A class of linear generalized equations

Solution stability of a class of linear generalized equations in finite dimensionalEuclidean spaces is investigated by means of generalized differentiation.. Exact formulasfor the Fr´ech

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 dimensionalEuclidean spaces is investigated by means of generalized differentiation Exact formulasfor the Fr´echet and the Mordukhovich coderivatives of the normal cone mappings of per-turbed Euclidean balls are obtained Necessary and sufficient conditions for the localLipschitz-like property of the solution maps of such linear generalized equations are de-rived from these coderivative formulas Since the trust-region subproblems in nonlinearprogramming can be regarded as linear generalized equations, these conditions lead tonew 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

1 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-valuedoperator of the generalized equation is affine and the accompanying set-valued map isthe 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] provedthat if a linear GE is monotone and the solution set is nonempty and bounded, then thesolution map is locally upper Lipschitzian with respect to the parameters describing theaffine 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 alsothe references therein)

In connection with the solution methods [12], [15] and the qualitative study [5] for thetrust-region subproblems, we are interested in the linear GEs of the form

‡ 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 > 0 areparameters, E(α) :={x ∈ IRn

by parameter α ∈ (0, +∞) Here E(α) is a ball centered at 0 with radius α

Ifx is a local solution of the optimization problem

min{f(x) = 12x>Ax + b>x| x ∈ E(α)}, (1.3)which is called the trust-region subproblem, then (1.1) holds due to the generalized Fermatrule (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 aLagrange multiplier λ≥ 0 such that

where I denotes the n× n unit matrix If x ∈ E(α) and there exists λ ≥ 0 satisfying(1.4), x is said to be a Karush-Kuhn-Tucker point (or a KKT point) of (1.3) and (x, λ) iscalled 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

hAx + b, y − xi ≥ 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 theperturbations of all the three components of its data set {A, b, α} Our main tools arethe Mordukhovich criterion (see [11, Theorem 4.10] and [14, Theorem 9.40]) for the localLipschitz-like property of multifunctions between finite dimensional normed spaces andsome lower and upper estimates for coderivatives of implicit multifunctions from [7] Ourresults 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 Fr´echet an the Mordukhovichcoderivatives of the normal cone mapping (x, α)7→ N(x; E(α)) Moreover, we complementthe 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 appliedwith 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 fromthat 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 charac-terize the local Lipschitz-like property of that map The latter is stronger than the innersemicontinuity 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 andgeneralized differentiation from [11] are recalled in Section 2 Section 3 provides exactformulas for the Fr´echet and the Mordukhovich coderivatives of the normal cone mapping(x, α) 7→ N(x, E(α)) In Section 4, necessary and sufficient conditions for the localLipschitz-like property of the solution maps (A, b, α)7→ S(A, b, α) of the linear generalizedequations (1.1) will be established We conclude the paper by four examples serving asillustrations for the obtained results

where x→ ¯x means x → ¯x with x ∈ Ω By convention, bΩ N (¯x; Ω) =∅ when ¯x 6∈ Ω

For a multifunction Φ : IRn ⇒ IRn, the sequential Painlev´e-Kuratowski upper limitwith respect to the norm topology of IRn is defined by

Given a point x0 in a normed space X and ρ > 0, we denote the open ball {x ∈

X kx − x0k < ρ} by B(x0, ρ), and the corresponding closed ball by ¯B(x0, ρ) We write

BX and ¯BX for B(0X, 1) and ¯B(0X, 1), respectively The norm in the product X × Y ofnormed spaces is given by k(x, y)k = kxk + kyk

The graph of a multifunctionF : IRn⇒ IRmis the set gphF :={(x, y) ∈ IRn×IRm| y ∈

F (x)} The kernel of F is defined by ker F := {x ∈ IRn| 0 ∈ F (x)} We say that F islocally closed around ¯z := (¯x, ¯y) ∈ gphF if there exists ρ > 0 such that the intersectiongphF ∩ ¯B(¯z, ρ) is closed in the product space IRn× IRm For every (¯x, ¯y)∈ gphF , we callthe multifunction bD∗F (¯x, ¯y) : IRm ⇒ IRn,

is said to be the Mordukhovich (or limiting/normal ) coderivative ofF at (¯x, ¯y) AlthoughgphD∗F (¯x, ¯y) might be nonconvex, D∗F (¯x, ¯y) is a multifunction of closed graph Onesays that F is graphically regular at (¯x, ¯y)∈ gphF if

D∗F (¯x, ¯y)(y0) = bD∗F (¯x, ¯y)(y0), ∀y0

∈ IRm.The last condition can be written equivalently as N ((¯x, ¯y); gphF ) = bN ((¯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 ` > 0 and neighborhoods U of ¯x, V of ¯y such that

3 Formulas for Coderivatives

The normal cone N (x; E(α)) can be computed explicitly Namely, we have

It is clear that

S(A, b, α) = eS(w, y) :=x ∈ IRn

y∈ G(x, w) + M(x, w) (3.4)Hence, the solution map

S : H(n)× IRn× IR ⇒ IRn, (A, b, α) 7→ S(A, b, α),

of (1.1) can be interpreted as the implicit multifunction

e

S : W × IRn⇒ IRn, (w, y)7→ eS(w, y), (3.5)where W := H(n)× IR with H(n) ⊂ IRn×n being the linear subspace of symmetric n× nmatrices 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 hu, vi = kuk · kvk cos [(u, v) For each pair u, v ∈ IRn with u = (u1, , un)> and

v = (v1, , vn)>, we define the vector −uv in IR→ n by setting −uv = (v→ 1− u1, , vn− un)>.For anyx, y, z ∈ IRn, we callxyz the angle between −d yx and −→ yz, provided the latter vectors→are nonzero

We are going to obtain exact formulas for the Fr´echet and the Mordukhovich tives of the normal cone mapping N (x, α) given by (3.2)

coderiva-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 kxk < α, then v = 0 and

b

D∗N (x, α, v)(v0

) ={(0IR n, 0IR)},for every v0

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 exactformula for computing the coderivative bD∗

N (x, α, v) in the situation kxk = α and v 6= 0

Theorem 3.1 If kxk = α and v 6= 0, then v = µx with µ = kvk · kxk−1 and, for every

Proof The property v = µx with µ = kvk · kxk−1 and the inclusion “ ⊂ ” of (3.6) areimmediate from Lemma 3.2.

To prove the opposite inclusion of (3.6), suppose to the contrary that there exists apair (x0, α0) belonging to the set described by the right-hand side of (3.6) with (x0, α0) 6∈b

by (3.2) and (3.1) we have vk =µkxk withµk> 0 As µk =kvkk · kxkk−1 and xk→ x, wemust have µk→ µ as k → ∞

If xk = x then αk = α and vk = µkxk = µkx Combining this with the properties

v = µx and hv0, xi = 0, we get Pk = 0, contradicting (3.7) We have thus shown that

xk 6= x for all k ∈ IN

It holds that limsupk→∞Rk ≤ 0 Indeed, otherwise there exist a subsequence {k`} of{k} and a constant ρ > 0 such that

Rk ` = hµv0

, xk `i − hv0

, vk `i

kxk `− xk + |αk `− α| + kvk `− vk ≥ ρ, ∀` ∈ IN. (3.8)Then we have

There is no loss of generality in assuming that kxk ` − xk−1(xk ` − x) → ξ with kξk = 1.Since µk ` → µ, we get

with the sphere ∂E(α) :={x ∈ IRn

kxk = α} by zk Letuk be the orthogonal projection

of x on the ray Oxk (Since xk → x 6= 0, uk is well defined for k ≥ N0 large enough.)Since xk 6= 0 for all k, we have

sufficient large From the above it follows that

δ

2 ≤ Qk ≤ α

0αk2α2 · kzk− xk

2

kuk− xk =

α0αk2α2 · kzk− xk

kuk− xk · kzk− xk−1

= α0αk2α2 · kzk− xksin [Ozkx <

α0

α · kzk− xksin [Ozkx.

Thus, for all k large enough,

0< δα2α0 < kzk− xk

Note that since the triangle Ozkx is isosceles and zk → x, the angle [Ozkx tends to π/2 as

k → ∞ Hence, from (3.10) we deduce that

0< δα2α0 ≤ 0,

an absurd Thus, the inclusion “⊃ ” of (3.6) is valid The proof is complete 2

FGGURE Nguyen Thanh Qui∗

Abstract This paper investigates

Key words Linear generalized equation, KKT point set map, normal cone mapping,

coderivative, local Lipschitz-like property

AMS subject classification 49J53, 49J52, 49J40

Figure 1: Illustration for the proof of Theorem 3.1

Remark 3.1 For the second part of the proof of Theorem 3.1, let us present anotherargument dealing with the case where α0 > 0 In this case we have

Remark 3.2 Formula (3.6) shows that if hv0, xi = 0 then the set bD∗

N (x, α, v)(v0) is astraight line in IRn × IR passing through the point (µv0, 0) To see this, it suffices toput first α0 = 0 to get x0 = µv0, then let α0 take an arbitrary real value and compute

The upper estimate for the Fr´echet coderivative value bD∗

N (x, α, v)(v0) provided byLemma 3.3 can be rewritten formally as

Example 3.1 Letn = 2 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(xk, αk, vk)gphN−→ (x, α, v) Select v0 = (1, 0)>,x0 = (−1, 0)>, α0

∈ IR, and observe that

hv0

, xi > 0 and x0

=γx,where γ =−1 However, (x0, α0)6∈ bD∗

N (x, α, v)(v0) To see this, it suffices to note thatlimsup

D∗N (x, α, v) in the case kxk = α and v = 0 as follows

Theorem 3.2 If kxk = α and v = 0, then

Proof Fix any v0

−γhx, xikxk =−γα.

Combining this with the condition α > 0, we get γ ≥ 0 Now, for every k ∈ IN, let

xk =αkα−1x and vk =v = 0, where αkwill be chosen so thatαk → α As (xk, αk, vk)gphN−→(x, α, v), by (3.15) we have

γhx, xk− xi + α0

(αk− α) ≤ ε(kxk− xk + |αk− α|), ∀k ≥ kε.Hence,

α ≥ γ − 2ε

α Since ε > 0 can be chosen arbitrary, it follows that

γ = −α0α−1 As γ ≥ 0 and α > 0, we must have α0 ≤ 0 Since x0 = γx = −α0α−1x byvirtue of (3.14), we have proved that

b

D∗

N (x, α, v)(v0)⊂

(x0, α0)∈ IRn+1

Let us check the opposite inclusion of (3.17) in the case hv0, xi ≥ 0 If one couldfind an element (x0, α0) from the set on the right-hand side of (3.17) with (x0, α0) 6∈b

≤ 0 and since v = 0, we have

We distinguish two cases: (i) hv0, xi = 0, (ii) hv0, xi > 0

Case (i): hv0, xi = 0 In this case Rk → 0 as k → ∞ Indeed, if vk = 0 for all large

k, then Rk = 0 for all k large enough; hence limk→∞Rk = 0 Otherwise, we may assumethat vk 6= 0 for all k For every k, since vk ∈ N (xk, αk)\ {0}, by (3.2) and (3.1) thereexists µk > 0 such that vk =µkxk Then, we havekvkk = µkkxkk 6= 0 Consequently,

If there exists a subsequence {vk `} of {vk} with vk ` 6= 0 for all ` ∈ IN, then vk ` =µk `xk `,where µk ` > 0 for all ` Since hv0, xk `i > 0 for all ` sufficiently large, we have

Since the last property of{Rk} is valid for any subsequence {vk `} of {vk} with vk ` 6= 0 forall `, we can assert that 0≤ Rk≤ kxk−1

hv0, xi + 1 for all k large enough

From the above analysis we see that, in both the cases (i) and (ii), there exists anindex k0 such thatRk≥ −δ/2 for all k ≥ k0 Then, by (3.18) and (3.19),

Qk =Pk+Rk ≥ δ + Rk ≥ δ

2, ∀k ≥ k0

3.2 The Mordukhovich Coderivative of N (x, α)

Based on the obtained formulas for bD∗N (x, α, v)(·), we provide exact formulas for theMordukhovich coderivative D∗

N (x, α, v)(·) of the normal cone mapping N (·) in variouscases 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 kxk < α, then v = 0 and

D∗N (x, α, v)(v0) = bD∗N (x, α, v)(v0) = {(0IR n, 0IR)},for every v0

∈ IRn

By Lemma 3.5, the normal cone mapping N (·) is graphically regular at any point(x, α, v) ∈ gphN with kxk < α The forthcoming theorem shows that N (·) is alsographically regular at any point (x, α, v)∈ gphN with kxk = α and v 6= 0

Theorem 3.3 If kxk = α and if v 6= 0, then we have

for every v0 ∈ IRn, where µ :=kvk · kxk−1

Proof Fix any v0

∈ IRn and let (x0, α0) ∈ D∗

N (x, α, v)(v0) be given arbitrary Bythe definition of the Mordukhovich coderivative, there exist sequences (xk, αk, vk) gphN−→(x, α, v) and (x0

k, α0

k, v0

k)→ (x0, α0, v0) such that(x0k, α0k)∈ bD∗N (xk, αk, vk)(v0k), ∀k ∈ IN (3.21)

Sincev 6= 0, we have vk 6= 0 for all k large enough For those k, according to Theorem 3.1,(3.21) holds if and only if kxkk = αk,

hv0k, xki = 0 and x0k=−α

0 k

0

.Thus (x0, α0)∈ bD∗

N (x, α, v)(v0) by Theorem 3.1 We have shown thatD∗

N (x, α, v)(v0)⊂b

D∗N (x, α, v)(v0) Since the reverse inclusion is obvious, combining this with (3.6) we

The case (x, α, v) ∈ gphN with kxk = α, and v = 0, is treated now Combining thefollowing theorem with Theorem 3.2, we see thatD∗N (x, α, v)(v0

)6= bD∗N (x, α, v)(v0

) forall v0 from the closed half-space {v0

∈ IRn| hv0, xi ≤ 0} So the multifunction N (·) isgraphically irregular at any point (x, α, v)∈ gphN where kxk = α and v = 0

Theorem 3.4 Suppose that kxk = α and v = 0 For every v0 ∈ IRn, the following hold(i) If hv0, xi 6= 0, then

(3.23)

(ii) If hv0, xi = 0, then

D∗N (x, α, v)(v0) =

(x0, α0)∈ IRn× IR

k, α0

k, v0

k)→ (x0, α0, v0) with(x0k, α0k)∈ bD∗N (xk, αk, vk)(v0k), ∀k ∈ IN (3.25)The condition hv0, xi < 0 implies that hv0

k, xki < 0 for large k Fix for a while such anindex k If kxkk = αk and if vk 6= 0, then bD∗N (xk, αk, vk)(v0

∅ by Theorem 3.2 and by the equality hv0

k, xki < 0 Therefore, the nonemptyness of