk∗ = → Therefore, F(·) is not SNC at (¯ x, p¯, v¯∗ ) In the case ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, and v¯∗ = 0: F(·) is also not SNC at (¯ x, p¯, v¯∗ ) Indeed, let (xk , pk , vk∗ ) = (¯ x, p¯, v¯∗ ), (x∗k , p∗k ) = (0, 0), and let vk∗∗ = (0, , 0, 1, 0, ) ∗∗ or vk = (0, , 0, −1, 0, ) such that vk∗∗ , ∇x ψ(¯ x, p¯) ≥ for every k ∈ IN Then it holds ∗ w that (xk , pk , vk∗ ) → (¯ x, p¯, v¯∗ ) and (x∗k , p∗k , vk∗∗ ) → (0, 0, 0) as k → ∞, and (x∗k , p∗k ) ∈ D∗ F(xk , pk , vk∗ )(vk∗∗ ), ∀k ∈ IN The last inclusion is due to Theorem 4.1(iii) Observe that (x∗k , p∗k , vk∗∗ ) = → as k → ∞ We have shown that in this case F(·) is also not SNC at (¯ x, p¯, v¯∗ ) ✷ 10 Remark 4.1 In the case ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, and v¯∗ = 0, the SNC property of F(·) at (¯ x, p¯, v¯∗ ) has not been asserted in the proof of Lemma 4.1 Thus, the SNC property of F(·) at (¯ x, p¯, v¯∗ ) in this case remains open Using the coderivative formulas of F(·) given in Theorem 4.1 and applying the results obtained in the previous section we can establish conditions for solution stability of the variational systems of the form: ∈ φ(x, p) + F(x, p) (4.10) Let S(·) be the solution map of (4.10), i.e., S : X ⇒ P is the implicit multifunction defined by setting S(x) = {p ∈ P | ∈ φ(x, p) + F(x, p)} (4.11) Due to Lemma 4.1, we will establish conditions for solution stability of (4.10) in finite dimensional spaces From now on, we suppose that X and P are finite dimensional spaces For each (¯ x, p¯, v¯∗ ) ∈ X × P × X ∗ with ∇x ψ(¯ x, p¯) = 0, we define the set x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = 0, ∆(¯ x, p¯) := (v ∗∗ , γ) ∈ X ∗∗ × IR ∇p φ(¯ ∇x ψ(¯ x, p¯)∗ v ∗∗ > 0, γ ∈ IR+ , and the matrix ∇x φ(¯ x, p¯)∗ + µ∇2xx ψ(¯ x, p¯)∗ ∇x ψ(¯ x, p¯) A(¯ x, p¯, v¯∗ ) := where µ = v¯∗ · ∇x ψ(¯ x, p¯) −1 ∇p φ(¯ x, p¯)∗ + µ∇2xp ψ(¯ x, p¯)∗ ∇p ψ(¯ x, p¯) , Theorem 4.2 Let ω ¯ := (¯ x, p¯) ∈ gphS and let v¯∗ := −φ(¯ x, p¯) ∈ F(¯ x, p¯) Then the assertions below are valid: (i) Consider the case: ψ(¯ x, p¯) < Assume that ker ∇φ(¯ x, p¯)∗ = {0} (4.12) Then S(·) is locally Lipschitz-like around ω ¯ if and only if rank∇p φ(¯ x, p¯)∗ = n (ii) Consider the case: ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, and v¯∗ = Assume that rankA(¯ x, p¯, v¯∗ ) = n + (4.13) Then, S(·) is locally Lipschitz-like around ω ¯ if and only if rank ∇p φ(¯ x, p¯)∗ + µ∇2xp ψ(¯ x, p¯)∗ ∇p ψ(¯ x, p¯) where µ := v¯∗ · ∇x ψ(¯ x, p¯) ∇x ψ(¯ x, p¯)∗ −1 = n + 1, (4.14) (iii) Consider the case: ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, and v¯∗ = Then the following hold: 11 (a) If S(·) is locally Lipschitz-like around ω ¯ , then ∆(¯ x, p¯) = ∅ and x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ =⇒ (v ∗∗ , γ) = (0, 0) ∇x ψ(¯ x, p¯)∗ v ∗∗ = ∗∗ v ∈ X ∗∗ , γ ∈ IR+ (4.15) (b) Assume that rankA(¯ x, p¯, v¯∗ ) = n + (4.16) Then, the conditions ∆(¯ x, p¯) = ∅, rank∇p φ(¯ x, p¯)∗ = n, and rank ∇p φ(¯ x, p¯)∗ ∇p ψ(¯ x, p¯) ∇x ψ(¯ x, p¯)∗ =n+1 (4.17) ensure that S(·) is locally Lipschitz-like around ω ¯ Proof We have ω ¯ = (¯ x, p¯) ∈ gphS and v¯∗ = −φ(¯ x, p¯) ∈ F(¯ x, p¯) Define the following sets Λ(F, φ, ω ¯ )(p∗ ) := x∗ ∈ X ∗ (x∗ , −p∗ ) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) , (4.18) v ∗∗ ∈X ∗∗ and Λ(F, φ, ω ¯ )(p∗ ) := x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) (4.19) x∗ ∈ X ∗ (x∗ , −p∗ ) ∈ ∇φ(¯ v ∗∗ ∈X ∗∗ We see that gphF is locally closed around (¯ x, p¯, v¯∗ ), and F(·) is obvious SNC at this point We now verify the assertions of the theorem For the case (i), we have D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = {(0, 0)}, hence the constraint qualification (0, 0) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) =⇒ v ∗∗ = is equivalent to ker ∇φ(¯ x, p¯)∗ = {0} Since F(·) is graphically regular at the point (¯ x, p¯, v¯∗ ) by (4.6), according to Theorem 3.3 one has Λ(F, φ, ω ¯ )(p∗ ) = D∗ S(¯ ω )(p∗ ) = D∗ S(¯ ω )(p∗ ) = Λ(F, φ, ω ¯ )(p∗ ), ∀p∗ ∈ P ∗ (4.20) Note that x∗ ∈ Λ(F, φ, ω ¯ )(0) if and only if there exists v ∗∗ ∈ X ∗∗ satisfying (x∗ , 0) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = ∇φ(¯ x, p¯)∗ v ∗∗ The later equality is due to D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = {(0, 0)} Thus, Λ(F, φ, ω ¯ )(0) = {0} if and only if the following constraint qualification holds: (x∗ , 0) = ∇φ(¯ x, p¯)∗ v ∗∗ =⇒ x∗ = 0, or equivalently as follows x∗ = ∇x φ(¯ x, p¯)∗ v ∗∗ = ∇p φ(¯ x, p¯)∗ v ∗∗ =⇒ x∗ = The latter is equivalent to that = ∇p φ(¯ x, p¯)∗ v ∗∗ =⇒ v ∗∗ = 12 Therefore, S(·) is locally Lipschitz-like around ω ¯ if and only if rank∇p φ(¯ x, p¯)∗ = n Consider the case (ii) In this case, F(·) is also graphically regular at (¯ x, p¯, v¯∗ ) and condition (4.13) ensures the constraint qualification (0, 0) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) =⇒ v ∗∗ = By Theorem 3.3, (4.20) is valid Hence, S(·) is locally Lipschitz-like around ω ¯ if and only if the constraint qualification below holds (x∗ , 0) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) =⇒ x∗ = Combining our assumptions with (4.3) and (4.7) we can rewrite the constraint qualification as follows ∗ x, p¯)∗ v ∗∗ x, p¯)∗ v ∗∗ + γ∇x ψ(¯ x, p¯) + µ∇2xx ψ(¯ x = ∇x φ(¯ 0 = ∇ φ(¯ x, p¯)∗ v ∗∗ x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) + µ∇2xp ψ(¯ p =⇒ x∗ = ∗∗ v , ∇ ψ(¯ x , p ¯ ) = x γ ∈ IR By the assumption (4.13), the latter is equivalent to that x, p¯) = x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯)∗ + µ∇2xp ψ(¯ ∇p φ(¯ ∗ ∗∗ ∇x ψ(¯ x, p¯) v = ∗∗ v ∈ X ∗∗ , γ ∈ IR =⇒ (v ∗∗ , γ) = (0, 0), or equivalently as follows x, p¯)∗ ∇p ψ(¯ x, p¯) ∇p φ(¯ x, p¯)∗ + µ∇2xp ψ(¯ ∇x ψ(¯ x, p¯) ∗ v ∗∗ γ = 0 =⇒ (v ∗∗ , γ) = (0, 0) This is equivalent to rank ∇p φ(¯ x, p¯)∗ + µ∇2xp ψ(¯ x, p¯)∗ ∇p ψ(¯ x, p¯) ∇x ψ(¯ x, p¯)∗ = n + We now verify (iii) First, we prove assertion (a) By Theorem 3.3 the following estimates hold: Λ(F, φ, ω ¯ )(p∗ ) ⊂ D∗ S(¯ ω )(p∗ ) ⊂ D∗ S(¯ ω )(p∗ ), ∀p∗ ∈ P ∗ (4.21) Suppose that S(·) is locally Lipschitz-like around ω ¯ This is equivalent to D∗ S(¯ ω )(0) = {0} Due to (4.21), the latter yields Λ(F, φ, ω ¯ )(0) = {0} This means that (x∗ , 0) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) =⇒ x∗ = By (4.8) and (4.4), the last constraint qualification means that ∗ x, p¯)∗ v ∗∗ + γ∇x ψ(¯ x, p¯) x = ∇x φ(¯ 0 = ∇ φ(¯ x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) p ∗∗ v , ∇x ψ(¯ x, p¯) ≥ γ ∈ IR+ 13 =⇒ x∗ = (4.22) This is equivalent to the two constraint qualifications below x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ =⇒ (v ∗∗ , γ) = (0, 0) ∇x ψ(¯ x, p¯)∗ v ∗∗ > γ ∈ IR+ and x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ ∗ ∗∗ ∇x ψ(¯ x, p¯) v = γ ∈ IR+ (4.23) =⇒ (v ∗∗ , γ) = (0, 0) Note that the last constraint qualification is (4.15), and (4.23) is equivalent to ∆(¯ x, p¯) = ∅ Thus, (4.22) is equivalent to both conditions ∆(¯ x, p¯) = ∅ and (4.15) We now prove assertion (b) Note that (4.16) ensures the constraint qualification (0, 0) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) =⇒ v ∗∗ = The condition Λ(F, φ, ω ¯ )(0) = {0} holds if and only if the constraint qualification below is satisfied (x∗ , 0) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) =⇒ x∗ = (4.24) If v ∗∗ , ∇x ψ(¯ x, p¯) < 0, using (4.9) we can rewrite (4.24) as follows (x∗ , 0) = ∇x φ(¯ x, p¯)∗ v ∗∗ , ∇p φ(¯ x, p¯)∗ v ∗∗ v ∗∗ , ∇x ψ(¯ x, p¯) < =⇒ x∗ = Since v ∗∗ , ∇x ψ(¯ x, p¯) < 0, we have v ∗∗ = Thus, by the assumption rank∇p φ(¯ x, p¯)∗ = n, we obtain = ∇p φ(¯ x, p¯)∗ v ∗∗ = (absurdity) Hence, (4.24) is equivalent to (x∗ , 0) = ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) v ∗∗ , ∇x ψ(¯ x, p¯) ≥ =⇒ x∗ = We can rewrite the last constraint qualification as follows x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ ∗ ∗∗ =⇒ (v ∗∗ , γ) = (0, 0) ∇x ψ(¯ x, p¯) v > γ ∈ IR+ and x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ ∇x ψ(¯ x, p¯)∗ v ∗∗ = ∗∗ v ∈ X ∗∗ , γ ∈ IR =⇒ (v ∗∗ , γ) = (0, 0) (4.25) (4.26) We see that (4.25) is equivalent to ∆(¯ x, p¯) = ∅, and (4.26) means ∇p φ(¯ x, p¯)∗ ∇p ψ(¯ x, p¯) ∗ ∇x ψ(¯ x, p¯) v ∗∗ γ = 0 =⇒ (v ∗∗ , γ) = (0, 0) which is equivalent to (4.17) Using our assumptions, we infer that Λ(F, φ, ω ¯ )(0) = {0}, which ensures the local Lipschitz-like property around ω ¯ of S(·) ✷ 14 Example 4.1 Consider the solution map S : IR2 ⇒ IR2 , S(x) = {p ∈ IR2 | ∈ φ(x, p) + F(x, p)}, of the variational system: ∈ φ(x, p) + F(x, p), where φ : IR2 × IR2 → IR2 and ψ : IR2 × IR2 → IR are respectively defined by φ(x, p) = (x21 + p1 − 5, x22 + p2 − 16) and ψ(x, p) = 5x21 − x1 + 3x2 + p1 + p22 Let x¯ = (1, 3) and p¯ = (−14, 1) Then we have φ(¯ x, p¯) = (−18, −6), ψ(¯ x, p¯) = 0, ∇x φ(¯ x, p¯) = 2¯ x1 0 2¯ x2 = ∇x ψ(¯ x, p¯) = (10¯ x1 − 1, 3) = (9, 3) = 0, , ∇p φ(¯ x, p¯) = , ∇p ψ(¯ x, p¯) = (1, 2¯ p2 ) = (1, 2) = 0, and 10 ∇2xx ψ(¯ x, p¯) = 0 , ∇2xp ψ(¯ x, p¯) = 0 0 Let µ = and let v¯∗ = −φ(¯ x, p¯) = µ∇x ψ(¯ x, p¯) ∈ F(¯ x, p¯) We have (¯ x, p¯, v¯∗ ) ∈ gphF and ω ¯ = (¯ x, p¯) ∈ gphS We see that ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, v¯∗ = 0, 22 x, p¯)∗ ∇x ψ(¯ x, p¯) ∇x φ(¯ x, p¯)∗ + µ∇2xx ψ(¯ = 3, = rank rank x, p¯)∗ ∇p ψ(¯ x, p¯) ∇p φ(¯ x, p¯)∗ + µ∇2xp ψ(¯ 1 and ∗ rank ∇p φ(¯ x, p¯) + µ∇2xp ψ(¯ x, p¯)∗ ∇x ψ(¯ x, p¯) ∇p ψ(¯ x, p¯) ∗ 1 = rank = According to Theorem 4.2(ii), we conclude that S(·) is locally Lipschitz-like around ω ¯ We now consider the parametric variational system q ∗ ∈ φ(x, p) + F(x, p), (4.27) where φ : X × P → X ∗ is a C -smooth function, and F(x, p) = N (x; C(p)) is given by (4.2) Denote S1 (·) the solution map of (4.27) Then, we have S1 : X × X ∗ ⇒ P with S1 (x, q ∗ ) = {p ∈ P | q ∗ ∈ φ(x, p) + F(x, p)} (4.28) Remark 4.2 Conditions (4.12), (4.13), and (4.16) can be removed if S(·) is replaced with the implicit multifunction S1 (·) in (4.28) This can be seen in the following theorem Theorem 4.3 Let ω ¯ := (¯ x, q¯∗ , p¯) ∈ gphS1 and let v¯∗ := q¯∗ − φ(¯ x, p¯) ∈ F(¯ x, p¯) Then the assertions below are valid: (i) Consider the case: ψ(¯ x, p¯) < Then, S1 (·) is locally Lipschitz-like around ω ¯ if and only if rank∇p φ(¯ x, p¯)∗ = n 15 (ii) Consider the case: ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, and v¯∗ = Then, S1 (·) is locally Lipschitz-like around ω ¯ if and only if (4.14) holds (iii) Consider the case: ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, and v¯∗ = We have (a) If S1 (·) is locally Lipschitz-like around ω ¯ , then ∆(¯ x, p¯) = ∅, and (4.15) holds (b) If ∆(¯ x, p¯) = ∅, rank∇p φ(¯ x, p¯)∗ = n, and (4.17) is satisfied, then S1 (·) is locally Lipschitz-like around ω ¯ Proof We have ω ¯ = (¯ p, q¯∗ , x¯) ∈ gphS1 and v¯∗ = q¯∗ − φ(¯ x, p¯) ∈ F(¯ x, p¯) Define the sets Λ1 (F, φ, ω ¯ )(p∗ ) := (x∗ , q ∗∗ ) ∈ X ∗ × X ∗∗ (x∗ , −p∗ , q ∗∗ ) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ × {−v ∗∗ } v ∗∗ ∈X ∗∗ +D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) × {0} , (4.29) and Λ1 (F, φ, ω ¯ )(p∗ ) := (x∗ , q ∗∗ ) ∈ X ∗ × X ∗∗ (x∗ , −p∗ , q ∗∗ ) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ × {−v ∗∗ } v ∗∗ ∈X ∗∗ +D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) × {0} (4.30) ∗ We see that F(·) is locally closed and SNC at (¯ x, p¯, v¯ ) Let us verify (i) Since F(·) is graphically regular at (¯ x, p¯, v¯∗ ), by Theorem 3.4 one has Λ1 (F, φ, ω ¯ )(p∗ ) = D∗ S1 (¯ ω )(p∗ ) = D∗ S1 (¯ ω )(p∗ ) = Λ1 (F, φ, ω ¯ )(p∗ ), (4.31) for every p∗ ∈ P ∗ Note that (x∗ , q ∗∗ ) ∈ Λ1 (F, φ, ω ¯ )(0) if and only if there exists v ∗∗ ∈ X ∗∗ satisfying (x∗ , 0, q ∗∗ ) ∈ ∇φ(¯ x, p¯)∗ v ∗∗ × {−v ∗∗ } + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) × {0} = ∇φ(¯ x, p¯)∗ v ∗∗ × {−v ∗∗ } The later equality is due to D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) = {(0, 0)} Thus, Λ1 (F, φ, ω ¯ )(0) = {(0, 0)} if and only if the following constraint qualification holds: ∗ x, p¯)∗ v ∗∗ x = ∇x φ(¯ =⇒ (x∗ , q ∗∗ ) = (0, 0) = ∇p φ(¯ x, p¯)∗ v ∗∗ ∗∗ q = −v ∗∗ The latter is equivalent to the condition rank∇p φ(¯ x, p¯)∗ = n Therefore, S1 (·) is locally ∗ Lipschitz-like around ω ¯ if and only if rank∇p φ(¯ x, p¯) = n We now prove (ii) In this case, F(·) is also graphically regular at (¯ x, p¯, v¯∗ ) According to Theorem 3.4, (4.31) is valid for this case Hence, S1 (·) is locally Lipschitz-like around ω ¯ if and only if the constraint qualification below holds (x∗ , 0) = ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) q ∗∗ = −v ∗∗ =⇒ (x∗ , q ∗∗ ) = (0, 0) Combining our assumptions with (4.3) and (4.7) we can rewrite the constraint qualification as follows x∗ = ∇x φ(¯ x, p¯)∗ v ∗∗ + γ∇x ψ(¯ x, p¯) + µ∇2xx ψ(¯ x, p¯)∗ v ∗∗ x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) + µ∇2xp ψ(¯ x, p¯)∗ v ∗∗ 0 = ∇p φ(¯ =⇒ (x∗ , q ∗∗ ) = (0, 0) v ∗∗ , ∇x ψ(¯ x, p¯) = v ∗∗ ∈ X ∗∗ , γ ∈ IR q ∗∗ = −v ∗∗ 16 The latter is equivalent to x, p¯) = x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯)∗ + µ∇2xp ψ(¯ ∇p φ(¯ ∗ ∗∗ ∇x ψ(¯ x, p¯) v = ∗∗ v ∈ X ∗∗ , γ ∈ IR =⇒ (v ∗∗ , γ) = (0, 0), or equivalently as follows ∇p φ(¯ x, p¯)∗ + µ∇2xp ψ(¯ x, p¯)∗ ∇p ψ(¯ x, p¯) ∇x ψ(¯ x, p¯) ∗ v ∗∗ γ = 0 =⇒ (v ∗∗ , γ) = (0, 0) This is equivalent to rank x, p¯)∗ ∇p ψ(¯ x, p¯) ∇p φ(¯ x, p¯)∗ + µ∇2xp ψ(¯ ∇x ψ(¯ x, p¯)∗ = n + Finally, we prove (iii) In accordance with our assumptions in this case, for every p∗ ∈ P ∗ the following estimates are valid: Λ1 (F, φ, ω ¯ )(p∗ ) ⊂ D∗ S1 (¯ ω )(p∗ ) ⊂ D∗ S1 (¯ ω )(p∗ ) ⊂ Λ1 (F, φ, ω ¯ )(p∗ ) (4.32) Assertion (a): Suppose that S1 (·) is locally Lipschitz-like around ω ¯ This is equivalent to ∗ the condition D S1 (¯ ω )(0) = {(0, 0)} By (4.32), the latter yields Λ1 (F, φ, ω ¯ )(0) = {(0, 0)} This means that (x∗ , 0) = ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) q ∗∗ = −v ∗∗ =⇒ (x∗ , q ∗∗ ) = (0, 0) By (4.8) and (4.4), the last constraint qualification means that x∗ = ∇x φ(¯ x, p¯)∗ v ∗∗ + γ∇x ψ(¯ x, p¯) ∗ ∗∗ x, p¯) v + γ∇p ψ(¯ x, p¯) 0 = ∇p φ(¯ ∗∗ =⇒ (x∗ , q ∗∗ ) = (0, 0), v , ∇x ψ(¯ x, p¯) ≥ γ ∈ IR+ q ∗∗ = −v ∗∗ which is equivalent to x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ ∗ ∗∗ ∇x ψ(¯ x, p¯) v ≥ γ ∈ IR+ =⇒ (v ∗∗ , γ) = (0, 0) The latter is equivalent to the two constraint qualifications below x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ ∗ ∗∗ =⇒ (v ∗∗ , γ) = (0, 0) ∇x ψ(¯ x, p¯) v > γ ∈ IR+ and x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ ∗ ∗∗ ∇x ψ(¯ x, p¯) v = γ ∈ IR+ 17 (4.33) =⇒ (v ∗∗ , γ) = (0, 0) (4.34) (4.35) We see that (4.34) is equivalent to ∆(¯ x, p¯) = ∅, hence (4.33) is equivalent to both conditions ∆(¯ x, p¯) = ∅ and (4.34) Assertion (b): The condition Λ1 (F, φ, ω ¯ )(0) = {(0, 0)} holds if and only if the constraint qualification below is satisfied (x∗ , 0) = ∇φ(¯ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) q ∗∗ = −v ∗∗ =⇒ (x∗ , q ∗∗ ) = (0, 0) (4.36) If v ∗∗ , ∇x ψ(¯ x, p¯) < 0, using (4.9) we can rewrite (4.36) as follows ∗ x, p¯)∗ v ∗∗ , ∇p φ(¯ x, p¯)∗ v ∗∗ (x , 0) = ∇x φ(¯ =⇒ (x∗ , q ∗∗ ) = (0, 0) v ∗∗ , ∇x ψ(¯ x, p¯) < ∗∗ q = −v ∗∗ The latter does not happen because rank∇p φ(¯ x, p¯)∗ = n by our assumptions and v ∗∗ = 0, ∗ ∗∗ thus = ∇p φ(¯ x, p¯) v = (absurdity) Therefore, (4.36) is equivalent to ∗ x, p¯)∗ v ∗∗ + D∗ F(¯ x, p¯, v¯∗ )(v ∗∗ ) (x , 0) = ∇φ(¯ =⇒ (x∗ , q ∗∗ ) = (0, 0) v ∗∗ , ∇x ψ(¯ x, p¯) ≥ ∗∗ q = −v ∗∗ We can rewrite the last constraint qualification as follows x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ ∗ ∗∗ =⇒ (v ∗∗ , γ) = (0, 0) ∇x ψ(¯ x, p¯) v > γ ∈ IR+ and x, p¯)∗ v ∗∗ + γ∇p ψ(¯ x, p¯) = ∇p φ(¯ ∗ ∗∗ ∇x ψ(¯ x, p¯) v = ∗∗ v ∈ X ∗∗ , γ ∈ IR =⇒ (v ∗∗ , γ) = (0, 0) (4.37) (4.38) We see that (4.37) is equivalent to ∆(¯ x, p¯) = ∅ and (4.38) is equivalent to rank ∇p φ(¯ x, p¯)∗ ∇p ψ(¯ x, p¯) ∇x ψ(¯ x, p¯)∗ = n + Therefore, using our assumptions we deduce that Λ1 (F, φ, ω ¯ )(0) = {(0, 0)} The last equality ensures that S1 (·) is locally Lipschitz-like around ω ¯ ✷ Example 4.2 Consider the parametric variational system q ∗ ∈ φ(x, p) + F(x, p), and its solution map S1 : IR2 × IR2 ⇒ IR2 , S1 (p, q ∗ ) = {x ∈ IR2 | q ∗ ∈ φ(x, p) + F(x, p)}, where φ : IR2 × IR2 → IR2 and ψ : IR2 × IR2 → IR are respectively defined by φ(x, p) = (x21 + p1 , x22 + p2 ) and ψ(x, p) = 5x21 − x1 + 3x2 + p1 + p22 18 (4.39) Let x¯ = (1, 3), p¯ = (−14, 1), and q¯∗ = (5, 16) Then we have φ(¯ x, p¯) = (−13, 10), ψ(¯ x, p¯) = 0, ∇x φ(¯ x, p¯) = 2¯ x1 0 2¯ x2 ∇x ψ(¯ x, p¯) = (10¯ x1 − 1, 3) = (9, 3) = 0, = 0 , ∇p ψ(¯ x, p¯) = (1, 2¯ p2 ) = (1, 2) = 0, and ∇2xx ψ(¯ x, p¯) = 10 0 Let µ = and v¯∗ = q¯∗ −φ(¯ x, p¯) = (18, 6) = µ∇x ψ(¯ x, p¯) ∈ F(¯ x, p¯) We have (¯ x, p¯, v¯∗ ) ∈ gphF and ω ¯ = (¯ p, q¯∗ , x¯) ∈ gphS1 As ψ(¯ x, p¯) = 0, ∇x ψ(¯ x, p¯) = 0, ∇p ψ(¯ x, p¯) = 0, v¯∗ = 0, and 22 ∇x φ(¯ x, p¯)∗ + µ∇2xx ψ(¯ x, p¯)∗ ∇x ψ(¯ x, p¯) rank = rank = 3, ∗ ∇x ψ(¯ x, p¯) we conclude that S1 (·) is Lipschitz-like 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Lee and N D Yen, Fr´echet and normal coderivatives of implicit multifunctions, Appl Anal 90 (2011), pp 1011–1027 [7] G M Lee and N D Yen, Coderivatives of a Karush-Kuhn-Tucker point set map and. .. implicit multifunction S1 (x, q) = {y ∈ Y | ∈ Q(x, y, q)} we obtain all the desired assertions ✷ Solution stability of variational systems Lipschitzian stability of (parametric) variational systems