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Existence and uniqueness results for variational...
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Existence and uniqueness results forvariational inequalities
Lecture 3
V.I. in quasiconvex optimization
Didier Aussel
Univ. de Perpignan
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Outline of lecture 3
I- About uniqueness
II- Introduction to QV optimization
III- Normal approacha- First definitionsb- Adjusted sublevel sets and normal operator
IV- Quasiconvex optimizationa- Optimality conditionsb- Convex constraint casec- Nonconvex constraint case
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I - About uniqueness
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Uniqueness
Global result:
If T is strictly monotone then, for anyK ⊂ X, there exists at most
one solution forV I(T,K)
Let us recall thatT is strictly monotone iff for anyx, y ∈ X x 6= y and anyx∗ ∈ T (x), y∗ ∈ T (y),
one has
〈y∗ − x∗, y − x〉 > 0.
Let x andy be two different solutions. Then there existsx∗ ∈ T (x) andy∗ ∈ T (y) such that
〈y∗, y − x〉 ≤ 0 and〈x∗, y − x〉 ≥ 0.
But this is a contradiction with〈y∗, y − x〉 > 〈x∗, y − x〉.
– p. 4
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Uniqueness
Theorem 1 If f : IRn → IRn is locally Lipschitz and the Clarke
generalized Jacobian off at every point ofC consists of positive
definite matrices only. Thenf is strictly monotone onC and thus
S(f, C) contains at most one solution.Recall that the Clarke generalized Jacobian off atx is
{ limi→∞
∇f(xi) : xi → x, ∇f(xi) exists}.
Local result:
x is said to be a local solution if there exists a neigh. ofx which do
not contains any other solution.
For local uniqueness (generalization of avove result), seeLuc,JMAA 268 (2002)
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II
Introduction to QV optimization
– p. 6
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Quasiconvexity
A function f : X → IR ∪ {+∞} is said to bequasiconvexonK
if, for all x, y ∈ K and allt ∈ [0, 1],
f(tx + (1 − t)y) ≤ max{f(x), f(y)}.
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Quasiconvexity
A function f : X → IR ∪ {+∞} is said to bequasiconvexonK
if, for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f(x) ≤ λ} is convex.
– p. 7
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Quasiconvexity
A function f : X → IR ∪ {+∞} is said to bequasiconvexonK
if, for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f(x) ≤ λ} is convex.
A functionf : X → IR ∪ {+∞} is said to besemistrictly
quasiconvexonK if, f is quasiconvex and for anyx, y ∈ K,
f(x) < f(y) ⇒ f(z) < f(y), ∀ z ∈ [x, y[.
– p. 7
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f differentiable
f is quasiconvex iffdf is quasimonotone
iff df(x)(y − x) > 0 ⇒ df(y)(y − x) ≥ 0
f is quasiconvex iff∂f is quasimonotone
iff ∃x∗ ∈ ∂f(x) : 〈x∗, y − x〉 > 0
⇒ ∀ y∗ ∈ ∂f(y), 〈y∗, y − x〉 ≥ 0
– p. 8
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Why not a subdifferential for quasiconvex programming?
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Why not a subdifferential for quasiconvex programming?
No (upper) semicontinuity of∂f if f is notsupposed to be Lipschitz
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Why not a subdifferential for quasiconvex programming?
No (upper) semicontinuity of∂f if f is notsupposed to be Lipschitz
No sufficient optimality condition
x ∈ Sstr(∂f, C) =⇒ x ∈ arg minC
fHHH ���
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III
Normal approach of quasiconvexanalysis
– p. 10
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III
Normal approach
a- First definitions
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A first approach
Sublevel set:
Sλ = {x ∈ X : f(x) ≤ λ}
S>λ = {x ∈ X : f(x) < λ}
Normal operator:
DefineNf (x) : X → 2X∗
by
Nf (x) = N(Sf(x), x)
= {x∗ ∈ X∗ : 〈x∗, y − x〉 ≤ 0, ∀ y ∈ Sf(x)}.
With the corresponding definition forN>f (x)
– p. 12
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But ...
Nf (x) = N(Sf(x), x) has no upper-semicontinuity properties
N>f (x) = N(S>
f(x), x) has no quasimonotonicity properties
– p. 13
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But ...
Nf (x) = N(Sf(x), x) has no upper-semicontinuity properties
N>f (x) = N(S>
f(x), x) has no quasimonotonicity properties
Example
Definef : R2 → R by
f (a, b) =
8
<
:
|a| + |b| , if |a| + |b| ≤ 1
1, if |a| + |b| > 1.
Thenf is quasiconvex. Considerx = (10, 0), x∗ = (1, 2), y = (0, 10) andy∗ = (2, 1).
We see thatx∗ ∈ N<(x) andy∗ ∈ N< (y) (since|a| + |b| < 1 implies(1, 2) · (a − 10, b) ≤ 0 and
(2, 1) · (a, b − 10) ≤ 0) while 〈x∗, y − x〉 > 0 and〈y∗, y − x〉 < 0. HenceN< is not
quasimonotone.
– p. 13
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But ...
Nf (x) = N(Sf(x), x) has no upper-semicontinuity properties
N>f (x) = N(S>
f(x), x) has no quasimonotonicity properties
These two operators are essentially adapted to the class ofsemi-strictly quasiconvex functions. Indeed in this case,for eachx ∈ domf \ arg min f , the setsSf(x) andS<
f(x) have the same
closure andNf (x) = N<f (x).
– p. 13
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III
Normal approach
b- Adjusted sublevel setsand
normal operator
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Definition
Adjusted sublevel set
For anyx ∈ domf , we define
Saf (x) = Sf(x) ∩ B(S<
f(x), ρx)
whereρx = dist(x, S<f(x)), if S<
f(x) 6= ∅
andSaf (x) = Sf(x) if S<
f(x) = ∅.
– p. 15
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Definition
Adjusted sublevel set
For anyx ∈ domf , we define
Saf (x) = Sf(x) ∩ B(S<
f(x), ρx)
whereρx = dist(x, S<f(x)), if S<
f(x) 6= ∅
andSaf (x) = Sf(x) if S<
f(x) = ∅.
Saf (x) coincides withSf(x) if cl(S>
f(x)) = Sf(x)
e.g.f is semistrictly quasiconvex
– p. 15
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Definition
Adjusted sublevel set
For anyx ∈ domf , we define
Saf (x) = Sf(x) ∩ B(S<
f(x), ρx)
whereρx = dist(x, S<f(x)), if S<
f(x) 6= ∅
andSaf (x) = Sf(x) if S<
f(x) = ∅.
Saf (x) coincides withSf(x) if cl(S>
f(x)) = Sf(x)
Proposition 2 Letf : X → IR ∪ {+∞} be any function, with
domaindomf . Then
f is quasiconvex⇐⇒ Saf (x) is convex,∀x ∈ domf.
– p. 15
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Proof
Let us suppose thatSaf
(u) is convex for everyu ∈ domf . We will show that for anyx ∈ domf ,
Sf(x) is convex.
If x ∈ arg min f thenSf(x) = Saf
(x) is convex by assumption.
Assume now thatx /∈ arg min f and takey, z ∈ Sf(x).
If both y andz belong toB“
S<f(x)
, ρx
”
, theny, z ∈ Saf
(x) thus[y, z] ⊆ Saf
(x) ⊆ Sf(x).
If both y andz do not belong toB“
S<f(x)
, ρx
”
, then
f(x) = f(y) = f(z), S<f(z)
= S<f(y)
= S<f(x)
andρy , ρz are positive. If, say,ρy ≥ ρz theny, z ∈ B“
S<f(y)
, ρy
”
thus
y, z ∈ Saf (y) and[y, z] ⊆ Sa
f (y) ⊆ Sf(y) = Sf(x).
– p. 16
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Proof
Finally, suppose that only one ofy, z, sayz, belongs toB(S<f(x)
, ρx) while y /∈ B(S<f(x)
, ρx). Then
f(x) = f(y), S<f(y)
= S<f(x)
andρy > ρx
so we havez ∈ B“
S<f(x)
, ρx
”
⊆ B“
S<f(y)
, ρy
”
and we deduce as before that
[y, z] ⊆ Saf
(y) ⊆ Sf(y) = Sf(x).
The other implication is straightforward.
– p. 17
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Adjusted normal operator
Adjusted sublevel set:
For anyx ∈ domf , we define
Saf (x) = Sf(x) ∩ B(S<
f(x), ρx)
whereρx = dist(x, S<f(x)), if S<
f(x) 6= ∅.
Ajusted normal operator:
Naf (x) = {x∗ ∈ X∗ : 〈x∗, y − x〉 ≤ 0, ∀ y ∈ Sa
f (x)}
– p. 18
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Example
x
��
��
– p. 19
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Example
x
��
��
B(S<f(x), ρx)
Saf (x) = Sf(x) ∩ B(S<
f(x), ρx)
– p. 19
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Example
Saf (x) = Sf(x) ∩ B(S<
f(x), ρx)
Naf (x) = {x∗ ∈ X∗ : 〈x∗, y − x〉 ≤ 0, ∀ y ∈ Sa
f (x)}
– p. 19
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Subdifferential vs normal operator
One can have
Naf (x) ⊆/ cone(∂f(x)) or cone(∂f(x)) ⊆/ Na
f (x)
Proposition 4f is quasiconvex andx ∈ domf
If there existsδ > 0 such that0 6∈ ∂Lf(B(x, δ)) then
[
cone(∂Lf(x)) ∪ ∂∞f(x)]
⊂ Naf (x).
– p. 20
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Basic properties ofNaf
Nonemptyness:
Proposition 6 Letf : X → IR ∪ {+∞} be lsc. Assume that rad.
continuous ondomf or domf is convex andintSλ 6= ∅,
∀λ > infX f . Then
f is quasiconvex⇔ Naf (x) \ {0} 6= ∅, ∀x ∈ domf \ arg min f.
Quasimonotonicity:The normal operatorNa
f is always quasimonotone
– p. 21
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Upper sign-continuity
• T : X → 2X∗
is said to beupper sign-continuousonK iff for any
x, y ∈ K, one have :
∀ t ∈ ]0, 1[, infx∗∈T (xt)
〈x∗, y − x〉 ≥ 0
=⇒ supx∗∈T (x)
〈x∗, y − x〉 ≥ 0
wherext = (1 − t)x + ty.
– p. 22
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Upper sign-continuity
• T : X → 2X∗
is said to beupper sign-continuousonK iff for any
x, y ∈ K, one have :
∀ t ∈ ]0, 1[, infx∗∈T (xt)
〈x∗, y − x〉 ≥ 0
=⇒ supx∗∈T (x)
〈x∗, y − x〉 ≥ 0
wherext = (1 − t)x + ty.
upper semi-continuous
⇓
upper hemicontinuous
⇓
upper sign-continuous – p. 22
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locally upper sign continuity
Definition 8 LetT : K → 2X∗
be a set-valued map.
T est calledlocally upper sign-continuousonK if, for anyx ∈ K
there exist a neigh.Vx of x and a upper sign-continuous set-valued
mapΦx(·) : Vx → 2X∗
with nonempty convexw∗-compact values
such thatΦx(y) ⊆ T (y) \ {0}, ∀ y ∈ Vx
– p. 23
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locally upper sign continuity
Definition 10 LetT : K → 2X∗
be a set-valued map.
T est calledlocally upper sign-continuousonK if, for anyx ∈ K
there exist a neigh.Vx of x and a upper sign-continuous set-valued
mapΦx(·) : Vx → 2X∗
with nonempty convexw∗-compact values
such thatΦx(y) ⊆ T (y) \ {0}, ∀ y ∈ Vx
Continuity of normal operator
Proposition 12Letf be lsc quasiconvex function such thatint(Sλ) 6= ∅, ∀λ > inf f .
ThenNf is locally upper sign-continuous ondomf \ arg min f .
– p. 23
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Proposition 14If f is quasiconvex such thatint(Sλ) 6= ∅, ∀λ > inf f
andf is lsc atx ∈ domf \ arg min f ,
ThenNaf is norm-to-w∗ cone-usc atx.
A multivalued map with conical valuedT : X → 2X∗
is said to becone-uscatx ∈ domT if there exists a neighbourhoodU of x and abaseC (u) of T (u), u ∈ U , such thatu → C (u) is usc atx.
– p. 24
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Integration of Naf
Let f : X → IR ∪ {+∞} quasiconvex
Question:Is it possible to characterize the functions
g : X → IR ∪ {+∞} quasiconvex such thatNaf = Na
g ?
– p. 25
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Integration of Naf
Let f : X → IR ∪ {+∞} quasiconvex
Question:Is it possible to characterize the functions
g : X → IR ∪ {+∞} quasiconvex such thatNaf = Na
g ?
A first answer:
Let C = {g : X → IR ∪ {+∞} cont. semistrictly quasiconvex
such that argminf is included in a closed hyperplane}
Then, for anyf, g ∈ C,
Naf = Na
g ⇔ g is Naf \ {0}-pseudoconvex
⇔ ∃x∗ ∈ Naf (x) \ {0} : 〈x∗, y − x〉 ≥ 0 ⇒ g(x) ≤ g(y).
– p. 25
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Integration of Naf
Let f : X → IR ∪ {+∞} quasiconvex
Question:Is it possible to characterize the functions
g : X → IR ∪ {+∞} quasiconvex such thatNaf = Na
g ?
General case:open question
– p. 25
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IV
Quasiconvex programming
a- Optimality conditions
– p. 26
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Quasiconvex programming
Let f : X → IR ∪ {+∞} andK ⊆ domf be a convex subset.
(P ) find x ∈ K : f(x) = infx∈K
f(x)
– p. 27
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Quasiconvex programming
Let f : X → IR ∪ {+∞} andK ⊆ domf be a convex subset.
(P ) find x ∈ K : f(x) = infx∈K
f(x)
Perfect case:f convex
f : X → IR ∪ {+∞} a proper convex function
K a nonempty convex subset ofX, x ∈ K + C.Q.
Then
f(x) = infx∈K
f(x) ⇐⇒ x ∈ Sstr(∂f,K)
– p. 27
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Quasiconvex programming
Let f : X → IR ∪ {+∞} andK ⊆ domf be a convex subset.
(P ) find x ∈ K : f(x) = infx∈K
f(x)
Perfect case:f convex
f : X → IR ∪ {+∞} a proper convex function
K a nonempty convex subset ofX, x ∈ K + C.Q.
Then
f(x) = infx∈K
f(x) ⇐⇒ x ∈ Sstr(∂f,K)
What aboutf quasiconvex case?
x ∈ Sstr(∂f(x),K) =⇒ x ∈ arg minK
fHHH ���
– p. 27
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Sufficient optimality condition
Proposition 16f : X → IR ∪ {+∞} quasiconvex, radially cont. ondomf
C ⊆ X such thatconv(C) ⊂ domf .
Suppose thatC ⊂ int(domf).
Thenx ∈ S(Naf \ {0}, C) =⇒ ∀x ∈ C, f(x) ≤ f(x).
wherex ∈ S(Naf \ {0},K) means that there existsx∗ ∈ Na
f (x) \ {0} such
that
〈x∗, c − x〉 ≥ 0, ∀ c ∈ C.
– p. 28
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Lemma 18 Letf : X → IR ∪ {+∞} be a quasiconvex function,
radially continuous ondomf . Thenf is Naf \ {0}-pseudoconvex on
int(domf), that is,
∃x∗ ∈ Naf (x) \ {0} : 〈x∗, y − x〉 ≥ 0 ⇒ f(y) ≥ f(x).
Proof.Let x, y ∈ int(domf). According to the quasiconvexity off , Na
f(x) \ {0} is nonempty.
Let us suppose that〈x∗, y − x〉 ≥ 0 with x∗ ∈ Naf(x) \ {0}. Let d ∈ X be such that
〈x∗, yn − x〉 > 0 for anyn, whereyn = y + 1n
d (∈ domf for n large enough).
This implies thatyn 6∈ S<f
(x) sincex∗ ∈ Naf(x) ⊂ N<
f(x).
It follows by the radial continuity off thatf(y) ≥ f(x).
– p. 29
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Necessary and Sufficient conditions
Proposition 20 LetC be a closed convex subset ofX, x ∈ C and
f : X → IR be continuous semistrictly quasiconvex such that
int(Saf (x)) 6= ∅ andf(x) > infX f .
Then the following assertions are equivalent:
i) f(x) = minC f
ii) x ∈ Sstr(Naf \ {0}, C)
iii) 0 ∈ Naf (x) \ {0} + NK(C, x).
– p. 30
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Proposition 22 LetC be a closed convex subset of an Asplund
spaceX andf : X → IR be a continuous quasiconvex function.
Suppose that eitherf is sequentially normally sub-compact orC is
sequentially normally compact.
If x ∈ C is such that
0 6∈ ∂Lf(x) and 0 6∈ ∂∞f(x) \ {0} + NK(C, x)
then the following assertions are equivalent:
i) f(x) = minC f
ii) x ∈ ∂Lf(x) \ {0} + NK(C, x)
iii) 0 ∈ Naf (x) \ {0} + NK(C, x)
– p. 31
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III
Quasiconvex optimization
b- Convex constraint case
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III
Quasiconvex optimization
b- Convex constraint case
(P ) Find x ∈ C such thatf(x) = infC
f.
with C convex set.
– p. 32
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Existence results with convex constraint set
Theorem 23 (Convex case)
f : X → IR ∪ {+∞} quasiconvex
+ continuous ondom(f)
+ for any λ > infX f , int(Sλ) 6= ∅.
+ C ⊆ int(domf) convex such thatC ∩ B(0, n)
is weakly compact for somen ∈ IN.
+ coercivity condition∃ ρ > 0, ∀x ∈ C \ B(0, ρ), ∃ y ∈ C with ‖y‖ < ‖x‖
such that∀x∗ ∈ Naf (x) \ {0}, 〈x∗, x − y〉 > 0
Then there existsx ∈ C such that∀x ∈ C, f(x) ≥ f(x).
– p. 33
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Existence for Stampacchia V.I.
Theorem 25C nonempty convex subset ofX.
T : C → 2X∗
quasimonotone
+ locally upper sign continuous onC
+ coercivity condition:
∃ ρ > 0, ∀x ∈ C \ B(0, ρ), ∃ y ∈ C with ‖y‖ < ‖x‖
such that∀x∗ ∈ T (x), 〈x∗, x − y〉 ≥ 0
and there existsρ′ > ρ such thatC ∩ B(0, ρ′) is weakly
compact (6= ∅).
Then S(T,C) 6= ∅.
– p. 34
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ProofIf arg min f ∩ K 6= ∅, we have nothing to prove.
Suppose thatarg min f ∩ K = ∅. ThenNa is quasimonotone and norm-to-w∗ cone-usc onK. Thus,
all assumptions of Theorem 25 hold for the operatorNa \ {0}, soSstr (Na \ {0} , K) 6= ∅. Finally,
using the sufficient condition, we infer thatf has a global minimum onK.
– p. 35
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ProofIf arg min f ∩ K 6= ∅, we have nothing to prove.
Suppose thatarg min f ∩ K = ∅. ThenNa is quasimonotone and norm-to-w∗ cone-usc onK. Thus,
all assumptions of Theorem 25 hold for the operatorNa \ {0}, soSstr (Na \ {0} , K) 6= ∅. Finally,
using the sufficient condition, we infer thatf has a global minimum onK.
Corollary 27 Assumptions onf andK as in Theorem 23. Assume that
there existsn ∈ IN such that for allx ∈ K, ‖x‖ > n, there existsy ∈ K,
‖y‖ < ‖x‖ such thatf (y) < f (x). Then there existsx0 ∈ K such that
∀x ∈ K, f(x) ≥ f(x0).
Proof. If f (y) < f(x) then for everyx∗ ∈ Na(x) ⊆ N< (x), 〈x∗, y − x〉 ≤ 0. Hence, coercivity
condition withT = Na holds. The corollary follows from Theorem 23.
– p. 35
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III
Quasiconvex optimization
c- Nonconvex constraint case(an example)
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Disjunctive programming
Let us consider the optimization problem:
min f(x)
s.t.
hi(x) ≤ 0 i = 1, . . . , l
minj∈J gj(x) ≤ 0
wheref, hi, gj : X → IR are quasiconvex
J is a (possibly infinite) index set.
– p. 37
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Disjunctive programming
Let us consider the optimization problem:
min f(x)
s.t.
hi(x) ≤ 0 i = 1, . . . , l
minj∈J gj(x) ≤ 0
wheref, hi, gj : X → IR are quasiconvex
J is a (possibly infinite) index set.
Example: g : X → IR is continuous concave and
min f(x)
s.t.
hi(x) ≤ 0 i = 1, . . . , l,
g(x) ≤ 0– p. 37
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Disjunctive programming
Let us consider the optimization problem:
min f(x)
s.t.
hi(x) ≤ 0 i = 1, . . . , l
x ∈ ∪j∈JCj
wheref, hi : X → IR are quasiconvex
Cj are convex subsets ofX
J is a (possibly infinite) index set.
– p. 37
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A little bit of history
Balas 1974: first paper about disjunctive prog.
Pure and mixed 0-1 linear programming
Min Z = d.x +∑m
k=1 ck
s.t.∨
j∈Jk
Yjk
Ajkx ≥ bjk
ck = γjk
, k ∈ K
0 ≤ x ≤ U, Yjk ∈ {0, 1}
– p. 38
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A little bit of history
Balas 1974: first paper about disjunctive prog.
Pure and mixed 0-1 linear programming
Min Z = d.x +∑m
k=1 ck
s.t.∨
j∈Jk
Yjk
Ajkx ≥ bjk
ck = γjk
, k ∈ K
0 ≤ x ≤ U, Yjk ∈ {0, 1}
puis Gugat, Grossmann, Borwein, Cornuejols-Lemaréchal,...
– p. 38
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Duality for disjunctive prog.
For linear Disjunctive program (Balas):
Primal α = inf c.x
s.t.∨
j∈J
Aj.x ≥ bj
x ≥ 0
– p. 39
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Duality for disjunctive prog.
For linear Disjunctive program (Balas):
Primal α = inf c.x
s.t.∨
j∈J
Aj.x ≥ bj
x ≥ 0
Dual β = sup w
s.t.∧
j∈J
w − ujbj ≤ 0
uj.Aj ≤ c
uj ≥ 0
– p. 39
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Duality theorem for linear disjunctive prog.
SetPj = {x : Aj .x ≥ bj , x ≥ 0}, Uj = {uj : uj .Aj ≤ c, uj ≥ 0}.
DenoteJ∗ = {j ∈ J : Pj 6= ∅} andJ∗∗ = {j ∈ J : Uj 6= ∅}
Theorem 29 (Balas 77)
If (P) and (D) satisfy the followingregularity assumption
J∗ 6= ∅, J \ J∗∗ 6= ∅ =⇒ J∗ \ J∗∗ 6= ∅,
Then
- either both problems are feasible, each has an optimal solution and
α = β
- or one is infeasible, the other one either is infeasible or has no finite
optimum.
– p. 40
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Duality theorem for linear disjunctive prog.
SetPj = {x : Aj .x ≥ bj , x ≥ 0}, Uj = {uj : uj .Aj ≤ c, uj ≥ 0}.
DenoteJ∗ = {j ∈ J : Pj 6= ∅} andJ∗∗ = {j ∈ J : Uj 6= ∅}
Theorem 30 (Balas 77)
If (P) and (D) satisfy the followingregularity assumption
J∗ 6= ∅, J \ J∗∗ 6= ∅ =⇒ J∗ \ J∗∗ 6= ∅,
Then
- either both problems are feasible, each has an optimal solution and
α = β
- or one is infeasible, the other one either is infeasible or has no finite
optimum.
Generalized by Borwein (JOTA 1980) to convex disjunctive program.
– p. 40
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Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
– p. 41
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Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
Aim: separatex from∪jPj or equivalently fromP = conv(∪jPj)
Problem: need of representation of the convex hull of the union of
polyhedral sets
– p. 41
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Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
Aim: separatex from∪jPj or equivalently fromP = conv(∪jPj)
Problem: need of representation of the convex hull of the union of
polyhedral sets
Solution:Lift-and-Project
1- representationP of the union of polyhedraP in a higher
dim. space2- projection back in the original space such thatproj P = P
– p. 41
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Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
Aim: separatex from∪jPj or equivalently fromP = conv(∪jPj)
Problem: need of representation of the convex hull of the union of
polyhedral sets
Proposition 32 (Balas 1998)
If Pj = {x ∈ IRn : Ajx ≥ bj} 6= ∅, j = 1, p then
P = {(x, (y1, y10), .., (y
p, yp0)) ∈ IRn+(n+1)p : x −
∑pj=1 yj = 0
Ajyj − yj0b
j ≥ 0
yj0 ≥ 0
∑pj=1 y
j0 = 1
– p. 41
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Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
Aim: separatex from∪jPj or equivalently fromP = conv(∪jPj)
Problem: need of representation of the convex hull of the union of
polyhedral sets
Grossmann review’s on disjunctive prog. techniques (Opt. and
Eng. 2002)Cornuejols-Lemaréchal (Math. Prog. 2006)
– p. 41
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Let us consider the problem:
(PC)inf f(x)
s.t. x ∈ C
wheref : X → IR ∪ {+∞} is quasiconvex lower semicontinuous
C ⊂ int(domf) is a locally finite union of closed convex sets,
C = ∪lfα∈ACα.
– p. 42
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Locally finite union
A subsetC of X is said to be alocally finite union of closed setsif
there exists a (possibly infinite) family{Cα : α ∈ A} of convex subsets
of X such that
C = ∪α∈ACα
for anyx ∈ C, there existρ > 0 and a finite subsetAx of A such that
B(x, ρ) ∩ C = B(x, ρ) ∩ [∪α∈AxCα]
and
∀α ∈ Ax, x ∈ Cα.
Notation:C = ∪lfα∈ACα
– p. 43
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Local mapping
For any subsetC of X, locally finite union of closed sets
C = ∪lfα∈ACα, a familyM = {(ρx, Ax) : x ∈ C} with ρx > 0 and
Ax satisfying
B(x, ρx) ∩ C = B(x, ρx) ∩ [∪α∈AxCα]
and
∀α ∈ Ax, x ∈ Cα.
is called a local mapping ofC.
– p. 44
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The following subset
C = {x ∈ X : g(x) ≤ 0}
is a locally finite union of convex sets, if
g is continuous concave and locally polyhedral
– p. 45
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The following subset
C = {x ∈ X : g(x) ≤ 0}
is a locally finite union of convex sets, if
g is continuous concave and locally polyhedral
or
g(x) = minj∈J gj(x) with eachgi : X → IR quasiconvex
coercive and
(H) ∀x ∈ X, ∃ ε > 0 andJx finite ⊂ J such that{j ∈ J : gj(u) = g(u)} ⊂ Jx, ∀,u ∈ B(x, ρ).
– p. 45
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Existence for lfuc
Theorem 34
Letf : X → IR ∪ {+∞} be a quasiconvex function, continuous ondomf .
Assume that
• for everyλ > infX f , int(Sλ) 6= ∅
subset ofC
• for anyα ∈ A, Cα ∩ B(0, n) is weakly compact for anyn ∈ IN
and the following coercivity condition holds
there existρ > 0 such that∀x ∈ Cα \ B(0, ρ),
∃ yx ∈ Cα ∩ B(0, ‖x‖) such thatf(yx) < f(x).
If there exists a local mappingM = {(ρx, Ax) : x ∈ C} of C such that the
set{x ∈ C : card(Ax) > 1} is included in a weakly compact subset ofC,
then problem(PC) admits a local solution.– p. 46
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Existence for disjunctive program
(DP )
min f(x)
s.t.
8
<
:
hi(x) ≤ 0 i = 1, . . . , l,
minj∈J gj(x) ≤ 0
– p. 47
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Existence for disjunctive program
Theorem 38
Letf : X → IR ∪ {+∞} be a quasiconvex function, continuous ondomf .
Assume that
• (H) holds and for everyλ > infX f , int(Sλ) 6= ∅
• for anyj, gj is lsc quasiconvex and coercive
• for anyi, hi is lsc quasiconvex
• for anyj and anyn ∈ IN, the subset
S0(gj) ∩ [∩iS0(hj)] ∩ B(0, n) is weakly compact
If there exists a local mappingM = {(ρx, Ax) : x ∈ C} of C such that the
set{x ∈ C : card(Ax) > 1} is included in a weakly compact subset ofC,
then problem(DP ) admits a local solution.
– p. 47
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References
D. A. & N. Hadjisavvas,On Quasimonotone Variational Inequalities,
JOTA 121(2004), 445-450.
D. Aussel & J. Ye,Quasiconvex programming on locally finite union
of convex sets, JOTA, to appear (2008).
D. Aussel & J. Ye,Quasiconvex programming with starshaped
constraint region and application to quasiconvex MPEC,
Optimization 55 (2006), 433-457.
– p. 48
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Appendix 5 - Normally compactness
A subsetC is said to besequentially normally compactatx ∈ C if
for any sequence(xk)k ⊂ C converging to x and any sequence
(x∗k)k, x∗
k ∈ NF (C, xk) weakly converging to 0, one has‖x∗k‖ → 0.
Examples:
X finite dimensional space
C epi-Lipschitz atx
C convex with nonempty interior
A functionf is said to besequentially normally subcompactat
x ∈ C if the sublevel setSf(x) is sequentially normally compact atx.Examples:
X finite dimensional space
f is locally Lipschitz aroundx and0 6∈ ∂Lf(x)
f is quasiconvex withint(Sf(x)) 6= ∅
– p. 49
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Appendix 2 - Limiting Normal Cone
Let C be any nonempty subset ofX andx ∈ C. The
limiting normal coneto C atx, denoted byNL(C, x) is defined by
NL(C, x) = lim supx′→x
NF (C, x′)
where the Frèchet normal coneNF (C, x′) is defined by
NF (C, x′) =
{
x∗ ∈ X∗ : lim supu→x′, u∈C
〈x∗, u − x′〉
‖u − x′‖≤ 0
}
.
The limiting normal cone is closed (but in general non convex)
– p. 50
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Appendix - Limiting subdifferential
One can define theLimiting subdifferential(in the sense of
Mordukhovich), and its asymptotic associated form, of a function
f : X → IR ∪ {+∞} by
∂Lf(x) = Limsupy
f→x
∂F f(y)
={
x∗ ∈ X∗ : (x∗,−1) ∈ NL(epif, (x, f(x)))}
∂∞f(x) = Limsupy
f→x
λց0
λ∂F f(y)
={
x∗ ∈ X∗ : (x∗, 0) ∈ NL(epif, (x, f(x)))}
.
– p. 51