Multivalued complementarity problems with asymptotically bounded - - PowerPoint PPT Presentation

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results

Multivalued complementarity problems with asymptotically bounded multifunctions

Fabián Flores-Bazán1

1Departamento de Ingeniería Matemática, Universidad de Concepción

VI Journées Franco-Chiliennes d’Optimisation

• U. Sud Toulon-Var, 19 - 21 Mai, 2008, Toulon - France.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results

Contents

1

Formulation Examples

2

Asymptotic bounded multifunctions Asymptotic bounded multifunctions Examples Preliminaries

3

The Basic Lemma Some Notations The Basic Lemma

4

Asymptotically well-behaved ... Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property

5

Sensitivity Results Sensitivity Results

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Examples

Motivation Consider m« ın

x≥0 h(x),

(1) h differentiable on Rn

+. The KKT optimality condition says: if

¯ x ≥ 0 is a solution to (1), then there exists λ ≥ 0 such that ∇h(¯ x) − λ = 0 λ, ¯ x = 0. Moreover, it is sufficient when h is pseudoconvex. By replacing ∇h(x) by F(x), the KKT condition leads ¯ x ≥ 0 : F(¯ x) ≥ 0, F(¯ x), ¯ x = 0.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Examples

Formulation Let q ∈ Rn, F : Rn

+ ֒

→ Rn be given: find ¯ x ≥ 0, ¯ y ∈ F(¯ x) : ¯ y + q ≥ 0, ¯ y + q, ¯ x = 0 (MCP(q, F)) find ¯ x ≥ 0, ¯ y ∈ F(¯ x) : ¯ y + q, x − ¯ x ≥ 0 ∀ x ≥ 0 (VIP(q, F)) S(q, F): set of solutions to (MCP(q, F)). F(x) = ∅ ∀ x ≥ 0; F(x) is convex, compact ∀ x ≥ 0; F is upper semicontinuous on Rn

+.

Karamardian S. 1972, 1976; Moré J.J 1974; García C.B. 1973; Saigal R. 1976; Isac G. 1989,...; Gowda S.M., Pang J.S. 1992; Crouzeix 1997; ....

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Examples

References Cottle R. W., Pang J. S., Stone R. E., The linear complementarity problem, Academic Press, Boston (1992). Harker P .T., Pang J.S., Finite-dimensional variational inequality and nonlinear compl. prob....Mathematical Prog. (1990); Ferris M.C., Pang J.S., Engineering and economic appl. of

• compl. prob. SIAM, Review (1997);

Facchinei F ., Pang J.S., Finite-dimensional var. ineq. and comp.

• prob. (two volumes) Springer-Verlag (2003).

Flores-Bazán F ., López R., The linear complementarity problem under asymptotic analysis, Mathematics of Operations Research (2005); Flores-Bazán F ., López R., Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems, ESAIM, Control Opt and Cal of Var (2006);

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Examples

Example 1. M ∈ Rn×n, A, B ∈ Rm×n, q ∈ Rn. Set F(x) = {Mu : Au + Bx ≤ 0}, x ≥ 0. The problem is to find (x, y) ∈ Rn × Rn, u ∈ Rn such that x ≥ 0, y = Mu + q ≥ 0, Mu + q, x = 0, Au + Bx ≤ 0. Example 2. F(x) = Mx + ∂h(x), M ∈ Rn×n, h(x) = sup

u∈C

x, u, C = ∅, convex, compact. If M is symmetric, the MCP is the stationary point problem of the following minimax problem m« ın

x≥0 sup u∈C

1 2x, Mx + x, q + u

• Flores-Bazán

Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Examples

Set R++ = ]0, +∞[. Let us consider C . =

• c : R++ → R++ :

l« ım

t→+∞ c(t) = +∞

• C0 .

=

• c ∈ C :

l« ım

t→+∞

c(λt) c(t) ∈ R for all λ > 0

• .

For c ∈ C0, set c∞(λ) . = l« ım

t→+∞

c(λt) c(t) ; c∞(1) = 1; c∞(ts) = c∞(t)c∞(s), t, s > 0. Examples c1(t) = tp(p > 0), c∞

1 (t) = tp,

c2(t) = tp ln(tγ+1), c2(t) = tp ln(tγ + 1), (p > 0, γ ≥ 1), c∞

2 (t) = tp.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

Asymptotic bounded multifunctions: Given a sequence of multifunctions F k : Rn

+ ֒

→ Rn, k ∈ N and c ∈ C, the c-asymptotic multifunction associated to {F k} is defined by l« ım sup

k ∞,cF k(v) .

=

• w ∈ Rn : λkm ↑ +∞, xkm ≥ 0, xkm

λkm → v, ykm ∈ F km(xkm), ykm c(λkm) → w

• .

Indeed, gph(l« ım sup

k ∞,cF k) = l«

ım sup

k ∞,c(gph F k),

where,

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

l« ım sup

k ∞,c(gph F k) .

=

• (v, w) ∈ Rn

+ × Rn : λkm ↑ +∞,

(xkm, ykm) ∈ gph F km, (xkm λkm , ykm c(λkm)) → (v, w)

• .

c-asymptotically bounded property The property of c-asymptotically bounded at v (c-AB) means: for every km → +∞, every λkm ↑ +∞, as m → +∞, every xkm ≥ 0, every ykm ∈ F km(xkm), such that {xkm/λkm} converges to v, the sequence {ykm/c(λkm)} is bounded. When F k = F, c(t) = tγ, γ > 0 it was discussed by Gowda-Pang, 92.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

Given F : Rn

+ ֒

→ Rn and c ∈ C, we set AF(x) . = sup

y∈F(x)

|y|. The above property holds iff l« ım sup

k ∞AF k,c(v) .

= sup

• l«

ım sup

m→+∞

1 c(λkm)AF km(λkmxkm) : km ↑ +∞, λkm ↑ +∞, xkm → v

• < +∞.

As before, in case F k = F for all k, we set F ∞

c (v) .

= l« ım sup

k ∞,cF k(v), A∞ F,c(v) .

= l« ım sup

k ∞AF k,c(v).

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

More precisely, one has the following lemma. Lemma: Flores-Bazán, 2007 Let F k : Rn

+ ֒

→ Rn with nonempty compact values, c ∈ C and v ≥ 0. Then, the following assertions are equivalent: (a) l« ım sup∞

k AF k,c(v) < +∞;

(b) {F k} satisfies the c-asymptotically bounded property at v.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

Examples: Take M, Mk ∈ Rn×n satisfying Mk → M, and ρ ∈ R: F k(x) = |x|γMkx − ρx (γ ≥ 0), c(t) = tγ+1 = c∞(t), then l« ım sup

k ∞AF k,c(v) = |v|γ|Mv|, l«

ım sup

k ∞,cF k(v) = |v|γMv.

F k(x) = ln(|x|γ + 1)Mkx − ρx (γ ≥ 1), c(t) = t ln(tγ + 1), then c∞(t) = t. l« ım sup

k ∞AF k,c(v) = |Mv|, l«

ım sup

k ∞,cF k(v) = Mv.

Take γ > 0 and ∅ = Ck ⊆ Rn, closed, converging (in the sense of Painlevé-Kuratowski) to a closed ∅ = C ⊆ Rn. F k(x) = |x|γCk, l« ım sup

k ∞,cF k(v) = |v|γC.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

Examples: Continued ... Take F(x) = ln(|x|γ + 1)H(x) or F(x) =

1 ln(|x|γ+1)H(x)

(γ ≥ 1), with H being positive homogeneous of degree p > 0, that is H(tx) = tpH(x) ∀ t > 0, x ≥ 0. Consider c(t) = tp ln(tγ + 1) in the first case, and c(t) =

tp ln(tγ+1) in

the second case. Here c∞(t) = tp. F(x) = M1x + ln(|x| + 1)M2x, where Mi ∈ Rn×n Here we use c(t) = t ln(t + 1). Here c∞(t) = t. Take F(x) = (M1x, x, ln(|x| + 1)M2x, x) ∈ R2, x ≥ 0. Here c(t) = t2 ln(t + 1) is useful.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

Theorem (Flores-Bazán, 2007): Let F k : Rn

+ ֒

→ Rn with nonempty compact values c ∈ C. The following assertions hold. (a) If l« ım sup∞

k AF k,c(v) < +∞ then l«

ım supk

∞,cF k(v) = ∅,

l« ım sup

k ∞AF k,c(v) = sup

• |w| : w ∈ l«

ım sup

k ∞,cF k(v)

• .

(b) If c ∈ C0 and c∞(λ) > 0 ∀ λ > 0, then l« ım sup

k ∞,cF k is c∞−homogeneous

(i.e., G(tx) = c∞(t)G(x), ∀ t > 0, x ≥ 0).

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

Proposition: F k, Gk : Rn

+ ֒

→ Rn be with nonempty values for k ∈ N; c ∈ C v ≥ 0. If l« ım sup

k ∞AGk,c(v) = 0 then

l« ım sup

k ∞,cHk(v) = l«

ım sup

k ∞,cF k(v),

where Hk = F k + Gk. Take Gk(x) = ∂σCk(x), k ∈ N, where Ck is any nonempty compact convex set in Rn which converges (in the sense of Painlevé-Kuratowski) to the nonempty compact convex set C, is a typical example satisfying the above condition for every c ∈ C. In fact, due to the convergence of Ck, these sets remain in a fixed bounded set.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

Theorem [Flores-Bazán, 2007] Let F k, F : Rn

+ ֒

→ Rn be multifunctions with nonempty values for k ∈ N; c ∈ C, v ≥ 0. Assume that F k

g

→ F, then F ∞

c (v) ⊆ l«

ım sup

k ∞,cF k(v).

If, in addition, each F k is c-subhomogeneous, then l« ım sup

k ∞,cF k(v) ⊆ F(v).

F k(tx) ⊆ c(t)F k(x) ∀ t > 0, ∀ x ≥ 0.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotic bounded multifunctions Examples Preliminaries

Example: Take {Hk} ⊆ X which is asymptotically equi-osc and converging pointwise to H ∈ X (See Sections E and F of Chap. 5 in [RW]). By Theorem 5.40 in [RW], Hk

g

→ H. Assume for some p > 0, Hk(tx) = tpHk(x) ∀ t > 0, ∀ x ≥ 0. Then H(tx) = tpH(x) ∀ t > 0, ∀ x ≥ 0. By setting F k(x) = ln(|x| + 1)Hk(x), as above, F k

g

→ F with F(x) = ln(|x| + 1)H(x), and l« ım sup

k ∞,cF k(v) = F ∞ c (v) = H(v), c(t) = tp ln(t + 1).

A model is given by Hk(x) = {Mky : Ax + By ≤ 0}, where Mk ∈ Rn×n, A, B ∈ Rm×n, p = 1.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Some Notations The Basic Lemma

Some notations:

given J ⊆ I . = {1, . . . , n} and d > 0 (component-wise); set ∆J = ∆J(d) . = co{ 1

di ei : i ∈ J}, where ei is the i-th column

• f the identity matrix in Rn×n. However, sometimes we will
• mit the dependence on d when no confusion arises;

denote ∆d . = {x ≥ 0 : d, x = 1} = ∆I; given x ∈ Rn, we set supp{x} . = {i ∈ I : xi = 0}; given k ∈ N, d > 0 (component-wise), we set Dk . = {x ≥ 0 : d, x ≤ σk}.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Some Notations The Basic Lemma

The Basic Lemma [FB-López, 2005; FB, 2007] F k, Gk ∈ X, and {(xk, yk, r k)} be a sequence of solutions to find xk ∈ Dk : yk ∈ F k(xk), r k ∈ Gk(xk), yk + r k + qk, x − xk ≥ 0 ∀ x ∈ Dk. (VIk) such that d, xk = σk and xk

σk → v as k → +∞. Then, there

exist subsequences {σkm} and {(xkm, ykm, r km)}, numbers k0, m0 ∈ N, and an index set ∅ = Jv ⊆ I such that (a) for all k ≥ k0, xk − σk

2 v ≥ 0 and 0 < d, xk − σk 2 v < d, xk;

(b) for all m ≥ m0,

1 σkm xkm ∈ ri(∆Jv), thus supp{xkm} = Jv

(hence supp{v} ⊆ Jv );

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Some Notations The Basic Lemma

The Basic Lemma: continued ... (c) for all m ≥ m0, z ∈ ∆Jv: ykm + r km + qkm, σkmz − xkm = 0. Moreover, for a given c ∈ C, (d) if l« ım sup

k ∞AF k,c(v) < +∞ and l«

ım sup

k ∞AGk,c(v) = 0,

then the subsequences {ykm}, σkm may be chosen in such a way that there is a vector w such that w, v ≤ 0,

1 c(σkm)ykm → w ∈ l«

ım sup∞,c

k

F k(v), w, y ≥ d, yw, v for all y ≥ 0, and w, z = w, v for all z ∈ ∆Jv.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Some Notations The Basic Lemma

Remark In the previous lemma we actually get v, d = 1. Additionally, by choosing y = ei, i = 1, . . . , n, in (d), and setting z . = w − w, vd ≥ 0, we obtain z, v = 0, w ∈ l« ım sup∞,c

k

F k(v). Therefore, if τ . = −w, v ≥ 0, then Jv = ∅, v ∈ ∆Jv, v = 0, v ∈ S(τd, l« ım sup

k ∞,cF k).

(straightforward)Theorem: Let d > 0, c ∈ C such that c∞(λ) > 0 ∀ λ > 0; let F ∈ X. If A∞

F,c(v) < +∞ ∀ v ∈ ∆d and

F ∞

c

is strictly copositive, then S(q, F + G) is nonempty and compact for all q ∈ Rn and all G ∈ X satisfying A∞

G,c(v) = 0

∀ v ∈ ∆d.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Some Notations The Basic Lemma

Proposition: Let d > 0, q ∈ Rn

+, c ∈ C, F, G ∈ X

A∞

F,c(v) < +∞ and A∞ G,c(v) = 0 ∀ v ∈ ∆d.

Then, (S(q, F + G))∞ ⊆ S(0, F ∞

c ).

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Some Notations The Basic Lemma

General existence theorem [FB, 2007]

Let d > 0, σk ↑ +∞ and q ∈ Rn; let F ∈ X, and {(xk, yk)} be a sequence of solutions to find xk ∈ Dk : yk ∈ F(xk), yk +q, x −xk ≥ 0 ∀ x ∈ Dk. (VI) such that d, xk = σk and xk

σk → v as k → +∞. Then, the

following assertions are equivalent: (a) ∃ m0 and {xkm} such that ykm + q, v ≥ 0 ∀ m ≥ m0; (b) ∃ m0 and {xkm} such that ∀ m ≥ m0 ∃ ukm ≥ 0, 0 < d, ukm < d, xkm and ykm + q, ukm − xkm ≤ 0. (c) ∃ m0 and {xkm} such that xkm ∈ S(q, F) ∀ m ≥ m0; (d) ∃ m0 and {xkm} such that ykm + q, v = 0 ∀ m ≥ m0.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Some Notations The Basic Lemma

Classes of Mappings

Let F : Rn

+ ֒

→ Rn be a multifunction with nonempty values. We say that F is a: (i) copositive mapping if x, y ≥ 0 ∀ (x, y) ∈ gph F; (ii) strictly copositive mapping if x, y > 0 ∀ (x, y) ∈ gph F, x = 0; (iii) (assuming 0 ∈ F(0)) semimonotone mapping if S(p, F) = {0} ∀ p > 0; (given p > 0) G(p)- mapping or shortly F ∈ G(p) if S(τp, F) = {0} ∀ τ > 0. In the case when F is c-homogeneous with c(t) = tγ, γ > 0, we get F ∈ G(p) ⇐ ⇒ S(p, F) = {0}, since v ∈ S(τp, F) ⇐ ⇒ τ −1/γv ∈ S(p, F).

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property

Definition

Given d > and c ∈ C, we say that F : Rn

+ ֒

→ Rn is asymptotically well-behaved T-mapping, if A∞

F,c(v) < +∞ ∀ v ∈ ∆d, and for any index set α ⊆ I, one has

v ≥ 0, w ≥ 0, w ∈ F ∞

c (v),

α = ∅, v ∈ ∆α, wα = 0

• =

⇒ v ∈ [F(pos+∆α)]∗. When c ∈ C0, F ∞

c

is c∞-homogeneous, so this definition is independent of d. Any F satisfying A∞

F,c(v) < +∞ ∀ v ∈ ∆d and S(0, F ∞) = {0}

is T....

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property

The Linear Case: F(x) = Mx [FB-López, 2005] Given d >, we say that M is a T-matrix, if one has 0 = v ≥ 0, Mv ≥ 0, (Mv)α = 0, supp{v} ⊆ α,

• =

⇒ (M⊤v)α ≥ 0. It generalizes the class of #-matrix introduced in [Gowda-Pang, 1993]. It contains properly the symmetric matrices, the copositives and those satisfying S(0, M) = {0}. We say that M is #-matrix if v ∈ S(0, M) = ⇒ (M + M⊤)(v) ≥ 0.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property

Theorem [FB, 2007] Let d > 0, c ∈ C0 such that 0 < c∞(t) ∀ t > 0. Let F k : Rn

+ ֒

→ Rn be a sequence asymptotically well-behaved T-mapping such that F ∞

c

∈ G(d). (a) If q ∈ [S(0, F ∞

c )]# then S(q, F + G) = ∅ and compact for

all G ∈ X copositive and zero-subhomogeneous. (b) If q ∈ [S(0, F ∞

c )]∗ \ [S(0, F ∞ c )]#, then S(q, F + G) = ∅ and

compact for all G ∈ X strictly copositive and zero-subhomogeneous. (c) If q ∈ [S(0, F ∞

c )]∗ then S(q, F) = ∅ (possibly unbounded).

Under the above assumptions, int[S(0, F ∞

c )]∗ = [S(0, F ∞ c )]#.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property

Remark One can exhibit instances showing that (a) in the previous theorem may be false, if either F ∈ T, or q ∈ [S(0, F ∞

c ))]# or

F ∞

c

is not a G-mapping. Also there is an example showing that the strict copositivity of G or the condition q ∈ [S(0, F ∞

c )]#

cannot be avoided to obtain the boundedness of the solution set.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property

Let d > 0, c ∈ C and F, ∈ X such that A∞

F,c(v) < +∞ ∀ v ∈ ∆d.

The system v ≥ 0, d, v = 1, w ∈ F ∞

c (v), w, v ≤ 0, w − w, vd ≥ 0, (1)

found in the basic Lemma (for Gk = G, and qk = q for all k), plays a fundamental role in characterizing the nonemptiness and boundedness of S(q, F + G) for all q ∈ Rn. When F ∞

c

is c∞-subhomogeneous the inconsistency of (1) is equivalent to the inconsistency of the following system 0 = v ≥ 0, z ∈ F ∞

c (v), τ ≥ 0, z + τd ≥ 0, z + τd, v = 0. (2)

When F(x) = Mx with M being a real matrix the previous system was introduced by Karamardian 1972, giving rise to regular matrices.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property

Asymptotically Regular-type mappings-Robustness property

Definition Given d > and c ∈ C0, we say that F is asymptotically (regular) R(d)-mapping, or shortly F ∈ R(d), if S(τd, F ∞

c ) = {0} ∀ τ ≥ 0.

That is, F ∈ R(d) if F ∞

c

∈ G(d) and S(0, F ∞

c ) = {0}.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property

Theorem [FB, 2007] Let d > 0, c ∈ C0 satisfying 0 < c∞(t) ∀ t > 0, and F ∈ X be c-subhomogeneous such that A∞

F,c(v) < +∞ ∀ v ∈ ∆d.

Assume in addition that F ∞

c

∈ G(d). The following assertions are equivalent: (a) S(q, F) is nonempty and compact for all q ∈ Rn; (b) S(q, F + G) is nonempty and compact for all q ∈ Rn and all G ∈ X copositive satisfying A∞

G,c(v) = 0 ∀ v ∈ ∆d;

(c) S(q, F + G) is nonempty and compact for all q ∈ Rn and all G ∈ X copositive and uniformly bounded; (d) S(q, F + G) is nonempty and compact for all q ∈ Rn and all G ∈ X copositive and zero-subhomogeneous; (e) S(0, F ∞

c ) = {0}.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Sensitivity Results

Sensitivity results

By H we denote the family of multifunctions with nonempty values, defined on Rn

+, satisfying

H(tx) = c∞(t)H(x) ∀ t > 0, ∀ x ≥ 0, with c∞ such that (c ∈ C0) c∞(1) = 1, c∞(ξ)c∞(λ) = c∞(ξλ), ξ > 0, λ > 0. For H ∈ H and d > 0, let us consider the outer norm [RW] |H|+

d

. = sup

• ||y|| : y ∈ H(x), x ∈ ∆d
• , |H|+

d < +∞ ∀ H ∈ H0.

On H0 . = {H ∈ H : H is loc. bound on ∆d, H(0) = 0}, |H1 − H2|+

d

. = sup{|y1 − y2| : y1 ∈ H1(x), y2 ∈ H2(x), x ∈ ∆d} becomes a metric.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Sensitivity Results

Given c ∈ C0, set X0 =

• F ∈ X : F ∞

c (0) = {0}, A∞ F,c(v) < +∞ ∀ v ∈ ∆d

• .

Thus, F ∈ X0 implies F ∞

c

∈ H0. Proposition: [FB, 2007] Let F k : Rn

+ ֒

→ Rn be any multifunctions with nonempty values, c ∈ C and v ≥ 0. If l« ım supk

∞AF k,c(v) < +∞ then

l« ım supk

∞,cF k is locally bounded at v.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Sensitivity Results

Proposition: [FB, 2007] Let d > 0, c ∈ C0, q0 ∈ Rn, F 0 ∈ X0. If q0 ∈ [S(0, (F 0)∞

c )]#,

then there exists ε > 0 such that for all q ∈ Rn, all F ∈ X0 satisfying ||q − q0|| + |F ∞

c

− (F 0)∞

c |d < ε

• ne has q ∈ [S(0, F ∞

c )]#.

Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Sensitivity Results

Theorem: [FB, 2007] Let d > 0, c ∈ C0, F 0 ∈ X0. If q0 ∈ [S(0, (F 0)∞

c )]#, then there

exists ε > 0 such that for all q ∈ Rn, all F ∈ X0, T − mappings, with F ∞

c

∈ G(d) satisfying ||q − q0|| + |F ∞

c

− (F 0)∞

c |d < ε,

• ne has S(q, F) is non-empty and compact.

Flores-Bazán Asymptotically bounded multifucntions and the MCP