A class of linear generalized equations

A CLASS OF LINEAR GENERALIZED EQUATIONS∗ Nguyen Thanh Qui† and Nguyen Dong Yen‡ June 24, 2012 Abstract Solution stability of a class of linear generalized equations in finite dimensional Euclidean spaces is investigated by means of generalized differentiation Exact formulas for the Fréchet and the Mordukhovich coderivatives of the normal cone mappings of perturbed Euclidean balls are obtained Necessary and sufficient conditions for the local Lipschitz-like property of the solution maps of such linear generalized equations are derived from these coderivative formulas Since the trust-region subproblems in nonlinear programming can be regarded as linear generalized equations, these conditions lead to new results on stability of the parametric trust-region subproblems Key words Linear generalized equation, trust-region subproblem, KKT point set map, normal cone mapping, coderivative, local Lipschitz-like property AMS subject classification 49J53, 49J52, 49J40 Introduction The concept of generalized equation introduced by Robinson [13] has been recognized as an efficient tool for dealing with various questions in optimization theory It is also a unified framework for studying equilibrium problems When the basic single-valued operator of the generalized equation is affine and the accompanying set-valued map is the normal cone operator of a fixed closed convex set called the constraint set, one has a linear generalized equation (linear GE for brevity) Robinson [13, Theorem 2] proved that if a linear GE is monotone and the solution set is nonempty and bounded, then the solution map is locally upper Lipschitzian with respect to the parameters describing the affine operator This important result has found many applications (see, e.g., [16]) Linear GEs with perturbed constraint sets have been studied in [4] and [8] (see also the references therein) In connection with the solution methods [12], [15] and the qualitative study [5] for the trust-region subproblems, we are interested in the linear GEs of the form ∈ Ax + b + N (x; E(α)), (1.1) ∗ Research of the authors was supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) † College of Information and Communication Technology, Can Tho University, Ly Tu Trong, Can Tho, Vietnam; email: ntqui@cit.ctu.edu.vn ‡ Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi 10307, Vietnam; email: ndyen@math.ac.vn where symmetric n × n matrix A ∈ IRn×n , vector b ∈ IRm , and real number α > are parameters, E(α) := {x ∈ IRn x ≤ α}, and N (x; E(α)) := {v ∈ IRn | v, y − x ≤ 0, ∀y ∈ E(α)}, if x ∈ E(α) ∅, if x ∈ E(α) (1.2) is the normal cone to E(α) at x The solution set of (1.1) is denoted by S(A, b, α) Note that (1.1) is a linear GE where the perturbation of the constraint set E(α) is described by parameter α ∈ (0, +∞) Here E(α) is a ball centered at with radius α If x is a local solution of the optimization problem min{f (x) = x Ax + b x| x ∈ E(α)}, (1.3) which is called the trust-region subproblem, then (1.1) holds due to the generalized Fermat rule (see, e.g., [4, p 85]) Here and in the sequel, the apex denotes matrix transposition It is well-known [10] that if x ∈ E(α) is a local minimum of (1.3), then there exists a Lagrange multiplier λ ≥ such that (A + λI)x = −b, λ( x − α) = 0, (1.4) where I denotes the n × n unit matrix If x ∈ E(α) and there exists λ ≥ satisfying (1.4), x is said to be a Karush-Kuhn-Tucker point (or a KKT point) of (1.3) and (x, λ) is called a KKT pair For each KKT point x, the Lagrange multiplier λ is defined uniquely (see, e.g., [5]) Recall [3] that x is a KKT point of (1.3) if and only if Ax + b, y − x ≥ 0, ∀y ∈ E(α) Thus, the solution set of (1.1) coincides with the Karush-Kuhn-Tucker point set of (1.3) The purpose of this paper is to investigate the stability of (1.1) with respect to the perturbations of all the three components of its data set {A, b, α} Our main tools are the Mordukhovich criterion (see [11, Theorem 4.10] and [14, Theorem 9.40]) for the local Lipschitz-like property of multifunctions between finite dimensional normed spaces and some lower and upper estimates for coderivatives of implicit multifunctions from [7] Our results develop furthermore the preceding work of Lee and Yen [6] on the stability of (1.1) To be more precise, we provide a complete solution for the open problems raised in [6, Remarks 3.6 and 3.13] by giving exact formulas for the Fréchet an the Mordukhovich coderivatives of the normal cone mapping (x, α) → N (x; E(α)) Moreover, we complement the sufficient conditions for stability of the solution set of (1.1) given in [6, Theorem 5.1] by a more comprehensive necessary and sufficient conditions for stability This paper shows how the generalized differentiation theory [11], [14] can be applied with a success for analyzing a typical polynomial optimization problem of the form (1.3) Our approach to the analysis of the parametric problem (1.3) is quite different from that one adopted by Lee, Tam and Yen [5] It is worthy to stress that the focus point of [5] is the lower semicontinuity of the solution map of (1.1), while our aim is to characterize the local Lipschitz-like property of that map The latter is stronger than the inner semicontinuity of the solution map, which is the basis for defining the above-mentioned lower semicontinuity It is still unclear to us whether the inner semicontinuity property [11, p 42] of a multifunction can be characterized by using coderivatives, or not The rest of the paper has three sections Several facts on variational analysis and generalized differentiation from [11] are recalled in Section Section provides exact formulas for the Fréchet and the Mordukhovich coderivatives of the normal cone mapping (x, α) → N (x, E(α)) In Section 4, necessary and sufficient conditions for the local Lipschitz-like property of the solution maps (A, b, α) → S(A, b, α) of the linear generalized equations (1.1) will be established We conclude the paper by four examples serving as illustrations for the obtained results Preliminaries Let us recall some facts from [11] The Fréchet normal cone to a set Ω ⊂ IRs at x¯ ∈ Ω is given by x , x − x¯ ≤0 , (2.1) N (¯ x; Ω) = x ∈ IRs limsup x − x¯ Ω x x→¯ Ω where x → x¯ means x → x¯ with x ∈ Ω By convention, N (¯ x; Ω) = ∅ when x¯ ∈ Ω For a multifunction Φ : IRn ⇒ IRn , the sequential Painlevé-Kuratowski upper limit with respect to the norm topology of IRn is defined by Limsup Φ(x) = x ∈ IRn ∃ xk → x¯ and xk → x with xk ∈ Φ(xk ) for k = 1, 2, x→¯ x If Ω is locally closed around x¯ ∈ Ω, the cone N (¯ x, Ω) := Limsup N (x; Ω) (2.2) x→¯ x is said to be the limiting (or basic/Mordukhovich) normal cone to Ω at x¯ ∈ Ω If x¯ ∈ Ω, N (¯ x; Ω) = ∅ by convention Given a point x0 in a normed space X and ρ > 0, we denote the open ball {x ∈ ¯ , ρ) We write X x − x0 < ρ} by B(x0 , ρ), and the corresponding closed ball by B(x ¯ ¯ BX and BX for B(0X , 1) and B(0X , 1), respectively The norm in the product X × Y of normed spaces is given by (x, y) = x + y The graph of a multifunction F : IRn ⇒ IRm is the set gphF := {(x, y) ∈ IRn ×IRm | y ∈ F (x)} The kernel of F is defined by ker F := {x ∈ IRn | ∈ F (x)} We say that F is locally closed around z¯ := (¯ x, y¯) ∈ gphF if there exists ρ > such that the intersection ¯ z , ρ) is closed in the product space IRn × IRm For every (¯ gphF ∩ B(¯ x, y¯) ∈ gphF , we call ∗ m n the multifunction D F (¯ x, y¯) : IR ⇒ IR , D∗ F (¯ x, y¯)(y ) = x ∈ IRn (x , −y ) ∈ N ((¯ x, y¯); gphF ) ∀y ∈ IRm , the Fréchet coderivative of F at (¯ x, y¯) It is not difficult to see that gphD∗ F (¯ x, y¯) is ∗ m n convex and closed The multifunction D F (¯ x, y¯) : IR ⇒ IR given by setting D∗ F (¯ x, y¯)(y ) = x ∈ IRn (x , −y ) ∈ N ((¯ x, y¯); gphF ) ∀y ∈ IRm is said to be the Mordukhovich (or limiting/normal ) coderivative of F at (¯ x, y¯) Although ∗ ∗ gphD F (¯ x, y¯) might be nonconvex, D F (¯ x, y¯) is a multifunction of closed graph One says that F is graphically regular at (¯ x, y¯) ∈ gphF if D∗ F (¯ x, y¯)(y ) = D∗ F (¯ x, y¯)(y ), ∀y ∈ IRm The last condition can be written equivalently as N ((¯ x, y¯); gphF ) = N ((¯ x, y¯); gphF ) One says that F is locally Lipschitz-like, or F has the Aubin property [2], around (¯ x, y¯) ∈ gphF if there exist > and neighborhoods U of x¯, V of y¯ such that F (x) ∩ V ⊂ F (u) + ¯IRm , ∀x, u ∈ U x−u B If F is locally closed around (¯ x, y¯) ∈ gphF then, by the Mordukhovich criterion [11, Theorem 4.10], F is locally Lipschitz-like around (¯ x, y¯) if and only if D∗ F (¯ x, y¯)(0) = {0} (2.3) Criterion (2.3) reduces the verification of the Lipschitz-like property of a multifunction to the computation of just one value of the Mordukhvich coderivative at a given point This criterion will play a central role in Section below Formulas for Coderivatives The normal cone N (x; E(α)) can be computed explicitly   if x {0}, N (x; E(α)) = {µx| µ ≥ 0}, if x   ∅, if x Namely, we have α (3.1) For every (x, α) ∈ IRn × IR, we put N (x, α) = N (x; E(α)), if α > ∅, if α ≤ 0, (3.2) where N (x; E(α)) is given by (1.2) Thus, N : IRn × IR ⇒ IRn is a multifunction with closed convex values It is called the normal cone mapping of the closed ball E(α) Setting y = −b, w = (A, α), G(x, w) = Ax, and M (x, w) = N (x, α), we can rewrite (1.1) equivalently as y ∈ G(x, w) + M (x, w) (3.3) It is clear that S(A, b, α) = S(w, y) := x ∈ IRn y ∈ G(x, w) + M (x, w) Hence, the solution map S : H(n) × IRn × IR ⇒ IRn , (A, b, α) → S(A, b, α), (3.4) of (1.1) can be interpreted as the implicit multifunction S : W × IRn ⇒ IRn , (w, y) → S(w, y), (3.5) where W := H(n) × IR with H(n) ⊂ IRn×n being the linear subspace of symmetric n × n matrices of IRn×n By (u, v) we denote the angle between nonzero vectors u and v in IRn , i.e., (u, v) ∈ [0, π] and u, v = u · v cos (u, v) For each pair u, v ∈ IRn with u = (u1 , , un ) and → in IRn by setting − → = (v − u , , v − u ) v = (v1 , , ) , we define the vector − uv uv 1 n n − → − → For any x, y, z ∈ IRn , we call xyz the angle between yx and yz, provided the latter vectors are nonzero We are going to obtain exact formulas for the Fréchet and the Mordukhovich coderivatives of the normal cone mapping N (x, α) given by (3.2) Fix any point (x, α, v) ∈ gphN 3.1 The Fr´ echet Coderivative of N (x, α) The following results are due to Lee and Yen [6] Lemma 3.1 (See [6, Lemma 3.1]) If x < α, then v = and D∗ N (x, α, v)(v ) = {(0IRn , 0IR )}, for every v ∈ IRn Lemma 3.2 (See [6, Lemma 3.2]) If x = α and v = 0, then v = µx for some µ > If (x , α ) ∈ D∗ N (x, α, v)(v ), then x =− α x + µv , α v , x = Lemma 3.1 describes the Fréchet coderivative D∗ N (x, α, v) in the case x < α Lemma 3.2 gives an upper estimate for the Fréchet coderivative value D∗ (x, α, v)(v ) in the case x = α, v = The first part of the open problem raised in [6, Remark 3.6] can be reformulated as follows: Is the upper estimate provided by Lemma 3.2 an exact one? The next statement, which answers this question in the affirmative, establishes an exact formula for computing the coderivative D∗ N (x, α, v) in the situation x = α and v = Theorem 3.1 If x = α and v = 0, then v = µx with µ = v · x −1 and, for every v ∈ IRn ,   (x , α ) ∈ IRn × IR x = − α x + µv , if v , x = α ∗ D N (x, α, v)(v ) = (3.6) ∅, if v , x = Proof The property v = µx with µ = v · x immediate from Lemma 3.2 −1 and the inclusion “ ⊂ ” of (3.6) are To prove the opposite inclusion of (3.6), suppose to the contrary that there exists a pair (x , α ) belonging to the set described by the right-hand side of (3.6) with (x , α ) ∈ D∗ N (x, α, v)(v ) Then (x , α , −v ) ∈ N ((x, α, v); gphN ) So there exist a sequence gphN (xk , αk , vk ) −→ (x, α, v) and a constant δ > such that Pk := x , xk − x + α (αk − α) − v , vk − v ≥ δ, xk − x + |αk − α| + vk − v (3.7) for all k ∈ IN := {1, 2, } By the choice of (x , α ), we have Pk = = = − αα x + µv , xk − x + α (αk − α) − v , vk − v xk − x + |αk − α| + vk − v − αα x, xk − x + α (αk − α) µv , xk − x − v , vk − v + xk − x + |αk − α| + vk − v xk − x + |αk − α| + vk − v − αα x, xk − x + α (αk − α) µv , xk − v , vk + xk − x + |αk − α| + vk − v xk − x + |αk − α| + vk − v = Qk + Rk , where Qk := − αα x, xk − x + α (αk − α) , xk − x + |αk − α| + vk − v Rk := µv , xk − v , vk xk − x + |αk − α| + vk − v Since v = µx = and vk → v, we may assume that vk = for all k Since vk ∈ N (xk , αk ), by (3.2) and (3.1) we have vk = µk xk with µk > As µk = vk · xk −1 and xk → x, we must have µk → µ as k → ∞ If xk = x then αk = α and vk = µk xk = µk x Combining this with the properties v = µx and v , x = 0, we get Pk = 0, contradicting (3.7) We have thus shown that xk = x for all k ∈ IN It holds that limsupk→∞ Rk ≤ Indeed, otherwise there exist a subsequence {k } of {k} and a constant ρ > such that Rk = µv , xk − v , vk ≥ ρ, xk − x + |αk − α| + vk − v ∀ ∈ IN (3.8) Then we have Rk ≤ µv , xk − v , vk µv , xk − v , µk xk = xk − x + v k − v xk − x + v k − v = (1 − µk µ−1 ) µv , xk xk − x + vk − v Since v , x = 0, it holds that Rk ≤ ≤ (1 − µk µ−1 ) µv , xk − x xk − x + vk − v (1 − µk µ−1 ) µv , xk − x xk − x = (1 − µk µ−1 ) µv , xk − x −1 (xk − x) There is no loss of generality in assuming that xk − x Since µk → µ, we get Rk ≤ (1 − µk µ−1 ) µv , −1 xk − x xk − x (xk − x) → ξ with ξ = →0 as → ∞ This contradicts (3.8), hence there must exist N0 > such that Rk ≤ δ/2 for all k ≥ N0 Since (3.7) is satisfied and Pk = Qk + Rk for all k ∈ IN , this implies that Qk ≥ δ/2 for all k ≥ N0 For each k ≥ N0 , we have − αα x, xk − x + α (αk − α) − αα x, xk + αα x, x + α (αk − α) δ ≤ Qk = = xk − x + |αk − α| + vk − v xk − x + |αk − α| + vk − v = = − αα x, xk + αα α2 + α αk − α α − αα x, xk + α αk = xk − x + |αk − α| + vk − v xk − x + |αk − α| + vk − v x,xk α α αk − xk − x + |αk − α| + vk − v (3.9) Hence, if α = then Qk = 0, which is impossible Thus α = If α < 0, then it follows k < Consequently, we have from (3.9) that αk − x,x α x · xk ααk x, xk ≤ = = αk , α α α αk < a contradiction It remains to consider the case where α > The subsequent analytical arguments are based on a geometrical construction Define the intersection of the ray Oxk with the sphere ∂E(α) := {x ∈ IRn x = α} by zk Let uk be the orthogonal projection of x on the ray Oxk (Since xk → x = 0, uk is well defined for k ≥ N0 large enough.) Since xk = for all k, we have k α αk − x,x α δ ≤ Qk = xk − x + |αk − α| + vk − v ≤ = α αk − x,xk α xk − x α αk − x,xk x · xk xk − x 2α αk sin2 = = α αk − xk − x = (x, xk ) xk − x x,xk ααk α αk − cos (x, xk ) xk − x 2α αk = zk −x α−1 xk − x = α αk zk − x · 2α2 xk − x The equality zk = x yields δ/2 ≤ Qk ≤ 0, an absurd Thus zk − x = for all k ≥ N0 sufficient large From the above it follows that δ α αk zk − x α αk zk − x ≤ Qk ≤ · = · 2 2α uk − x 2α uk − x · zk − x = α αk zk − x α zk − x · < · 2α α sin Ozk x sin Ozk x −1 FGGURE Nguyen Thanh Qui∗ Thus, for all k large enough, 0< Abstract This paper investigates δα zk − x < 2α sin Ozk x (3.10) Note that since the triangle Ozk x is isosceles and zk → x, the angle Ozk x tends to π/2 as Key words Linear generalized equation, KKT point set map, normal cone mapping, k→ ∞ Hence, from (3.10) we deduce that coderivative, local Lipschitz-like property δα AMS subject classification 49J53, 049J52, < 49J40 ≤ 0, 2α an absurd Thus, the inclusion “ ⊃ ” of (3.6) is valid The proof is complete αk xk O uk zk uk x α O zk ✷ α k xk α x Figure 1: Illustration for the proof of Theorem 3.1 ωk ωk Remark 3.1 For the second part of the proof of Theorem 3.1, let us present another argument dealing with the case where α > In this case we have k α αk − x,x α δ α α−1 (ααk − x, xk ) ≤ Qk = ≤ xk − x + |αk − α| + vk − v xk − x It follows that δα 2(ααk − x, xk ) 2( x · xk − x, xk ) ≤ = α xk − x xk − x xk − x, xk − x + x · xk − x − xk xk − Can x Tho University, Ly Tu and Communication Technology, = College of Information Tho, Vietnam; email: ntqui@cit.ctu.edu.vn ∗ = xk − x − Since xk − x xk − x · xk − x xk − x Trong, Can (3.11) ≤ xk − x , we have 0≤ xk − x · xk − x xk − x ≤ xk − x Therefore, passing k to infinity, from (3.11) we deduce that < at a contradiction δα α ≤ We have arrived Remark 3.2 Formula (3.6) shows that if v , x = then the set D∗ N (x, α, v)(v ) is a straight line in IRn × IR passing through the point (µv , 0) To see this, it suffices to put first α = to get x = µv , then let α take an arbitrary real value and compute x = − αα x + µv for each α ∈ IR In the case where x = α and v = 0, the following result has been obtained in [6] Lemma 3.3 (See [6, Lemma 3.3]) If x = α, v = 0, and (x , α ) ∈ D∗ N (x, α, v)(v ), then v , x ≥ 0, and there exists γ ∈ IR such that x = γx The upper estimate for the Fréchet coderivative value D∗ N (x, α, v)(v ) provided by Lemma 3.3 can be rewritten formally as D∗ N (x, α, v)(v ) ⊂ (x , α ) ∈ IRn × IR x = γx for some γ ∈ IR (3.12) when v , x ≥ 0, and D∗ N (x, α, v)(v ) = ∅ if v , x < The estimate (3.12) may be strict Example 3.1 Let n = In this case, N is a multifunction between IR2 ×IR and IR2 For α = 1, x = (1, 0) , and v = (0, 0) , we have (x, α, v) ∈ gphN because v ∈ N (x; E(α)) Choosing αk = α = 1, xk = (1 − k −1 , 0) , and vk = v = (0, 0) , we see at once that gphN (xk , αk , vk ) −→ (x, α, v) Select v = (1, 0) , x = (−1, 0) , α ∈ IR, and observe that v , x > and x = γx, where γ = −1 However, (x , α ) ∈ D∗ N (x, α, v)(v ) To see this, it suffices to note that limsup gphN (x,α,v) −→ (x,α,v) ≥ limsup k→∞ x , x − x + α (α − α) − v , v − v x − x + |α − α| + v − v x , xk − x + α (αk − α) − v , vk − v = > 0; xk − x + |αk − α| + vk − v hence (x , α , −v ) ∈ N ((x, α, v); gphN ) Tightening the estimate (3.12) we can get an exact formula for the coderivative D N (x, α, v) in the case x = α and v = as follows ∗ Theorem 3.2 If x = α and v = 0, then   (x , α ) ∈ IRn × IR x = − α x, α ≤ , if α ∗ D N (x, α, v)(v ) = ∅, if for every v ∈ IRn v ,x ≥ v , x < 0, (3.13) Proof Fix any v ∈ IRn If v , x < 0, then D∗ N (x, α, v)(v ) = ∅ by Lemma 3.3 If v , x ≥ and if (x , α ) ∈ D∗ N (x, α, v)(v ), then by Lemma 3.3 we can select a γ ∈ IR such that x = γx (3.14) We are going to show that γ = − αα and α ≤ Since (x , α ) ∈ D∗ N (x, α, v)(v ), we have limsup gphN (x,α,v) −→ (x,α,v) x , x − x + α (α − α) − v , v − v ≤ x − x + |α − α| + v − v (3.15) Choosing αk = α, xk = (1 − k −1 )x, and vk = for every k ∈ IN , we can infer that gphN (xk , αk , vk ) −→ (x, α, v) Hence, in accordance with (3.15) and (3.14), ≥ lim k→∞ = lim k→∞ x , xk − x + α (αk − α) − v , vk − v xk − x + |αk − α| + vk − v x , xk − x γx, (1 − k −1 )x − x −γ x, x = lim = = −γα −1 k→∞ xk − x (1 − k )x − x x Combining this with the condition α > 0, we get γ ≥ Now, for every k ∈ IN , let gphN xk = αk α−1 x and vk = v = 0, where αk will be chosen so that αk → α As (xk , αk , vk ) −→ (x, α, v), by (3.15) we have limsup k→∞ x , xk − x + α (αk − α) ≤ xk − x + |αk − α| Thus, for any ε > 0, there exists kε ∈ IN satisfying x , xk − x + α (αk − α) ≤ ε( xk − x + |αk − α|), ∀k ≥ kε Since x = γx, the latter implies that γ x, xk − x + α (αk − α) ≤ ε( xk − x + |αk − α|), ∀k ≥ kε Hence, α (αk − α) ≤ γ( x, x − x, xk ) + ε( xk − x + |αk − α|) = γ(α2 − x, αk α−1 x ) + ε( αk α−1 x − x + |αk − α|) = γ(α2 − αk α) + 2ε|αk − α| Therefore, − α 2ε (α − αk ) ≤ γ(α − αk ) + |αk − α| α α (3.16) Letting αk ↑ α as k → ∞, from (3.16) we get − αα ≤ γ + 2ε Letting αk ↓ α as k → ∞, α α 2ε from (3.16) we obtain − α ≥ γ − α Since ε > can be chosen arbitrary, it follows that γ = −α α−1 As γ ≥ and α > 0, we must have α ≤ Since x = γx = −α α−1 x by virtue of (3.14), we have proved that D∗ N (x, α, v)(v ) ⊂ (x , α ) ∈ IRn+1 x = − 10 α x, α ≤ α (3.17) Let us check the opposite inclusion of (3.17) in the case v , x ≥ If one could find an element (x , α ) from the set on the right-hand side of (3.17) with (x , α ) ∈ D∗ N (x, α, v)(v ), then there would exist a sequence (xk , αk , vk ) and a constant δ > such gphN that (xk , αk , vk ) −→ (x, α, v) as k → ∞ and Pk := x , xk − x + α (αk − α) − v , vk − v ≥ δ, xk − x + |αk − α| + vk − v ∀k ∈ IN (3.18) Since x = −α α−1 x with α ≤ and since v = 0, we have Pk = Qk − Rk (3.19) where Qk := − αα x, xk − x + α (αk − α) , xk − x + |αk − α| + vk Rk := v , vk xk − x + |αk − α| + vk We distinguish two cases: (i) v , x = 0, (ii) v , x > Case (i): v , x = In this case Rk → as k → ∞ Indeed, if vk = for all large k, then Rk = for all k large enough; hence limk→∞ Rk = Otherwise, we may assume that vk = for all k For every k, since vk ∈ N (xk , αk ) \ {0}, by (3.2) and (3.1) there exists µk > such that vk = µk xk Then, we have vk = µk xk = Consequently, |Rk | = ≤ | v , vk | xk − x + |αk − α| + vk | v , xk | | v ,x | | v , vk | = → = (as k → ∞) vk xk x We have seen that Rk → as k → ∞ Case (ii): v , x > Since xk → x, this strict inequality yields v , xk > for all k ¯ then Rk = for all k ≥ k ¯ large enough If there is k¯ ∈ IN such that vk = for all k ≥ k, If there exists a subsequence {vk } of {vk } with vk = for all ∈ IN , then vk = µk xk , where µk > for all Since v , xk > for all sufficiently large, we have v , vk = v , µk xk = µk v , xk > ( is large enough) This implies that ≤ Rk = ≤ v , vk xk − x + |αk − α| + vk v , vk vk = v , xk xk → v ,x x (as → ∞) Since the last property of {Rk } is valid for any subsequence {vk } of {vk } with vk = for all , we can assert that ≤ Rk ≤ x −1 v , x + for all k large enough From the above analysis we see that, in both the cases (i) and (ii), there exists an index k0 such that Rk ≥ −δ/2 for all k ≥ k0 Then, by (3.18) and (3.19), δ Qk = Pk + Rk ≥ δ + Rk ≥ , 11 ∀k ≥ k0 Since α ≤ and xk ≤ αk , for each k ≥ k0 we have − αα x, xk − x + α (αk − α) − αα x, xk + αα α2 + α αk − α α δ ≤ Qk = = xk − x + |αk − α| + vk xk − x + |αk − α| + vk = ≤ α αk − αα x, xk α αk − αα x · xk ≤ xk − x + |αk − α| + vk xk − x + |αk − α| + vk α αk − α αk = xk − x + |αk − α| + vk This contradiction completes the proof of the opposite inclusion in (3.17), hence establishes (3.13) ✷ Remark 3.3 Theorem 3.13 gives a complete solution for the second part of the open problem raised in [6, Remark 3.6] 3.2 The Mordukhovich Coderivative of N (x, α) Based on the obtained formulas for D∗ N (x, α, v)(·), we provide exact formulas for the Mordukhovich coderivative D∗ N (x, α, v)(·) of the normal cone mapping N (·) in various cases In the next two lemmas, we recall some existing results Lemma 3.4 (See [6, Lemma 4.4]) The set gphN is locally closed in the product space IRn × IR × IRn Lemma 3.5 (See [6, Lemma 3.7]) If x < α, then v = and D∗ N (x, α, v)(v ) = D∗ N (x, α, v)(v ) = {(0IRn , 0IR )}, for every v ∈ IRn By Lemma 3.5, the normal cone mapping N (·) is graphically regular at any point (x, α, v) ∈ gphN with x < α The forthcoming theorem shows that N (·) is also graphically regular at any point (x, α, v) ∈ gphN with x = α and v = Theorem 3.3 If x = α and if v = 0, then we have D∗ N (x, α, v)(v ) = D∗ N (x, α, v)(v )   (x , α ) ∈ IRn × IR x = − α x + µv , if α = ∅, if for every v ∈ IRn , where µ := v · x −1 v ,x = (3.20) v ,x = Proof Fix any v ∈ IRn and let (x , α ) ∈ D∗ N (x, α, v)(v ) be given arbitrary By gphN the definition of the Mordukhovich coderivative, there exist sequences (xk , αk , vk ) −→ (x, α, v) and (xk , αk , vk ) → (x , α , v ) such that (xk , αk ) ∈ D∗ N (xk , αk , vk )(vk ), 12 ∀k ∈ IN (3.21) Since v = 0, we have vk = for all k large enough For those k, according to Theorem 3.1, (3.21) holds if and only if xk = αk , vk , xk = and xk = − αk vk xk + v αk xk k (3.22) Passing (3.22) to limit as k → ∞ and remembering that xk → x, αk → α, vk → v, xk → x , αk → α , and vk → v , we obtain v , x = and x = − v α x+ v α x Thus (x , α ) ∈ D∗ N (x, α, v)(v ) by Theorem 3.1 We have shown that D∗ N (x, α, v)(v ) ⊂ D∗ N (x, α, v)(v ) Since the reverse inclusion is obvious, combining this with (3.6) we obtain (3.20) for every v ∈ IRn ✷ The case (x, α, v) ∈ gphN with x = α, and v = 0, is treated now Combining the following theorem with Theorem 3.2, we see that D∗ N (x, α, v)(v ) = D∗ N (x, α, v)(v ) for all v from the closed half-space {v ∈ IRn | v , x ≤ 0} So the multifunction N (·) is graphically irregular at any point (x, α, v) ∈ gphN where x = α and v = Theorem 3.4 Suppose that x = α and v = For every v ∈ IRn , the following hold (i) If v , x = 0, then D∗ N (x, α, v)(v ) = =  D∗ N (x, α, v)(v ), if v ,x > {(0 if v ,x < IRn , 0IR )}, (3.23)   (x , α ) ∈ IRn+1 x = − α x, α ≤ , if α v ,x > {(0 v , x < IRn , 0IR )}, if (ii) If v , x = 0, then D∗ N (x, α, v)(v ) = (x , α ) ∈ IRn × IR x = − α x, α ∈ IR α (3.24) Proof Let (x, α, v) ∈ gphN , x = α, v = 0, and v ∈ IRn (i) If v , x < 0, then D∗ N (x, α, v)(v ) = {(0IRn , 0IR )} Indeed, for such v , given (x , α ) ∈ D∗ N (x, α, v)(v ) one gphN can find sequences (xk , αk , vk ) −→ (x, α, v) and (xk , αk , vk ) → (x , α , v ) with (xk , αk ) ∈ D∗ N (xk , αk , vk )(vk ), ∀k ∈ IN (3.25) The condition v , x < implies that vk , xk < for large k Fix for a while such an index k If xk = αk and if vk = 0, then D∗ N (xk , αk , vk )(vk ) = ∅ by Theorem 3.1 and by the equality vk , xk < If xk = αk and if vk = 0, then D∗ N (xk , αk , vk )(vk ) = ∅ by Theorem 3.2 and by the equality vk , xk < Therefore, the nonemptyness of 13 D∗ N (xk , αk , vk )(vk ) shown in (3.25) yields xk < αk By Lemma 3.1, the latter implies that D∗ N (xk , αk , vk )(vk ) = {(0IRn , 0IR )} Consequently, (xk , αk ) = (0IRn , 0IR ) for all k large enough; hence (x , α ) = limk→∞ (xk , αk ) = (0IRn , 0IR ) This justifies the inclusion D∗ N (x, α, v)(v ) ⊂ {(0IRn , 0IR )} To get the reverse inclusion, choose xk = (1 − k −1 )x, αk = α, vk = 0, xk = 0, αk = 0, and vk = v for every k ∈ IN Then, (xk , αk , vk ) ∈ gphN , (xk , αk , vk ) → (x, α, v), and (xk , αk , vk ) → (0IRn , 0IR , v ) as k → ∞ The choice of xk and αk yields xk < x = α = αk Hence, by Lemma 3.1 we have (xk , αk ) ∈ D∗ N (xk , αk , vk )(vk ) = {(0IRn , 0IR )}, ∀k ∈ IN This gives (0IRn , 0IR ) ∈ D∗ N (x, α, v)(v ) and thus establishes the desired equality D∗ N (x, α, v)(v ) = {(0IRn , 0IR )} Suppose now that v , x > Due to the inclusion D∗ N (x, α, v)(v ) ⊂ D∗ N (x, α, v)(v ) and Theorem 3.2, the proof of the equalities D∗ N (x, α, v)(v ) = D∗ N (x, α, v)(v ) = (x , α ) ∈ IRn+1 x = − α x, α ≤ , α which are stated in (3.23), reduces to checking the fulfilment of the inclusion D∗ N (x, α, v)(v ) ⊂ D∗ N (x, α, v)(v ) (3.26) gphN For any (x , α ) ∈ D∗ N (x, α, v)(v ), there are sequences (xk , αk , vk ) −→ (x, α, v) and (xk , αk , vk ) → (x , α , v ) such that (3.25) holds (a) Consider the situation vk = for all k sufficiently large If xk < αk for all large k, then by Lemma 3.5 we get (xk , αk ) = (0, 0) for large indexes k Hence, (x , α ) = (0, 0) ∈ D∗ N (x, α, v)(v ) (the last inclusion is ready by Theorem 3.2 and the assumptions x = α, v = 0, and v , x > 0) If there exists a subsequence {k } of {k} such that xk = αk for all ∈ IN , then from (3.25) and Theorem 3.2 we can infer that vk , xk Taking the limits as ≥ 0, xk = − αk xk , αk αk ≤ → ∞, we obtain v , x ≥ 0, x =− α x, α α ≤ By virtue of (3.13), this yields (x , α ) ∈ D∗ N (x, α, v)(v ) (b) Suppose now that there is a subsequence {k } of {k} such that vk = for all ∈ IN Then, xk = αk for all From (3.25) and Theorem 3.1 we obtain vk , xk = and xk = − αk vk xk + v αk xk k for all This obviously leads to v , x = 0, a contradiction with the assumption that v , x > The inclusion (3.26) has been proved 14 Thus, if v , x = 0, then we get (3.23) (ii) Suppose that v , x = For any (x , α ) ∈ D∗ N (x, α, v)(v ), there exist sequences gphN (xk , αk , vk ) −→ (x, α, v) and (xk , αk , vk ) → (x , α , v ) such that (3.25) holds If vk = for all k large enough then, arguing similarly as in the subcase (a) of the proof of assertion (i), we obtain α v , x = 0, x = − x, α ≤ α Hence (x , α ) belongs to the set on the right-hand side of (3.24) If there is a subsequence {k } of {k} such that vk = for all ∈ IN , then by repeating the arguments of subcase (b) of the proof of assertion (i) we have xk = αk , vk , xk and xk = − = 0, αk vk xk + v , αk xk k for all Since v = 0, this implies that v , x = 0, x =− α x, α and α ∈ IR Thus, the inclusion “ ⊂ ” in (3.24) is valid To verify the inclusion “ ⊃ ” in (3.24), fix any element (x , α ) from the set on the right-hand side of (3.24) We have to show that (x , α ) ∈ D∗ N (x, α, v)(v ) For every k ∈ IN , choose xk = x, αk = α, and vk = µk xk , where µk := k −1 It is clear that (xk , αk , vk ) ∈ gphN and (xk , αk , vk ) → (x, α, v) as k → ∞ For every k ∈ IN , putting vk = v , αk = α , and xk = − we have vk , xk = v , x = 0, xk = − α x + k −1 v , α αk xk + µk vk αk Hence, in accordance with Theorem 3.1, (xk , αk ) ∈ D∗ N (xk , αk , vk )(vk ) for all k Observing (xk , αk , vk ) → (x , α , v ) as k → ∞, we obtain the inclusion (x , α ) ∈ D∗ N (x, α, v)(v ) which completes the proof of (3.24) ✷ Necessary and Sufficient Conditions for Stability Conditions for stability of the solution map (A, b, α) → S(A, b, α) of the linear GE of the form (1.1) are obtained in this section 4.1 Coderivatives of the KKT point set map As in Section 1, we put G(x, w) = Ax and M (x, w) = N (x, α) for every x ∈ IRn and ¯ ¯b, α w = (A, α) ∈ W with W = H(n) × IR Fix a triplet (A, ¯ ) ∈ H(n) × IRn × IR Put ¯ α ¯ ¯b, α w¯ = (A, ¯ ), y¯ = −¯b, and let x¯ ∈ S(A, ¯ ) Then we have x¯ ∈ S(w, ¯ y¯) with S(w, ¯ y¯) being ¯ ¯ given by (3.4) Let v¯ = y¯ − G(¯ x, w) ¯ = −b − A¯ x We will need two more lemmas of [6] 15 Lemma 4.1 (See [6, Lemma 4.1]) The Mordukhovich coderivative D∗ M (¯ x, w, ¯ v¯) : IRn ⇒ IRn × H(n)∗ × IR of the multifunction M : IRn × W ⇒ IRn , where H(n)∗ is the dual space of H(n), is computed by the formula D∗ M (¯ x, w, ¯ v¯)(v ) = (x , 0H(n)∗ , α ) (x , α ) ∈ D∗ N (¯ x, α ¯ , v¯)(v ) for every v ∈ IRn Lemma 4.2 (See [6, Lemma 4.3]) For every v ∈ IRn , ¯ } × {τ (v , x¯)} × {0IR }, ∇G(¯ x, w) ¯ ∗ (v ) = {Av where τ (v , x¯) := (vi x¯j ) is the n × n matrix whose ij-th element is vi x¯j Remark 4.1 Similarly as in Lemma 4.1, the Fréchet coderivative D∗ M (¯ x, w, ¯ v¯) : IRn ⇒ IRn × H(n)∗ × IR of the multifunction M : IRn × W ⇒ IRn is computed as follows D∗ M (¯ x, w, ¯ v¯)(v ) = (x , 0H(n)∗ , α ) (x , α ) ∈ D∗ N (¯ x, α ¯ , v¯)(v ) For each x ∈ IRn , we put ΩG,¯y (x ) = v ∈IRn (w , y ) ∈ W × IRn (−x , w , y ) ∈ ∇G(¯ x, w) ¯ ∗ (v ) × {0IRn } −{0IRn } × {0W } × {v } + D∗ M (¯ x, w, ¯ v¯)(v ) × {0IRn } , and ΩG,¯y (x ) = v ∈IRn (w , y ) ∈ W × IRn (−x , w , y ) ∈ ∇G(¯ x, w) ¯ ∗ (v ) × {0IRn } −{0IRn } × {0W } × {v } + D∗ M (¯ x, w, ¯ v¯)(v ) × {0IRn } , ¯x where v¯ = y¯ − G(¯ x, w) ¯ = −¯b − A¯ Since M : IRn ×W ⇒ IRn has a locally closed graph by Lemma 3.4, the next statement is an immediate corollary of [7, Theorem 4.3] Theorem 4.1 The inclusions ¯ y¯, x¯)(x ) ⊂ D∗ S(w, ¯ y¯, x¯)(x ) ⊂ ΩG,¯y (x ) ΩG,¯y (x ) ⊂ D∗ S(w, (4.1) ¯ y¯, x¯)(x ) = D∗ S(w, ¯ y¯, x¯)(x ) = ΩG,¯y (x ) ΩG,¯y (x ) = D∗ S(w, (4.2) hold for all x ∈ IRn If, in addition, M (·) is graphically regular at (¯ x, w, ¯ v¯) ∈ gphM , then for every x ∈ IRn Combining (4.2) with Lemmas 4.1 and 4.2, Remark 4.1, Lemma 3.5, Theorem 3.3, and the first assertion of Theorem 3.4, we obtain exact formulas for computing the Fréchet and the Mordukhovich coderivatives of S(w, y) = S(A, b, α) at the point (w, ¯ y¯, x¯) ∈ gphS with the property that M (x, w) = N (x, α) is graphically regular at ω ¯ := (¯ x, w, ¯ v¯) ∈ gphM Similarly, invoking (4.1), Lemmas 4.1 and 4.2, Remark 4.1, Theorems 3.2 and 3.4, we get explicit estimates for the Fréchet and the Mordukhovich coderivatives of S(·) at the point ¯x = (w, ¯ y¯, x¯) ∈ gphS where x¯ = α ¯ , v¯ = −¯b − A¯ 16 4.2 The Lipschitz-like property Since gphN is locally closed in the product space IRn × IR × IRn by Lemma 3.4, gphM is also locally closed in IRn × W × IRn So, both gphS and gphS are respectively locally closed in the product spaces H(n) × IRn × IR × IRn and W × IRn × IRn Therefore, by the Mordukhovich criterion [11, Theorem 4.10] we can assert that S(·) is locally Lipschitz-like around (w, ¯ y¯, x¯) if and only if D∗ S(w, ¯ y¯, x¯)(0) = {0} (4.3) By (3.4) we have ¯ ¯b, α D∗ S(A, ¯ , x¯)(0) = {0} ⇐⇒ D∗ S(w, ¯ y¯, x¯)(0) = {0} ¯ ¯b, α This implies that S(·) is locally Lipschitz-like around (A, ¯ , x¯) if and only if S(·) is locally Lipschitz-like around (w, ¯ y¯, x¯) If M (·) is graphically regular at ω ¯ , then by (4.2) we see that (4.3) holds if and only if ΩG,¯y (0) = ΩG,¯y (0) = {0} In the case where M (·) is graphically irregular at ω ¯ , by (4.1) we can infer that ΩG,¯y (0) = {0} =⇒ (4.3) =⇒ ΩG,¯y (0) = {0} (4.4) ¯ ¯b, α Theorem 4.2 For any (A, ¯ , x¯) ∈ gphS, the following assertions hold: ¯ ¯b, α (i) If x¯ < α ¯ , then the map S(·) is locally Lipschitz-like around (A, ¯ , x¯) if and only ¯ if detA = ¯x + ¯b = 0, then S(·) is locally Lipschitz-like around (A, ¯ ¯b, α (ii) If x¯ = α ¯ and A¯ ¯ , x¯) if ¯ ¯ and only if detQ(A, b, α ¯ , x¯) = 0, where A¯ + µI − α1¯ x¯ x¯ ¯ ¯b, α Q(A, ¯ , x¯) := (4.5) with µ being the unique Lagrange multiplier associated to x¯ Proof (i) Suppose that x¯ < α ¯ By Lemma 3.5, N (·) is graphically regular at (¯ x, α ¯ , v¯) Hence, M (·) is also graphically regular at ω ¯ = (¯ x, w, ¯ v¯) According to (4.2), S(·) is locally Lipschitz-like around ω ¯ if and only if ΩG,¯y (0) = {0} We see that (w , y ) = (A , α , y ) belongs to ΩG,¯y (0) if and only if there exists v ∈ IRn such that ¯ , A − (vi x¯j ), α , y + v − Av ∈ D∗ M (¯ x, w, ¯ v¯)(v ) × {0IRn } According to Lemma 4.1, this is equivalent to ¯ , α , A − (v x¯j ), y + v − Av i ∈ D∗ N (¯ x, α ¯ , v¯)(v ) × {0H(n)∗ } × {0IRn } Since D∗ N (¯ x, α ¯ , v¯)(v ) = {(0IRn , 0IR )} by Lemma 3.5, the last inclusion means that ¯ , α , A − (v x¯j ), y + v − Av i 17 = (0IRn , 0IR , 0H(n)∗ , 0IRn ) (4.6) So, the equality ΩG,¯y (0) = {0} holds if and only if the fulfilment of (4.6) for some v ∈ IRn yields A = 0H(n)∗ , α = 0IR , and y = 0IRn The latter means that detA¯ = Indeed, if ¯ = Setting A = τ (v , x¯) = (v x¯j ), α = 0, detA¯ = 0, then there is v = such that −Av i and y = −v , we get (w , y ) = (A , α , y ) = (0H(n)∗ , 0IR , 0IRn ) satisfying (4.6) Thus, there exists v ∈ IRn such that the fulfilment of (4.6) does not yield (w , y ) = (0W , 0IRn ) ¯ = 0; hence v = Substituting Conversely, if detA¯ = 0, then (4.6) implies that −Av v = into (4.6) yields A = 0, α = 0, and y = ¯x + ¯b = As in the case (i), S(·) is locally Lipschitz(ii) Suppose that x¯ = α ¯ and A¯ like around ω ¯ if and only if ΩG,¯y (0) = {0} Moreover, (w , y ) ∈ ΩG,¯y (0) if and only if there exists v ∈ IRn such that ¯ , α , A − (v x¯j ), y + v − Av i ∈ D∗ N (¯ x, α ¯ , v¯)(v ) × {0H(n)∗ } × {0IRn } ¯x = 0, Theorem 3.3 tells us that the last inclusion can be rewritten Since v¯ = −¯b − A¯ equivalently as  ¯ = − α x¯ + µv −Av  α ¯    v , x¯ = (4.7) α ∈ IR, A = (vi x¯j )    y + v = 0IRn ¯ ¯b, α with µ := v¯ · x¯ −1 If λ is the Lagrange multiplier corresponding to x¯ ∈ S(A, ¯ ), −1 ¯ ¯ ¯ ¯ then (A¯ x + λI)¯ x = −b So, λ¯ x = −b − A¯ x = v¯ It follows that λ = v¯ · x¯ Thus, µ is the Lagrange multiplier corresponding to x¯ Clearly, ΩG,¯y (0) = {0} if and only if from (4.7), with v ∈ IRn being chosen arbitrarily, it follows that A = 0H(n)∗ , α = 0IR , and y = 0IRn The latter is equivalent to saying that  α ¯  (A + µI)v − α¯ x¯ = v =0 =⇒ (4.8) x¯ v =  α =  n v ∈ IR , α ∈ IR Since (4.8) can be rewritten equivalently as A¯ + µI − α1¯ x¯ x¯ v α = 0 =⇒ v =0 α = 0, ¯ ¯b, α ¯ ¯b, α condition ΩG,¯y (0) = {0} means that detQ(A, ¯ , x¯) is nonzero, where Q(A, ¯ , x¯) has been defined by (4.5) ✷ The proof of the theorem is complete ¯ ¯b, α ¯x + ¯b = Then, the Theorem 4.3 Let (A, ¯ , x¯) ∈ gphS be such that x¯ = α ¯ and A¯ following hold ¯ ¯b, α (i) If S(·) is locally Lipschitz-like around (A, ¯ , x¯), then the constraint qualification below is satisfied  α ¯  Av − α¯ x¯ = v =0 =⇒ (4.9) v , x¯ ≥  α =  n v ∈ IR , α ≤ 18 ¯ ¯b, α (ii) If detA¯ = 0, detQ1 (A, ¯ , x¯) = 0, where ¯ ¯b, α Q1 (A, ¯ , x¯) := A¯ x¯ − α1¯ x¯ , (4.10) ¯ ¯b, α and (4.9) is satisfied, then S(·) is locally Lipschitz-like around (A, ¯ , x¯) ¯ ¯b, α Proof (i) Suppose that S(·) is locally Lipschitz-like around (A, ¯ , x¯) Then we have ∗ ∗ ¯ ¯ D S(A, b, α ¯ , x¯)(0) = {0} Thus, D S(w, ¯ y¯, x¯)(0) = {0} By (4.4), the latter implies that ΩG,¯y (0) = {0} Observe that (w , y ) ∈ ΩG,¯y (0) if and only if there exists v ∈ IRn such that ¯ , A − (v x¯j ), α , y + v ∈ D∗ M (¯ − Av x, w, ¯ v¯)(v ) × {0IRn } i Due to Remark 4.1, the last inclusion means that ¯ , α , A − (vi x¯j ), y + v − Av ∈ D∗ N (¯ x, α ¯ , v¯)(v ) × {0H(n)∗ } × {0IRn } By virtue of Theorem 3.2, this means that  ¯ = − α x¯ −Av  α ¯    v , x¯ ≥  α ≤ 0, A = (vi x¯j )    y = −v (4.11) Therefore, the condition ΩG,¯y (0) = {0} is equivalent to saying that (4.9) holds ¯ ¯b, α ¯ ¯b, α (ii) Suppose that detA¯ = 0, detQ1 (A, ¯ , x¯) = with Q1 (A, ¯ , x¯) given by (4.10), and (4.9) is satisfied As we have seen in the proof of Theorem 4.2(i), (w , y ) ∈ ΩG,¯y (0) if and only if there exists v ∈ IRn such that ¯ , A − (v x¯j ), α , y + v − Av i ∈ D∗ M (¯ x, w, ¯ v¯)(v ) × {0IRn } or, equivalently, ¯ , α , A − (vi x¯j ), y + v − Av ∈ D∗ N (¯ x, α ¯ , v¯)(v ) × {0H(n)∗ } × {0IRn } (4.12) ¯ , α ) ∈ D∗ N (¯ If v , x¯ < 0, then (−Av x, α ¯ , v¯)(v ) Indeed, the inequality v , x¯ < yields ¯ ¯ v = Hence −Av = because detA = By Theorem 3.4(i) and by the condition ¯ , α ) ∈ D∗ N (¯ v , x¯ < 0, we have (−Av x, α ¯ , v¯)(v ) Therefore, (4.12) is equivalent to  v , x¯ ≥    (−Av ¯ , α ) ∈ D∗ N (¯ x, α ¯ , v¯)(v )  A = (vi x¯j )    y = −v So, the equality ΩG,¯y (0) = {0} holds if and only if  v , x¯ >    (−Av ¯ , α ) ∈ D∗ N (¯ x, α ¯ , v¯)(v )  A = (vi x¯j )    y = −v 19   A = =⇒ α =   y =0 (4.13) and  v , x¯ =    (−Av ¯ , α ) ∈ D∗ N (¯ x, α ¯ , v¯)(v )  A = (vi x¯j )    y = −v By Theorem 3.4(i), (4.13) means that  α ¯  Av − α¯ x¯ = v , x¯ >   v ∈ IRn , α ≤ =⇒   A = =⇒ α =   y = (4.14) v =0 α = (4.15) Since (4.9) is satisfied by our assumptions, (4.15) is valid By virtue of Theorem 3.4(ii), (4.14) is equivalent to  α ¯  Av − α¯ x¯ = v =0 =⇒ (4.16) x¯ v =  α =  v ∈ IRn , α ∈ IR or, equivalently, A¯ x¯ − α1¯ x¯ v α = 0 =⇒ v =0 α = ¯ ¯b, α ¯ ¯b, α The latter holds because detQ1 (A, ¯ , x¯) = where Q1 (A, ¯ , x¯) is given by (4.10) We have shown that under the assumptions made, the equality ΩG,¯y (0) = {0} holds This ¯ ¯b, α implies that S(·) is locally Lipschitz-like around (A, ¯ , x¯), and completes the proof ✷ We now analyze Theorems 4.2 and 4.3 by four examples The first two show how Theorems 4.2 can recognize stability/instability of S(·) in the situation where x¯ = α ¯ ¯ ¯ and A¯ x + b = The third example illustrates a good association of the necessary stability condition and the sufficient stability condition provided by Theorem 4.3 for the case where ¯x + ¯b = The last example shows that the necessary stability condition x¯ = α ¯ and A¯ given in Theorem 4.3(i) can recognize instability of many linear GEs Example 4.1 Following [15] and [6], we consider the problem f (x) = −4x22 + x1 x = (x1 , x2 ) ∈ IR2 , x21 + x22 ≤ Here we have A¯ = 0 , −8 ¯b = , α ¯ = Using the necessary optimality condition (1.4) we find that √ √ ¯ ¯b, α S(A, ¯ ) = (−1, 0) , (−1/8, 63/8) , (−1/8, − 63/8) 20 (4.17) Note that x := (−1, 0) is the unique local-nonglobal solution of (4.17) The Lagrange √ multiplier corresponding to the KKT point x¯ := (−1/8, 63/8) is λ = Hence,   8 √ 63 ¯ ¯b, α detQ(A, ¯ , x¯) = det  √0 − 863  = − , 63 − 81 ¯x+¯b = Thanks to and we see that the stability criterion (4.5) is satisfied Observe that A¯ ¯ ¯b, α Theorem 4.2(ii), we can infer that S(·) is locally Lipschitz-like around√(A, ¯ , x¯) ∈ gphS 63/8) is also valid By symmetry, we see at one that the assertions made for x ¯ = (−1/8, √ for the KKT point (−1/8, − 63/8) For the local-nonglobal solution x = (−1, 0) and ¯ + ¯b = and the associated Lagrange multiplier λ = 1, we find that Ax   1 ¯ ¯b, α detQ(A, ¯ , x) = det  −7 0 = −7 −1 0 ¯ ¯b, α Thus, by Theorem 4.2(ii), S(·) is also locally Lipschitz-like around (A, ¯ , x) ∈ gphS Example 4.2 As in [15] and [6], we consider the problem f (x) = −4(x22 + x23 ) + x1 x = (x1 , x2 , x3 ) ∈ IR3 , x21 + x22 + x23 ≤ ¯ ¯b, α with the data tube (A, ¯ ) given by   0 A¯ = 0 −8  , 0 −8   ¯b = 0 , (4.18) α ¯ = Using (1.4) we obtain ¯ ¯b, α S(A, ¯ ) = (−1, 0, 0) ∪ (−1/8, x2 , x3 ) x22 + x23 = 63/64 It can be verified that x := (−1, 0, 0) is the unique local-nonglobal minimizer of (4.18) A computation similar to that given in Example 4.1 shows that S(·) is locally Lipschitz-like ¯ ¯b, α around the point (A, ¯ , x) ∈ gphS To complete the stability analysis, fix any t ∈ [0, 2π) and consider the KKT point √ √ xt := − 1/8, ( 63/8) sin t, ( 63/8) cos t with the associated Lagrange multiplier λ = Note that   0 √  63 0 − sin t    √8 ¯ ¯b, α detQ(A, ¯ , xt ) = det   = 63 − cos t  √ √ 63 − 18 sin t 863 cos t ¯xt + ¯b = for any t ∈ [0, 2π), applying Theorem 4.2(ii) we deduce that the map Since A¯ ¯ ¯b, α S(·) is not locally Lipschitz-like around any point (A, ¯ , xt ) for t ∈ [0, 2π) 21 √ √ Example 4.3 Consider (1.1) with n = 2, A¯ = I, ¯b = −(1, 1) , α ¯ = 1, and x¯ = ( 2, 2) ¯x = The necessary condition for stability of S(·) provided by We have v¯ := −¯b − A¯ Theorem 4.3(i) is as follows:  √  v     v − α √2 = v =0 =⇒  α = v1 + v2 ≥    α ≤ 0, v = (v , v ) ∈ IR2 It is not difficult to see that this condition is satisfied Since detA¯ = 0, the sufficient ¯ ¯b, α stability condition from Theorem 4.3(ii) reduces to detQ1 (A, ¯ , x¯) = 0, where the matrix ¯ ¯b, α Q1 (A, ¯ , x¯) is given by (4.10) We have √   −√2 ¯ ¯b, α detQ1 (A, ¯ , x¯) = √0 √1 − 2 = = 2 ¯ ¯b, α Thus S(·) is locally Lipschitz-like around (A, ¯ , x¯) ∈ gphS Example 4.4 (A class of unstable problems) Let A¯ ∈ H(n) be not positive definite, ¯1, λ ¯2, , λ ¯ n } of A¯ there is an element λ ¯ i ≤ Since i.e., among the eigenvalue set {λ ¯ i I) = 0, there exists x¯ ∈ IRn with x¯ = such that (A¯ − λ ¯ i I)¯ det(A¯ − λ x = Let 0 ¯b = −A¯ ¯x We claim that S(·) is not locally Lipschitz-like around (A, ¯ ¯b, α ¯ , x¯) ∈ gphS To see this, it suffices to check that (4.9) is violated For v := x¯, we have v = and ¯ i ≤ The condition Av ¯ − α x¯ = in (4.9) is equivalent to v , x¯ > Choose α = λ α ¯ ¯ i I)¯ (A¯ − λ x = Since the latter is guaranteed by the choice of x¯, we conclude that (4.9) fails to holds Our claim has been proved References [1] A R Conn, N I M Gould, and P L Toint, Trust-Region Methods, MPSSIAM Ser Optim., Philadelphia, 2000 [2] A L Dontchev and R T Rockafellar, Implicit Functions and Solution Mappings, Springer, Dordrecht, 2009 [3] H A Le Thi, T Pham Dinh, and N D Yen, Properties of two DC algorithms in quadratic programming, J Global Optim., 49 (2011), pp 481–495 [4] G M Lee, N N Tam, and N D Yen, Quadratic programming and affine variational inequalities: A qualitative study, Springer-Verlag, New York, 2005 [5] G M Lee, N N Tam, and N D Yen, Stability of linear-quadratic minimization over Euclidean balls, SIAM Journal on Optimization (Accepted for publication) [6] G M Lee and N D Yen, Coderivatives of a Karush-Kuhn-Tucker point set map and applications (Submitted) 22 [7] G M Lee and N D Yen, Fréchet and normal coderivatives of implicit multifunctions, Appl Anal., 90 (2011), pp 1011–1027 [8] S Lu and S M Robinson, Variational inequalities over perturbed polyhedral convex sets, Math Oper Res., 33 (2008), pp 689–711 [9] S Lucidi, L Palagi, and M Roma, On some properties of quadratic programs with a convex quadratic constraint, SIAM J Optim., (1998), pp 105–122 [10] J M Martinez, Local minimizers of quadratic functions on Euclidean balls and spheres, SIAM J Optim., (1994), pp 159–176 [11] B S Mordukhovich, Variational Analysis and Generalized Differentiation, Vol I: Basic Theory and Vol II: Applications, Springer-Verlag, Berlin, 2006 [12] T Pham Dinh and H A Le Thi, A d.c optimization algorithm for solving the trust-region subproblem, SIAM J Optim., (1998), pp 476–505 [13] S M Robinson, Generalized equations and their solutions I Basic theory, Math Program Stud., 10 (1979), pp 128–141 [14] R T Rockafellar and R J.-B Wets, Variational Analysis, Springer-Verlag, Berlin, 1998 [15] H N Tuan and N D Yen, Convergence of Pham Dinh–Le Thi’s algorithm for the trust-region subproblem, J Global Optim., Online First, DOI: 10.1007/s10898011-9820-0, 21 December 2011 [16] N D Yen and J.-C Yao, Monotone affine vector variational inequalities, Optimization 60 (2011), pp 53–68 23 [...]... programming and affine variational inequalities: A qualitative study, Springer-Verlag, New York, 2005 [5] G M Lee, N N Tam, and N D Yen, Stability of linear- quadratic minimization over Euclidean balls, SIAM Journal on Optimization (Accepted for publication) [6] G M Lee and N D Yen, Coderivatives of a Karush-Kuhn-Tucker point set map and applications (Submitted) 22 [7] G M Lee and N D Yen, Fréchet and normal... coderivatives of implicit multifunctions, Appl Anal., 90 (2011), pp 1011–1027 [8] S Lu and S M Robinson, Variational inequalities over perturbed polyhedral convex sets, Math Oper Res., 33 (2008), pp 689–711 [9] S Lucidi, L Palagi, and M Roma, On some properties of quadratic programs with a convex quadratic constraint, SIAM J Optim., 8 (1998), pp 105–122 [10] J M Martinez, Local minimizers of quadratic... Euclidean balls and spheres, SIAM J Optim., 4 (1994), pp 159–176 [11] B S Mordukhovich, Variational Analysis and Generalized Differentiation, Vol I: Basic Theory and Vol II: Applications, Springer-Verlag, Berlin, 2006 [12] T Pham Dinh and H A Le Thi, A d.c optimization algorithm for solving the trust-region subproblem, SIAM J Optim., 8 (1998), pp 476–505 [13] S M Robinson, Generalized equations and their... ¯b, α implies that S(·) is locally Lipschitz-like around (A, ¯ , x¯), and completes the proof ✷ We now analyze Theorems 4.2 and 4.3 by four examples The first two show how Theorems 4.2 can recognize stability/instability of S(·) in the situation where x¯ = α ¯ ¯ ¯ and A x + b = 0 The third example illustrates a good association of the necessary stability condition and the sufficient stability condition... completes the proof of (3.24) ✷ 4 Necessary and Sufficient Conditions for Stability Conditions for stability of the solution map (A, b, α) → S (A, b, α) of the linear GE of the form (1.1) are obtained in this section 4.1 Coderivatives of the KKT point set map As in Section 1, we put G(x, w) = Ax and M (x, w) = N (x, α) for every x ∈ IRn and ¯ ¯b, α w = (A, α) ∈ W with W = H(n) × IR Fix a triplet (A, ¯ ) ∈ H(n)... [1] A R Conn, N I M Gould, and P L Toint, Trust-Region Methods, MPSSIAM Ser Optim., Philadelphia, 2000 [2] A L Dontchev and R T Rockafellar, Implicit Functions and Solution Mappings, Springer, Dordrecht, 2009 [3] H A Le Thi, T Pham Dinh, and N D Yen, Properties of two DC algorithms in quadratic programming, J Global Optim., 49 (2011), pp 481–495 [4] G M Lee, N N Tam, and N D Yen, Quadratic programming... α, v) and (xk , αk , vk ) → (x , α , v ) such that (3.25) holds If vk = 0 for all k large enough then, arguing similarly as in the subcase (a) of the proof of assertion (i), we obtain α v , x = 0, x = − x, α ≤ 0 α Hence (x , α ) belongs to the set on the right-hand side of (3.24) If there is a subsequence {k } of {k} such that vk = 0 for all ∈ IN , then by repeating the arguments of subcase (b) of the... I Basic theory, Math Program Stud., 10 (1979), pp 128–141 [14] R T Rockafellar and R J.-B Wets, Variational Analysis, Springer-Verlag, Berlin, 1998 [15] H N Tuan and N D Yen, Convergence of Pham Dinh–Le Thi’s algorithm for the trust-region subproblem, J Global Optim., Online First, DOI: 10.1007/s10898011-9820-0, 21 December 2011 [16] N D Yen and J.-C Yao, Monotone affine vector variational inequalities,... with Lemmas 4.1 and 4.2, Remark 4.1, Lemma 3.5, Theorem 3.3, and the first assertion of Theorem 3.4, we obtain exact formulas for computing the Fréchet and the Mordukhovich coderivatives of S(w, y) = S (A, b, α) at the point (w, ¯ y¯, x¯) ∈ gphS with the property that M (x, w) = N (x, α) is graphically regular at ω ¯ := (¯ x, w, ¯ v¯) ∈ gphM Similarly, invoking (4.1), Lemmas 4.1 and 4.2, Remark 4.1,... verified that x := (−1, 0, 0) is the unique local-nonglobal minimizer of (4.18) A computation similar to that given in Example 4.1 shows that S(·) is locally Lipschitz-like ¯ ¯b, α around the point (A, ¯ , x) ∈ gphS To complete the stability analysis, fix any t ∈ [0, 2π) and consider the KKT point √ √ xt := − 1/8, ( 63/8) sin t, ( 63/8) cos t with the associated Lagrange multiplier λ = 8 Note that  

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