Download Lecture 2

Existence and uniqueness results for
variational inequalities
Lecture 2
Basis results (end) and existence results
Didier Aussel
Univ. de Perpignan
– p. 1
Outline of lecture 2
I- Set-valued or not? (end)
a- The pointwise case
b- The local case
II- First Existence results
a- The linear case
b- The finite dimensional case
III- Other classes of V.I.
– p. 2
Notations
X a topological vector space
X ∗ its topological dual (w∗ -top.)
h·, ·i the duality product
– p. 3
Notations
X a topological vector space
X ∗ its topological dual (w∗ -top.)
h·, ·i the duality product
Stampacchia variational inequality :
X∗
Let T : X → 2 be a map and C be a nonempty subset of X.
Find x̄ ∈ C such that there exists x̄∗ ∈ T (x̄) for which
hx̄∗ , y − x̄i ≥ 0,
∀ y ∈ C.
Notation : S(T, C) set of solutions (⊂ C).
– p. 3
I - Set-valued or not? (cont.)
The pointwise case
– p. 4
Pointwise Single-valuedness
Theorem 1 (Kenderov 1975)
X∗
Let X be a Banach space and T : X → 2 be a monotone
set-valued map.
If T is lower semicontinuous at a point x0 ,
then T is single-valued at x0 .
T is lsc at x0 if, for every open V such that V ∩ T (x0 ) 6= ∅, there exists a
neigh. U of x0 such that V ∩ T (x) 6= ∅, for any x ∈ U .
– p. 5
X∗
Definition 2 A set-valued map T : K → 2 is said to be
single-directional at x ∈ dom T if, T (x) ⊆ R+ {x∗ } for some
x∗ ∈ T (x).
X∗
Proposition 3 Let T : X → 2 be a quasimonotone set-valued
map. If T is lower semicontinuous at x ∈ X, then T is
single-directional at x;
– p. 6
Recovering Kenderov’73 (monotone lsc)
Proposition 4 (D.A., J.-N. Corvellec & M. Lassonde ’94)
X∗
A map T : X → 2 is monotone if and only if, for any α∗ ∈ X ∗ ,
the map T + {α∗ } is quasimonotone
– p. 7
Recovering Kenderov’73 (monotone lsc)
Proposition 6 (D.A., J.-N. Corvellec & M. Lassonde ’94)
X∗
A map T : X → 2 is monotone if and only if, for any α∗ ∈ X ∗ ,
the map T + {α∗ } is quasimonotone
X∗
Proposition 7 Let T : X → 2 be a set-valued map and x be a
point of its domain dom T be such that, for any α∗ ∈ X ∗ , the map
T + α∗ is single-directional at x, then T is single-valued at x.
– p. 7
Recovering Kenderov’73 (monotone lsc)
Hyp : T is monotone and lsc at x.
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is quasimonotone lsc at x
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is single-directional at x
⇒ T is single-valued at x.
– p. 8
Recovering Kenderov’73 (monotone lsc)
Hyp : T is monotone and lsc at x.
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is quasimonotone lsc at x
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is single-directional at x
⇒ T is single-valued at x.
Theorem 9 (Kenderov 1975)
X∗
Let X be a Banach space and T : X → 2 be a monotone
set-valued map.
If T is lower semicontinuous at a point x0 ,
then T is single-valued at x0 .
– p. 8
I - Set-valued or not? (cont.)
The local case
– p. 9
Local Single-valuedness
Theorem 10 (Dontchev and Hager 1994)
X∗
Let T : X → 2 be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y) ∈ gr T ,
then T is single-valued in a neigh. of x0 .
T : X → 2Y is Lipschitz-like around (x, y) ∈ gr T if it exist a neighb. U of x, a
neighb. V of y and l > 0 such that
T (u) ∩ V ⊂ T (u′ ) + lku′ − ukB Y (0, 1),
∀ u, u′ ∈ U
If T is single-valued
T is Lipschitz-like around (x, T (x)) ⇔ T is loc. Lipschitz at x
– p. 10
X∗
Definition 11 A set-valued map T : K → 2
is said to be
- single-directional at x ∈ dom T if, T (x) ⊆ R+ {x∗ } for some
x∗ ∈ T (x).
- locally single-directional at x ∈ dom T if it exists a neigh. U of
x such that, for any x′ ∈ U , T (x′ ) is single-directional at x′ .
Theorem 12
∗
Let T : X → 2X be a quasimonotone set-valued map. If T satisfies
the Lipschitz-like property around a point (x, x∗ ) of its graph and
x∗ 6= 0, then T is alocally single-directional at x.
– p. 11
Proposition 13
X∗
Let T : X → 2 be a quasimonotone set-valued map with convex
values and satisfying the Lipschitz-like property around a point
(x, x∗ ) of its graph.
Then T is locally single-directional at x.
– p. 12
Proofs (quasimonotone Lipschitz-like case)
direct proof
thanks to the following results
∗
Proposition 14 Let T : X → 2X be a set-valued map satisfying the
Lipschitz-like property around a point (x, x∗ ) of its graph (with U and V
the corresponding neighborhoods). Then the mapping TU ×V , whose graph
is given by gr TU ×V = gr T ∩ (U × V ) is lower semicontinuous at x.
∗
Y
2
be a set-valued map with convex
Proposition 15 Let T : Y →
values. If T admits a single-directional localization around a point (y, y ∗ )
of its graph, then T is locally single-valued at y.
– p. 13
Recovering Dontchev-Hager’94 (monotone Lipschitz-like)
Hyp : T is monotone and Lipschitz-like around (x, x∗ ).
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is quasimonotone and Lipschitz-like
around (x, x∗ )
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is locally single-directional at x
⇒ T is single-valued on a neigh. of x.
– p. 14
Recovering Dontchev-Hager’94 (monotone Lipschitz-like)
Hyp : T is monotone and Lipschitz-like around (x, x∗ ).
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is quasimonotone and Lipschitz-like
around (x, x∗ )
⇒ ∀ α∗ ∈ X ∗ , T + {α∗ } is locally single-directional at x
⇒ T is single-valued on a neigh. of x.
In the case of a convex valued operator...
Theorem 17 (Dontchev and Hager 1994)
X∗
Let T : X → 2 be a monotone set-valued map. If T is
Lipschitz-like around (x0 , y) ∈ gr T ,
then T is single-valued in a neigh. of x0 .
– p. 14
References
1- Facchinei, F. and Pang, J.S. Finite Dimensional Variational Inequalities, Springer, (2003).
2- Kenderov, P., Semi-continuity of set-valued mappings, Fund. Math., 88 (1975), 61-69.
3- Dontchev, A.L. and Hager, W., Implicit functions, Lipschitz maps, and stability in optimization,
Math. Oper. Reasearch, 19 (1994), 753-768.
5- Aussel, D., Garcia Y. and Hadjisavvas, N., Single-directional property of multivalued
maps and variational systems, SIAM J. Optim (2009).
6- Aussel, D. and Garcia Y., Single-valuedness of quasimonotone and monotone set-valued maps,
preprint, (2009).
– p. 15
II
First Existence results
a- The linear case
b- The finite dimension case
c- V.I. under monotonicity
d- Other assumptions
– p. 16
II - a - The linear case
Let H be an Hilbert space, f an element of H ∗ = H and
a : H × H → H a bilinear application.
Theorem 18 (Stampacchia 64) Let C be a nonempty closed
convex subset of H. If a is continuous and satisfies the following
property
∃ α > 0 such that a(v, v) ≥ αkvk2 , ∀ v ∈ H,
then
there exists x̄ ∈ C such that a(x̄, y − x̄) ≥ hf, y − x̄i,
∀ y ∈ C.
D. Kinderlehrer and G. Stampacchia, An introduction to Variational Inequalities and Their
Applications, 1980, Academic Press
– p. 17
Theorem 19 (Lax-Milgram) If a is continuous and satisfies the
following property
∃ α > 0 such that a(v, v) ≥ αkvk2 , ∀ v ∈ H,
then there exists a unique x̄ ∈ H such that
a(x̄, u) = hf, ui,
∀ u ∈ H.
Proof. From the Theorem of Stampacchia, there exists u ∈ H such that
a(u, v − u) ≥ hf, v − ui,
or in other words ϕu (v) ≥ ϕu (u),
∀v ∈ H
∀ v ∈ H where
ϕu (v) = a(u, v) − hf, vi.
But ϕu is linear and therefore ϕu ≡ 0.
– p. 18
II - b - The finite dimension case
Let f : X → X ∗ be a function and C be a nonempty subset of X.
Find x̄ ∈ C such that
hf (x̄), y − x̄i ≥ 0,
∀ y ∈ C.
First property :
S(f, C) ∩ int(C) = {x ∈ int(C) : f (x) = 0}
– p. 19
II - b - The finite dimension case
Proposition 20 (Stampacchia 1966) Let C be a nonempty convex
compact subset of IRn and f : C → IRn a continuous function. Then
S(f, C) is nonempty, that is,
there exists x̄ ∈ C such that hf (x̄), y − x̄i ≥ 0,
∀ y ∈ C.
– p. 20
II - b - The finite dimension case
Proposition 21 (Stampacchia 1966) Let C be a nonempty convex
compact subset of IRn and f : C → IRn a continuous function. Then
S(f, C) is nonempty, that is,
there exists x̄ ∈ C such that hf (x̄), y − x̄i ≥ 0,
∀ y ∈ C.
Proof: Let us consider the application
ψ:
C
→
C
x
7→
ψ(x) = PC ◦ (Id − f )(x) = PC (x − f (x))
ψ is continuous on the convex compact set C and therefore, according to the
Brouwer fixed point theorem, there exists a point x̄ ∈ C such that
x̄ = ψ(x̄)
⇔
x̄ = PC (x̄ − f (x̄))
⇔
hx̄ − f (x̄) − x̄, y − x̄i ≤ 0, ∀ y ∈ C
⇔
x̄ ∈ S(f, C).
– p. 20
Without compacity
The compacity hypothesis can be replaced by a "coercivity
condition":
Proposition 22 Let C be a nonempty closed convex subset of IRn
and f : C → IRn a continuous function. The following conditions
are equivalent:
- S(f, C) is nonempty.
- ∃ r > 0 such that S(f, C ∩ B(0, r)) ∩ B(0, r) 6= ∅,
– p. 21
Without compacity 2
The compacity hypothesis can be replaced by a "coercivity
condition" in term of the function:
Proposition 23 Let C be a nonempty closed convex subset of IRn
and f : C → IRn a continuous function. If f is coercive, that is,
hf (x) − f (x0 ), x − x0 i
∃ x0 ∈ C such that lim
= +∞
kx − x0 k
then S(f, C) is nonempty.
– p. 22
Multivalued form
Theorem 24 Let C be a nonempty convex compact subset of IRn
IRn
and F : C → 2 a upper semicontinuous map with convex
compact values. Then S(F, C) is nonempty, that is,
there exist x̄ ∈ C and x̄∗ ∈ F (x̄)
such that hx̄∗ , y − x̄i ≥ 0,
∀ y ∈ C.
– p. 23
Multivalued form
Theorem 24 Let C be a nonempty convex compact subset of IRn
IRn
and F : C → 2 a upper semicontinuous map with convex
compact values. Then S(F, C) is nonempty, that is,
there exist x̄ ∈ C and x̄∗ ∈ F (x̄)
such that hx̄∗ , y − x̄i ≥ 0,
∀ y ∈ C.
can be found, e.g., Borwein, J. and Lewis, A., Convex Analysis and Nonlinear Optimization, 2000
– p. 24
Multivalued form
Theorem 24 Let C be a nonempty convex compact subset of IRn
IRn
and F : C → 2 a upper semicontinuous map with convex
compact values. Then S(F, C) is nonempty, that is,
there exist x̄ ∈ C and x̄∗ ∈ F (x̄)
such that hx̄∗ , y − x̄i ≥ 0,
∀ y ∈ C.
F is upper semicontinuous at x if
∀ V neighb. of F (x), ∃ U neighb. of x such that F (U ) ⊂ V .
– p. 24
Multivalued form: proof
Proof. Since F is cusco, its range F (C) is compact and the map
T = (Id − F ) ◦ PC is also cusco since PC is continuous (Exercise).
Now one can observe that T (co(C − F (C))) = co(C − F (C)). Therefore
T has a fixed point (say x0 ) on C − F (C), that is
x0 ∈ T (x0 ) ⇔ x0


⇔



⇔

∈ (Id − F ) ◦ PC (x0 )
x̄ = PC (x0 )
x̄∗ = x̄ − x0 ∈ F (x̄)
hx0 − x̄, y − x̄i ≤ 0,
∀y ∈ C
x̄∗ = x̄ − x0 ∈ F (x̄)
and thus x̄ ∈ S(F, C).
– p. 25
Equivalence
Actually the above existence theorem 24 is equivalent to the Kakutani Fixed point theorem. Indeed, let
C be a nonempty convex compact subset of IRn and T : C → 2C be a cusco map. Observing that the
set-valued map F = Id − T is also cusco, Theorem 24 implies the solvability of the set-valued
Stampacchia variational inequality defined by F and C, that is there exists x̄ ∈ C such that
x̄ ∈ S(Id − T, C)
⇔
⇔
8
< ∃ u∗ ∈ (Id − T )(x̄) such that
: hu∗ , y − x̄i ≥ 0, ∀ y ∈ C
8
< ∃ x̄∗ ∈ T (x̄) such that
: hx̄ − x̄∗ , y − x̄i ≥ 0, ∀ y ∈ C
Taking y = x̄∗ , one has
hx̄ − x̄∗ , x̄∗ − x̄i ≥ 0
showing that x̄ is a fixed point of T .
– p. 26
II
Existence of solution
c - V.I. under monotonicity
– p. 27
Monotone case
Theorem 27 (1966) Let X be a reflexive Banach space, C be a
weakly compact convex nonempty subset of X. If f : C → X ∗ is a
monotone function, continuous on finite dimension subsets, then
S(f, C) is nonempty.
Hartmann, G. and Stampacchia, G., Acta Mathematica, 115 (1966), 271-310
– p. 28
Monotone case
Theorem 28 (1966) Let X be a reflexive Banach space, C be a
weakly compact convex nonempty subset of X. If f : C → X ∗ is a
monotone function, continuous on finite dimension subsets, then
S(f, C) is nonempty.
Theorem 30 Let X be a reflexive Banach space, C be a nonempty
X∗
convex and weakly compact subset of X. If F : C → 2 is a
monotone map with convex weakly compact values and continuous
on finite dimension subsets, then S(F, C) is nonempty.
can be found, e.g., in Shih, M.H. and Tan, K.K., JMAA, 134 (1988), 431-440
– p. 29
Monotonicities
T is monotone iff ∀ x, y ∈ X, ∀ x∗ ∈ T (x) and ∀ y ∗ ∈ T (x)
hy ∗ − x∗ , y − xi ≥ 0.
T is pseudomonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi ≥ 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
T is quasimonotone iff
∃ x∗ ∈ T (x) : hx∗ , y − xi > 0 ⇒ hy ∗ , y − xi ≥ 0, ∀ y ∗ ∈ T (y).
monotone
⇓
pseudomonotone
⇓
quasimonotone
– p. 30
Weakening monotonicty: why?
Let f : X → IR be differentiable
f is convex iff df is monotone
iff df (x)(y − x) > 0 ⇒ df (y)(y − x) ≥ 0
f is pseudoconvex iff df is pseudomonotone
iff df (x)(y − x) > 0 ⇒ df (y)(y − x) ≥ 0
f is quasiconvex iff df is quasimonotone
iff df (x)(y − x) > 0 ⇒ df (y)(y − x) ≥ 0
Let f : X → IR ∪ {+∞} be lsc
f is convex iff ∂f is monotone
iff ∀ x∗ ∈ ∂f (x), y ∗ ∈ ∂f (y), hy ∗ − x∗ , y − xi ≥ 0
f is pseudoconvex iff ∂f is pseudomonotone
iff ∃ x∗ ∈ ∂f (x) : hx∗ , y − xi ≥ 0
⇒ ∀ y ∗ ∈ ∂f (y), hy ∗ , y − xi ≥ 0
f is quasiconvex iff ∂f is quasimonotone
iff ∃ x∗ ∈ ∂f (x) : hx∗ , y − xi > 0
⇒ ∀ y ∗ ∈ ∂f (y), hy ∗ , y − xi ≥ 0
– p. 31
monotone
⇓
pseudomonotone
⇓
quasimonotone
Some previous attempts:
J.C. Yao (1994) with pseudomonotone map on reflexive space
N. Hadjisavvas and S. Schaible (1996) with quasimonotone +
existence of inner points
Dinh The Luc (2001) with singlevalued densely
pseudomonotone maps
– p. 32
Existence: simplified form
Theorem 32
C nonempty convex compact subset of X.
X∗
T : C → 2 quasimonotone
+ upper hemi-continuous
+ T (x) 6= ∅ convex w∗ -compact.
Then S(T, C) 6= ∅.
T is upper hemicontinuous on X if the restriction of T to any line segment
is usc with respect to the w∗ -topology
– p. 33
Knaster-Kuratowski-Mazurkiewicz (1929) :
C nonempty subset of X.
ϕ : C × C → IR is said to be a KKM-application if
∀ x1 , . . . , xn ∈ C, ∀ x ∈ co{x1 , . . . , xn }
∃ i ∈ 1, . . . , n such that ϕ(xi , x) ≥ 0.
Theorem 34
C nonempty closed convex subset of X.
ϕ : C × C → IR KKM-application
+ ϕ(x, .) usc quasiconcave, ∀ x
+ ∃ x̃ ∈ C with {y ∈ C : ϕ(x̃, y) ≥ 0} compact.
Then
∃ ȳ ∈ C such that ϕ(x, ȳ) ≥ 0, ∀ x ∈ C.
– p. 34
A very simple proof:
Let us define
ϕ : C × C → IR
(x, y) → inf x∗ ∈T (x) hx∗ , x − yi
The function ϕ is continuous concave relatively to y and
{y ∈ C : ϕ(x, y) ≥ 0} is compact, ∀ x ∈ C.
C AS 1 : ϕ is KKM
T h.KKM
=⇒
∃ ȳ ∈ C such that ϕ(x, ȳ) ≥ 0, ∀ x ∈ C.
C AS 2 : ∃ x1 , . . . , xn ∈ C, ∃ ȳ ∈ co{x1 , . . . , xn } such that
ϕ(xi , ȳ) < 0,
∀ i = 1, . . . , n
that is
∀ i = 1, . . . , n, ∃ x∗i ∈ T (xi ) : hx∗i , ȳ − xi i > 0
and, for some ρ > 0 and all z ∈ B(ȳ, ρ) ∩ C
∀ i, ∃ x∗i ∈ T (xi ) : hx∗i , z − xi i > 0.
– p. 35
By quasimonotonicity,
∀ z ∈ B(ȳ, ρ) ∩ C, ∀ z ∗ ∈ T (z), ∀ i = 1, . . . , n, hz ∗ , z − xi i ≥ 0.
In both cases,
∀ z ∈ B(ȳ, ρ) ∩ C, ∀ z ∗ ∈ T (z), hz ∗ , z − ȳi ≥ 0.
(1)
Let us prove that ȳ ∈ S(T, C):
Let x ∈ C (x 6= ȳ), t0 > 0 be such that zt = ȳ + t(x − ȳ) ∈ B(ȳ, ρ) ∩ C
for all t ∈]0, t0 [. From (1), one have,
hzt∗ , x − ȳi ≥ 0,
∀ t ∈]0, t0 [, ∀ zt∗ ∈ T (zt ).
(2)
Let us define the open subset
W = {y ∗ ∈ X ∗ : hy ∗ , x − ȳi < 0}.
– p. 36
If T (ȳ) ⊂ W , by upper hemicontinuity, T (zt ) ⊂ W , for any t sufficiently
closed to 0, which is a contradiction with (2).
Finally ∃ ȳ ∗ ∈ T (ȳ) : hȳ ∗ , x − ȳi ≥ 0, that is
inf
sup hy ∗ , x − ȳi ≥ 0
x∈C y ∗ ∈T (ȳ)
thus ȳ ∈ S(T, C) by a classical minimax theorem (Sion).
– p. 37
Upper sign-continuity
X∗
• T : X → 2 is said to be upper sign-continuous on C iff for any
x, y ∈ C, one have :
∀ t ∈ ]0, 1[,
∗
inf
hx
, y − xi ≥ 0
∗
x ∈T (xt )
=⇒ sup hx∗ , y − xi ≥ 0
x∗ ∈T (x)
where xt = (1 − t)x + ty.
– p. 38
Upper sign-continuity
X∗
• T : X → 2 is said to be upper sign-continuous on C iff for any
x, y ∈ C, one have :
∀ t ∈ ]0, 1[,
∗
inf
hx
, y − xi ≥ 0
∗
x ∈T (xt )
=⇒ sup hx∗ , y − xi ≥ 0
x∗ ∈T (x)
where xt = (1 − t)x + ty.
upper semi-continuous
⇓
upper hemicontinuous
⇓
upper sign-continuous
– p. 38
locally upper sign continuity
X∗
Definition 35 Let T : C → 2
be a set-valued map.
T est called locally upper sign-continuous on C if, for any x ∈ C
there exist a neigh. Vx of x and a upper sign-continuous set-valued
X∗
map Φx : Vx → 2 with nonempty convex w ∗ -compact values such
that Φx (y) ⊆ T (y) \ {0}, ∀ y ∈ Vx
– p. 39
Existence: complete form
C nonempty convex compact subset of X.
X∗
T : C → 2 quasimonotone
+ T (x) 6= ∅ convex w∗ -compact
+ upper hemicontinuous
Then
S(T, C) 6= ∅.
– p. 40
Existence: complete form
C nonempty convex compact subset of X.
X∗
T : C → 2 quasimonotone
+ T (x) 6= ∅ convex w∗ -compact
+ upper sign-continuous
Then
S(T, C) 6= ∅.
– p. 40
Existence: complete form
C nonempty convex compact subset of X.
X∗
T : C → 2 quasimonotone
+ T (x) 6= ∅ convex w∗ -compact
+ locally upper sign continuous on C
Then
S(T, C) 6= ∅.
– p. 40
Existence: complete form
C nonempty convex subset of X.
X∗
T : C → 2 quasimonotone
+ locally upper sign continuous on C
+ coercivity condition:
∃ ρ > 0, ∀ x ∈ C \ B(0, ρ), ∃ y ∈ C with kyk < kxk
such that ∀ x∗ ∈ T (x), hx∗ , x − yi ≥ 0
and there exists ρ′ > ρ such that C ∩ B(0, ρ′ ) is weakly
compact (6= ∅).
Then S(T, C) 6= ∅.
– p. 40
II - d - Other assumptions
Hartman and Stampacchia : sum of monotone operator and
continuous function
J.C. Yao, N.D. Yen, B.T. Kien and many others: continuous maps
with degre technics
– p. 41
III
Other classes of V.I.
– p. 42
a - A general framework
X, Y l.c.t.v.s., C subset of X
T : C → Y set-valued map
Coupling function : Φ : Y × C → IR
• Variational inequality:
Find x̄ ∈ C and x̄∗ ∈ T (x̄) such that
Φ(x̄∗ , y) − Φ(x̄∗ , x̄) ≥ 0,
∀ y ∈ C.
– p. 43
a - A general framework
X, Y l.c.t.v.s., C subset of X
T : C → Y set-valued map
Coupling function : Φ : Y × C → IR
• Variational inequality:
Find x̄ ∈ C and x̄∗ ∈ T (x̄) such that
Φ(x̄∗ , y) − Φ(x̄∗ , x̄) ≥ 0,
∀ y ∈ C.
Example : Mixed var. ineq.
Find x̄ ∈ C and x̄∗ ∈ T (x̄) such that
hx̄∗ , y − x̄i + h(y) − h(x̄) ≥ 0,
Also called hemi-variational inequality
∀ y ∈ C.
– p. 43
a - A general framework
X, Y l.c.t.v.s., C subset of X
T : C → Y set-valued map
Coupling function : Φ : Y × C → IR
• Variational inequality:
Find x̄ ∈ C and x̄∗ ∈ T (x̄) such that
Φ(x̄∗ , y) − Φ(x̄∗ , x̄) ≥ 0,
∀ y ∈ C.
Example : General var. ineq.
Find x0 ∈ X such that
hf (x0 ), g(x) − g(x0 )i ≥ 0,
∀ x ∈ X avec g(x) ∈ C.
– p. 43
a - A general framework
X, Y l.c.t.v.s., C subset of X
T : C → Y set-valued map
Coupling function : Φ : Y × C → IR
• Variational inequality:
Find x̄ ∈ C and x̄∗ ∈ T (x̄) such that
Φ(x̄∗ , y) − Φ(x̄∗ , x̄) ≥ 0,
∀ y ∈ C.
Example : Equilibrium problem
Find x0 ∈ C such that
f (x0 , x) + h(x) − h(x0 ) ≥ 0,
∀ x ∈ C.
– p. 43
a - A general framework
X, Y l.c.t.v.s., C subset of X
T : C → Y set-valued map
Coupling function : Φ : Y × C → IR
• Variational inequality:
Find x̄ ∈ C and x̄∗ ∈ T (x̄) such that
Φ(x̄∗ , y) − Φ(x̄∗ , x̄) ≥ 0,
∀ y ∈ C.
• Quasimonotonicity –> Φ-quasimonotonicity
• Upper-sign continuity –> Φ-upper sign-continuity
– p. 43
a - A general framework
X, Y l.c.t.v.s., C subset of X
T : C → Y set-valued map
Coupling function : Φ : Y × C → IR
• Variational inequality:
Find x̄ ∈ C and x̄∗ ∈ T (x̄) such that
Φ(x̄∗ , y) − Φ(x̄∗ , x̄) ≥ 0,
∀ y ∈ C.
• Quasimonotonicity –> Φ-quasimonotonicity
• Upper-sign continuity –> Φ-upper sign-continuity
Hyp.: Φ(x∗ , ·) convex continue and Φ(·, x) concave.
See Aussel-Luc, Bull. Austral. Math. Soc. (2005)
– p. 43
b - Minty and local Minty V. I.
Let C ⊆ X and T : C →
∗
X
2
be a map.
We denote by
• Sw (T, C) set of weak solutions of Stampacchia V.I.
Find x̄ ∈ C such that
∀ y ∈ C, ∃ x̄∗ ∈ T (x̄) : hx̄∗ , y − x̄i ≥ 0
– p. 44
b - Minty and local Minty V. I.
Let C ⊆ X and T : C →
∗
X
2
be a map.
We denote by
• Sw (T, C) set of weak solutions of Stampacchia V.I.
Find x̄ ∈ C such that
∀ y ∈ C, ∃ x̄∗ ∈ T (x̄) : hx̄∗ , y − x̄i ≥ 0
• M (T, C) set of solutions of the Minty V.I.
Find x̄ ∈ C such that
∀ y ∈ C and ∀ y ∗ ∈ T (y) : hy ∗ , y − x̄i ≥ 0.
– p. 44
b - Minty and local Minty V. I.
Let C ⊆ X and T : C →
∗
X
2
be a map.
We denote by
• Sw (T, C) set of weak solutions of Stampacchia V.I.
Find x̄ ∈ C such that
∀ y ∈ C, ∃ x̄∗ ∈ T (x̄) : hx̄∗ , y − x̄i ≥ 0
• M (T, C) set of solutions of the Minty V.I.
Find x̄ ∈ C such that
∀ y ∈ C and ∀ y ∗ ∈ T (y) : hy ∗ , y − x̄i ≥ 0.
• LM (T, C) set of local solutions of Minty V.I., i.e.
x̄ ∈ LM (T, C) if it exists a neighb. V of x̄ such that x̄ ∈ M (T, V ∩ C).
– p. 44
Links ?
X∗
Proposition 37 C nonempty convex subset of X and T : C → 2
a map.
i) If T is pseudomonotone, then LM (T, C) = M (T, C).
ii) If T is locally upper sign-continuous at x then
LM (T, C) ⊆ Sw (T, C).
iii) If, additionally to ii), the maps Sx are convex valued,
then LM (T, C) ⊆ Sw (T, C) = S(T, C).
– p. 45
Proper Quasimonotonicity:
X∗
Let C ⊆ X and T : C → 2
.
• T is properly quasimonotone if, for any x1 , . . . , xn ∈ dom T ,
and any x ∈ co {x1 , x2 , . . . , xn }, there exists i ∈ {1, 2, . . . , n} such
that
∀x∗ ∈ T (xi ) : hx∗ , xi − xi ≥ 0.
monotone
⇓
pseudomonotone
⇓
properly quasimonotone
⇓
quasimonotone
– p. 46
Existence of Minty V.I.
Proposition 39
X∗
Let T : X → 2 be a set-valued map. Then the following are
equivalent:
i) T is properly quasimonotone
ii) ∀ C nonempty convex compact, M (T, C) 6= ∅.
R. John, Proc. of the GCM7, Ed. Kluwer (1999)
– p. 47
Existence of Minty V.I.
Proposition 41
X∗
Let T : X → 2 be a set-valued map. Then the following are
equivalent:
i) T is properly quasimonotone
ii) ∀ C nonempty convex compact, M (T, C) 6= ∅.
The proof is based on the Ky Fan theorem:
Let C be a closed convex set of X and D ⊂ C . If {Cx : x ∈ D} is a KKM family
of closed convex subsets of C and one of them is compact, then ∩{Cx : x ∈ D} is
nonempty.
A family {Cx : x ∈ D} is a KKM if ∀ x1 , . . . , xn ∈ D, conv{x1 , . . . , xn } ⊂ ∪x∈D Cx
– p. 47
Existence of Minty V.I.
Proof of i) ⇒ ii).
Let us define the set-valued map G : C → C by
G(y) = {x ∈ C : ∀ y ∗ ∈ T (y), hy ∗ , y − xi ≥ 0}.
By hypothesis, G(y) is nonempty, convex and compact. Now reformulating the proper
quasimonotonicity of T , we have that
x1 , . . . , xn ∈ C and ∀ x ∈ conv{x1 , . . . , xn },
x ∈ ∪i=1,n G(xi )
thus showing that the family {G(x) : x ∈ C} is a KKM family. By applying Ky Fan’s theorem, we
have
∩y∈C G(y) 6= ∅.
But any element of this intersection (say x̄) satisfies
∀ y ∈ C, ∀ y ∗ ∈ T (y),
hy ∗ , y − x̄i ≥ 0
and is therefore a solution of the Minty variational inequality.
– p. 48
Existence of Minty V.I.
Proof of ii) ⇒ i).
Let x1 , . . . , xn be elements of X. Then conv{x1 , . . . , xn } is a nonempty convex compact subset of
X and consequently, M (T, conv{x1 , . . . , xn }) is non empty. This means that
∃ x̄ ∈ conv{x1 , . . . , xn } such that ∀ x ∈ conv{x1 , . . . , xn }, ∀ x∗ ∈ T (x),
hx∗ , x − x̄i ≥ 0
and thus
∃ x̄ ∈ conv{x1 , . . . , xn } such that ∀ i = 1, . . . , n, x̄ ∈ G(xi ).
As a conclusion
∀ x1 , . . . , xn ∈ X, ∃ x̄ ∈ conv{x1 , . . . , xn } such that x̄ ∈ ∩i=1,n G(xi )
or in other word, the family of closed convex {G(x) : x ∈ X} satisfies the Finite Intersection
Property. It can be shown by induction on n (a good exercise) that the family {G(x) : x ∈ X} is a
KKM family.
– p. 49
Existence of Minty V.I.
Proposition 43
X∗
Let T : X → 2 be a set-valued map. Then the following are
equivalent:
i) T is quasimonotone
ii) ∀ x, y ∈ X, M (T, [x, y]) 6= ∅.
Indeed
T is quasimonotone
⇔
its restriction to any line segment is quasimonotone
⇔
its restriction to any line segment is properly
quasimonotone
– p. 50
Existence of local solutions of Minty V.I.
Proposition 45 Let C be a nonempty convex subset of X and
X∗
T : C → 2 be quasimonotone. Then one of the following
assertions holds:
i) T is properly quasimonotone
ii) LM (T, C) 6= ∅.
If, C is weakly compact, then LM (T, C) 6= ∅ in both cases.
– p. 51
Existence of local solutions of Minty V.I.
Proof. Suppose that i) does not hold. Thus there exists x1 , . . . , xn ∈ C,
x∗i ∈ T (xi ) and x ∈ conv{x1 , . . . , xn } such that
hx∗i , x − xi i > 0,
∀ i.
By continuity of the x∗i ’s, there exists a neighb. U of x for which
hx∗i , x − xi i > 0,
∀y ∈ C ∩ U
and thus, according to the quasimonotonicity of T , for any y ∗ ∈ T (y),
hy ∗ , y − xi i ≥ 0.
Now since x ∈ conv{x1 , . . . , xn }, for any y ∈ C ∩ U and any y ∗ ∈ T (y),
one has hy ∗ , y − xi ≥ 0 and x ∈ LM (T, C).
– p. 52
c - Quasivariational inequalities
X Banach space
X ∗ its topological dual (w∗ -top.)
h·, ·i the duality product
Stampacchia Quasivariational inequality:
X∗
Let C be a nonempty subset of X and T : X → 2 and ϕ : C → 2C be
set-valued maps.
Find x̄ ∈ C such that x̄ ∈ ϕ(x̄) and there exists x̄∗ ∈ T (x̄) for which
hx̄∗ , y − x̄i ≥ 0,
∀ y ∈ ϕ(x̄).
– p. 53
Theorem 47 Let C a convex compact and nonempty subset of X
Assume that
i) ϕ : C → 2C is usc with convex compact nonempty values
ii) f : C × C → IR is a function such that



 ∀ y ∈ C, f (·, y)is lsc
∀ x ∈ C, f (x, ·)is quasiconcave



∀ x ∈ C, f (x, x) ≤ 0
iii) the subset {x ∈ C : supy∈ϕx f (x, y) > 0} is open
Then there exists x̄ ∈ C such that x̄ ∈ ϕ(x̄) and
supy∈ϕx̄ f (x̄, y) ≤ 0.
can be found in Lassonde, Master course, UAG
– p. 54
References
1- D. A. & N. Hadjisavvas, On Quasimonotone Variational Inequalities,
JOTA 121 (2004), 445-450.
2- D. A. & D.T. Luc, Existence Conditions in General Quasimonotone
Variational Inequalities, Bull. Austral. Math. Soc. 71 (2005), 285-303.
3- J.-C. Yao, Multivalued variational inequalities with K-pseudomonotone
operators, JOTA (1994).
4- J.-P. Crouzeix, Pseudomonotone Variational Inequalities Problems:
Existence of Solutions, Math. Programming (1997).
5- D. T. Luc, Existence results for densely pseudomonotone variational
inequality, J. Math. Anal. Appl. (2001).
6- R. John, A note on Minty Variational Inequality and Generalized
Monotonicity, Proc. GCM7 (2001).
– p. 55
Brouwer fixed point theorem
Theorem 48
Let C be a nonempty compact convex subset of IRn and f : C → IRn
a continuous function. Then f admits on C at least a fixed point.
– p. 56
Kakutani fixed point theorem
Theorem 49
Let C be a nonempty compact convex subset of IRn and
IRn
F : C → 2 a cusco map. Then F admits on C at least a
fixed point.
– p. 57