Existence and uniqueness results for
variational inequalities
Lecture 3
V.I. in quasiconvex optimization
Didier Aussel
Univ. de Perpignan
– p. 1
Outline of lecture 3
I- About uniqueness
II- Introduction to QV optimization
III- Normal approach
a- First definitions
b- Adjusted sublevel sets and normal operator
IV- Quasiconvex optimization
a- Optimality conditions
b- Convex constraint case
c- Nonconvex constraint case
– p. 2
I - About uniqueness
– p. 3
Uniqueness
Global result:
If T is strictly monotone then, for any K ⊂ X, there exists at most
one solution for V I(T, K)
Let us recall that T is strictly monotone iff for any x, y ∈ X x 6= y and any x̄∗ ∈ T (x̄), ȳ ∗ ∈ T (ȳ),
one has
hȳ ∗ − x̄∗ , y − xi > 0.
Let x̄ and ȳ be two different solutions. Then there exists x̄∗ ∈ T (x̄) and ȳ ∗ ∈ T (ȳ) such that
hȳ ∗ , y − xi ≤ 0 and hx̄∗ , y − xi ≥ 0.
But this is a contradiction with hȳ ∗ , y − xi > hx̄∗ , y − xi.
– p. 4
Uniqueness
Theorem 1 If f : IRn → IRn is locally Lipschitz and the Clarke
generalized Jacobian of f at every point of C consists of positive
definite matrices only. Then f is strictly monotone on C and thus
S(f, C) contains at most one solution.
Recall that the Clarke generalized Jacobian of f at x is
{ lim ∇f (xi ) : xi → x, ∇f (xi ) exists}.
i→∞
Local result:
x is said to be a local solution if there exists a neigh. of x which do
not contains any other solution.
For local uniqueness (generalization of avove result), see Luc,
JMAA 268 (2002)
– p. 5
II
Introduction to QV optimization
– p. 6
Quasiconvexity
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K
if, for all x, y ∈ K and all t ∈ [0, 1],
f (tx + (1 − t)y) ≤ max{f (x), f (y)}.
– p. 7
Quasiconvexity
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K
if, for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
– p. 7
Quasiconvexity
A function f : X → IR ∪ {+∞} is said to be quasiconvex on K
if, for all λ ∈ IR, the sublevel set
Sλ = {x ∈ X : f (x) ≤ λ} is convex.
A function f : X → IR ∪ {+∞} is said to be semistrictly
quasiconvex on K if, f is quasiconvex and for any x, y ∈ K,
f (x) < f (y) ⇒ f (z) < f (y),
∀ z ∈ [x, y[.
– p. 7
f differentiable
f is quasiconvex iff df is quasimonotone
iff df (x)(y − x) > 0 ⇒ df (y)(y − x) ≥ 0
f is quasiconvex iff ∂f is quasimonotone
iff ∃ x∗ ∈ ∂f (x) : hx∗ , y − xi > 0
⇒ ∀ y ∗ ∈ ∂f (y), hy ∗ , y − xi ≥ 0
– p. 8
Why not a subdifferential for quasiconvex programming?
– p. 9
Why not a subdifferential for quasiconvex programming?
No (upper) semicontinuity of ∂f if f is not
supposed to be Lipschitz
– p. 9
Why not a subdifferential for quasiconvex programming?
No (upper) semicontinuity of ∂f if f is not
supposed to be Lipschitz
No sufficient optimality condition
x̄ ∈ arg min f
H
x̄ ∈ Sstr (∂f, C) H
=⇒
H
C
– p. 9
III
Normal approach of quasiconvex
analysis
– p. 10
III
Normal approach
a- First definitions
– p. 11
A first approach
Sublevel set:
Sλ = {x ∈ X : f (x) ≤ λ}
Sλ> = {x ∈ X : f (x) < λ}
Normal operator:
X∗
Define Nf (x) : X → 2
by
Nf (x) = N (Sf (x) , x)
= {x∗ ∈ X ∗ : hx∗ , y − xi ≤ 0, ∀ y ∈ Sf (x) }.
With the corresponding definition for Nf> (x)
– p. 12
But ...
Nf (x) = N (Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N (Sf>(x) , x) has no quasimonotonicity properties
– p. 13
But ...
Nf (x) = N (Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N (Sf>(x) , x) has no quasimonotonicity properties
Example
Define f : R2 → R by
8
< |a| + |b| ,
f (a, b) =
:
1,
if |a| + |b| ≤ 1
if |a| + |b| > 1
.
Then f is quasiconvex. Consider x = (10, 0), x∗ = (1, 2), y = (0, 10) and y ∗ = (2, 1).
We see that x∗ ∈ N < (x) and y ∗ ∈ N < (y) (since |a| + |b| < 1 implies (1, 2) · (a − 10, b) ≤ 0 and
(2, 1) · (a, b − 10) ≤ 0) while hx∗ , y − xi > 0 and hy ∗ , y − xi < 0. Hence N < is not
quasimonotone.
– p. 13
But ...
Nf (x) = N (Sf (x) , x) has no upper-semicontinuity properties
Nf> (x) = N (Sf>(x) , x) has no quasimonotonicity properties
These two operators are essentially adapted to the class of
semi-strictly quasiconvex functions. Indeed in this case, for each
x ∈ dom f \ arg min f , the sets Sf (x) and Sf<(x) have the same
closure and Nf (x) = Nf< (x).
– p. 13
III
Normal approach
b- Adjusted sublevel sets
and
normal operator
– p. 14
Definition
Adjusted sublevel set
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅
and Sfa (x) = Sf (x) if Sf<(x) = ∅.
– p. 15
Definition
Adjusted sublevel set
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅
and Sfa (x) = Sf (x) if Sf<(x) = ∅.
Sfa (x) coincides with Sf (x) if cl(Sf>(x) ) = Sf (x)
e.g. f is semistrictly quasiconvex
– p. 15
Definition
Adjusted sublevel set
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅
and Sfa (x) = Sf (x) if Sf<(x) = ∅.
Sfa (x) coincides with Sf (x) if cl(Sf>(x) ) = Sf (x)
Proposition 2 Let f : X → IR ∪ {+∞} be any function, with
domain dom f . Then
f is quasiconvex ⇐⇒ Sfa (x) is convex , ∀ x ∈ dom f.
– p. 15
Proof
Let us suppose that Sfa (u) is convex for every u ∈ dom f . We will show that for any x ∈ dom f ,
Sf (x) is convex.
If x ∈ arg min f then Sf (x) = Sfa (x) is convex by assumption.
Assume now that x ∈
/ arg min f and take y, z ∈ Sf (x) .
”
“
<
If both y and z belong to B Sf (x) , ρx , then y, z ∈ Sfa (x) thus [y, z] ⊆ Sfa (x) ⊆ Sf (x) .
”
“
<
If both y and z do not belong to B Sf (x) , ρx , then
f (x) = f (y) = f (z), Sf<(z) = Sf<(y) = Sf<(x)
”
“
<
and ρy , ρz are positive. If, say, ρy ≥ ρz then y, z ∈ B Sf (y) , ρy thus
y, z ∈ Sfa (y) and [y, z] ⊆ Sfa (y) ⊆ Sf (y) = Sf (x) .
– p. 16
Proof
/ B(Sf<(x) , ρx ). Then
Finally, suppose that only one of y, z, say z, belongs to B(Sf<(x) , ρx ) while y ∈
f (x) = f (y), Sf<(y) = Sf<(x) and ρy > ρx
”
”
“
“
<
<
so we have z ∈ B Sf (x) , ρx ⊆ B Sf (y) , ρy and we deduce as before that
[y, z] ⊆ Sfa (y) ⊆ Sf (y) = Sf (x) .
The other implication is straightforward.
– p. 17
Adjusted normal operator
Adjusted sublevel set:
For any x ∈ dom f , we define
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
where ρx = dist(x, Sf<(x) ), if Sf<(x) 6= ∅.
Ajusted normal operator:
Nfa (x) = {x∗ ∈ X ∗ : hx∗ , y − xi ≤ 0, ∀ y ∈ Sfa (x)}
– p. 18
Example
x
– p. 19
Example
x
B(Sf<(x) , ρx )
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
– p. 19
Example
Sfa (x) = Sf (x) ∩ B(Sf<(x) , ρx )
Nfa (x) = {x∗ ∈ X ∗ : hx∗ , y − xi ≤ 0, ∀ y ∈ Sfa (x)}
– p. 19
Subdifferential vs normal operator
One can have
Nfa (x) ⊆/ cone(∂f (x))
or
cone(∂f (x)) ⊆/ Nfa (x)
Proposition 4
f is quasiconvex and x ∈ dom f
If there exists δ > 0 such that 0 6∈ ∂ L f (B(x, δ)) then
L
∞
cone(∂ f (x)) ∪ ∂ f (x) ⊂ Nfa (x).
– p. 20
Basic properties of Nfa
Nonemptyness:
Proposition 6 Let f : X → IR ∪ {+∞} be lsc. Assume that rad.
continuous on dom f or dom f is convex and intSλ 6= ∅,
∀ λ > inf X f . Then
f is quasiconvex ⇔ Nfa (x) \ {0} =
6 ∅, ∀ x ∈ dom f \ arg min f.
Quasimonotonicity:
The normal operator Nfa is always quasimonotone
– p. 21
Upper sign-continuity
X∗
• T : X → 2 is said to be upper sign-continuous on K iff for any
x, y ∈ K, 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. 22
Upper sign-continuity
X∗
• T : X → 2 is said to be upper sign-continuous on K iff for any
x, y ∈ K, 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. 22
locally upper sign continuity
X∗
Definition 8 Let T : K → 2
be a set-valued map.
T est called locally upper sign-continuous on K if, for any x ∈ K
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. 23
locally upper sign continuity
X∗
Definition 10 Let T : K → 2
be a set-valued map.
T est called locally upper sign-continuous on K if, for any x ∈ K
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
Continuity of normal operator
Proposition 12
Let f be lsc quasiconvex function such that int(Sλ ) 6= ∅, ∀ λ > inf f .
Then Nf is locally upper sign-continuous on dom f \ arg min f .
– p. 23
Proposition 14
If f is quasiconvex such that int(Sλ ) 6= ∅, ∀ λ > inf f
and f is lsc at x ∈ dom f \ arg min f ,
Then Nfa is norm-to-w∗ cone-usc at x.
∗
A multivalued map with conical valued T : X → 2X is said to be
cone-usc at x ∈ dom T if there exists a neighbourhood U of x and a
base C (u) of T (u), u ∈ U , such that u → C (u) is usc at x.
– p. 24
Integration of Nfa
Let f : X → IR ∪ {+∞} quasiconvex
Question: Is it possible to characterize the functions
g : X → IR ∪ {+∞} quasiconvex such that Nfa = Nga ?
– p. 25
Integration of Nfa
Let f : X → IR ∪ {+∞} quasiconvex
Question: Is it possible to characterize the functions
g : X → IR ∪ {+∞} quasiconvex such that Nfa = Nga ?
A first answer:
Let C = {g : X → IR ∪ {+∞} cont. semistrictly quasiconvex
such that argmin f is included in a closed hyperplane}
Then, for any f, g ∈ C,
Nfa = Nga ⇔ g is Nfa \ {0}-pseudoconvex
⇔ ∃ x∗ ∈ Nfa (x) \ {0} : hx∗ , y − xi ≥ 0 ⇒ g(x) ≤ g(y).
– p. 25
Integration of Nfa
Let f : X → IR ∪ {+∞} quasiconvex
Question: Is it possible to characterize the functions
g : X → IR ∪ {+∞} quasiconvex such that Nfa = Nga ?
General case: open question
– p. 25
IV
Quasiconvex programming
a- Optimality conditions
– p. 26
Quasiconvex programming
Let f : X → IR ∪ {+∞} and K ⊆ dom f be a convex subset.
(P )
find x̄ ∈ K : f (x̄) = inf f (x)
x∈K
– p. 27
Quasiconvex programming
Let f : X → IR ∪ {+∞} and K ⊆ dom f be a convex subset.
(P )
find x̄ ∈ K : f (x̄) = inf f (x)
x∈K
Perfect case: f convex
f : X → IR ∪ {+∞} a proper convex function
K a nonempty convex subset of X, x̄ ∈ K + C.Q.
Then
f (x̄) = inf f (x)
x∈K
⇐⇒
x̄ ∈ Sstr (∂f, K)
– p. 27
Quasiconvex programming
Let f : X → IR ∪ {+∞} and K ⊆ dom f be a convex subset.
(P )
find x̄ ∈ K : f (x̄) = inf f (x)
x∈K
Perfect case: f convex
f : X → IR ∪ {+∞} a proper convex function
K a nonempty convex subset of X, x̄ ∈ K + C.Q.
Then
f (x̄) = inf f (x)
x∈K
⇐⇒
x̄ ∈ Sstr (∂f, K)
What about f quasiconvex case?
x̄ ∈ arg min f
H
x̄ ∈ Sstr (∂f (x̄), K)H
=⇒
H
K
– p. 27
Sufficient optimality condition
Proposition 16
f : X → IR ∪ {+∞} quasiconvex, radially cont. on dom f
C ⊆ X such that conv(C) ⊂ dom f .
Suppose that C ⊂ int(dom f ).
Then x̄ ∈ S(Nfa \ {0}, C)
=⇒
∀ x ∈ C, f (x̄) ≤ f (x).
where x̄ ∈ S(Nfa \ {0}, K) means that there exists x̄∗ ∈ Nfa (x̄) \ {0} such
that
hx̄∗ , c − xi ≥ 0,
∀ c ∈ C.
– p. 28
Lemma 18 Let f : X → IR ∪ {+∞} be a quasiconvex function,
radially continuous on dom f . Then f is Nfa \ {0}-pseudoconvex on
int(dom f ), that is,
∃ x∗ ∈ Nfa (x) \ {0} : hx∗ , y − xi ≥ 0 ⇒ f (y) ≥ f (x).
Proof.
Let x, y ∈ int(dom f ). According to the quasiconvexity of f , Nfa (x) \ {0} is nonempty.
Let us suppose that hx∗ , y − xi ≥ 0 with x∗ ∈ Nfa (x) \ {0}. Let d ∈ X be such that
1
hx∗ , yn − xi > 0 for any n, where yn = y + n
d (∈ dom f for n large enough).
This implies that yn 6∈ Sf< (x) since x∗ ∈ Nfa (x) ⊂ Nf< (x).
It follows by the radial continuity of f that f (y) ≥ f (x).
– p. 29
Necessary and Sufficient conditions
Proposition 20 Let C be a closed convex subset of X, x̄ ∈ C and
f : X → IR be continuous semistrictly quasiconvex such that
int(Sfa (x̄)) 6= ∅ and f (x̄) > inf X f .
Then the following assertions are equivalent:
i) f (x̄) = minC f
ii) x̄ ∈ Sstr (Nfa \ {0}, C)
iii) 0 ∈ Nfa (x̄) \ {0} + N K(C, x̄).
– p. 30
Proposition 22 Let C be a closed convex subset of an Asplund
space X and f : X → IR be a continuous quasiconvex function.
Suppose that either f is sequentially normally sub-compact or C is
sequentially normally compact.
If x̄ ∈ C is such that
0 6∈ ∂ L f (x̄)
and
0 6∈ ∂ ∞ f (x̄) \ {0} + N K(C, x̄)
then the following assertions are equivalent:
i) f (x̄) = minC f
ii) x̄ ∈ ∂ L f (x̄) \ {0} + N K(C, x̄)
iii) 0 ∈ Nfa (x̄) \ {0} + N K(C, x̄)
– p. 31
III
Quasiconvex optimization
b- Convex constraint case
– p. 32
III
Quasiconvex optimization
b- Convex constraint case
(P )
Find x̄ ∈ C such that f (x̄) = inf f.
C
with C convex set.
– p. 32
Existence results with convex constraint set
Theorem 23 (Convex case)
f : X → IR ∪ {+∞} quasiconvex
+ continuous on dom (f )
+ for any λ > inf X f , int(Sλ ) 6= ∅.
+ C ⊆ int(dom f ) convex such that C ∩ B(0, n)
is weakly compact for some n ∈ IN.
+ coercivity condition
∃ ρ > 0, ∀ x ∈ C \ B(0, ρ), ∃ y ∈ C with kyk < kxk
such that ∀ x∗ ∈ Nfa (x) \ {0}, hx∗ , x − yi > 0
Then there exists x̄ ∈ C such that ∀ x ∈ C, f (x) ≥ f (x̄).
– p. 33
Existence for Stampacchia V.I.
Theorem 25
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. 34
Proof If arg min f ∩ K 6= ∅, we have nothing to prove.
Suppose that arg min f ∩ K = ∅. Then N a is quasimonotone and norm-to-w∗ cone-usc on K. Thus,
all assumptions of Theorem 25 hold for the operator N a \ {0}, so Sstr (N a \ {0} , K) 6= ∅. Finally,
using the sufficient condition, we infer that f has a global minimum on K.
– p. 35
Proof If arg min f ∩ K 6= ∅, we have nothing to prove.
Suppose that arg min f ∩ K = ∅. Then N a is quasimonotone and norm-to-w∗ cone-usc on K. Thus,
all assumptions of Theorem 25 hold for the operator N a \ {0}, so Sstr (N a \ {0} , K) 6= ∅. Finally,
using the sufficient condition, we infer that f has a global minimum on K.
Corollary 27 Assumptions on f and K as in Theorem 23. Assume that
there exists n ∈ IN such that for all x ∈ K, kxk > n, there exists y ∈ K,
kyk < kxk such that f (y) < f (x). Then there exists x0 ∈ K such that
∀ x ∈ K,
f (x) ≥ f (x0 ).
Proof. If f (y) < f (x) then for every x∗ ∈ N a (x) ⊆ N < (x), hx∗ , y − xi ≤ 0. Hence, coercivity
condition with T = N a holds. The corollary follows from Theorem 23.
– p. 35
III
Quasiconvex optimization
c- Nonconvex constraint case
(an example)
– p. 36
Disjunctive programming
Let us consider the optimization problem:
min f (x)

 h (x) ≤ 0 i = 1, . . . , l
i
s.t.
 minj∈J gj (x) ≤ 0
where f, hi , gj : X → IR are quasiconvex
J is a (possibly infinite) index set.
– p. 37
Disjunctive programming
Let us consider the optimization problem:
min f (x)

 h (x) ≤ 0 i = 1, . . . , l
i
s.t.
 minj∈J gj (x) ≤ 0
where f, hi , gj : X → IR are quasiconvex
J is a (possibly infinite) index set.
Example: g : X → IR is continuous concave and
min f (x)

 h (x) ≤ 0
i
s.t.
 g(x) ≤ 0
i = 1, . . . , l,
– p. 37
Disjunctive programming
Let us consider the optimization problem:
min f (x)

 h (x) ≤ 0 i = 1, . . . , l
i
s.t.
 x ∈ ∪j∈J Cj
where f, hi : X → IR are quasiconvex
Cj are convex subsets of X
J is a (possibly infinite) index set.
– p. 37
A little bit of history
Balas 1974: first paper about disjunctive prog.
Pure and mixed 0-1 linear programming
M in Z = d.x +

s.t.
Pm
k=1 ck
Yjk



jk  , k ∈ K
 jk
j∈Jk  A x ≥ b

ck = γjk
W
0 ≤ x ≤ U, Yjk ∈ {0, 1}
– p. 38
A little bit of history
Balas 1974: first paper about disjunctive prog.
Pure and mixed 0-1 linear programming
M in Z = d.x +

s.t.
Pm
k=1 ck
Yjk



jk  , k ∈ K
 jk
j∈Jk  A x ≥ b

ck = γjk
W
0 ≤ x ≤ U, Yjk ∈ {0, 1}
puis Gugat, Grossmann, Borwein, Cornuejols-Lemaréchal,...
– p. 38
Duality for disjunctive prog.
For linear Disjunctive program (Balas):
P rimal
α = inf c.x
s.t.
W
j∈J


Aj .x ≥ bj
x≥0


– p. 39
Duality for disjunctive prog.
For linear Disjunctive program (Balas):
P rimal
α = inf c.x
W
s.t.
Dual
j∈J


Aj .x ≥ bj
x≥0


β = sup w
s.t.



j∈J 
V
w − uj bj ≤ 0


u .A ≤ c 

uj ≥ 0
j
j
– p. 39
Duality theorem for linear disjunctive prog.
Set Pj = {x : Aj .x ≥ bj , x ≥ 0}, Uj = {uj : uj .Aj ≤ c, uj ≥ 0}.
Denote J ∗ = {j ∈ J : Pj 6= ∅} and J ∗∗ = {j ∈ J : Uj 6= ∅}
Theorem 29 (Balas 77)
If (P) and (D) satisfy the following regularity 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
Duality theorem for linear disjunctive prog.
Set Pj = {x : Aj .x ≥ bj , x ≥ 0}, Uj = {uj : uj .Aj ≤ c, uj ≥ 0}.
Denote J ∗ = {j ∈ J : Pj 6= ∅} and J ∗∗ = {j ∈ J : Uj 6= ∅}
Theorem 30 (Balas 77)
If (P) and (D) satisfy the following regularity 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
Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
– p. 41
Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
Aim: separate x from ∪j Pj or equivalently from P = conv(∪j Pj )
Problem: need of representation of the convex hull of the union of
polyhedral sets
– p. 41
Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
Aim: separate x from ∪j Pj or equivalently from P = conv(∪j Pj )
Problem: need of representation of the convex hull of the union of
polyhedral sets
Solution: Lift-and-Project
1- representation P̃ of the union of polyhedra P in a higher
dim. space
2- projection back in the original space such that proj P̃ = P
– p. 41
Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
Aim: separate x from ∪j Pj or equivalently from P = conv(∪j Pj )
Problem: need of representation of the convex hull of the union of
polyhedral sets
Proposition 32 (Balas 1998)
If Pj = {x ∈ IRn : Aj x ≥ bj } =
6 ∅, j = 1, p then
P̃ =
{(x, (y 1 , y01 ), .., (y p , y0p ))
∈ IRn+(n+1)p : x −
Pp
j
y
j=1
Aj y j − y0j bj
y0j
Pp
j
y
j=1 0

=0 




≥0 
≥0 





=1
– p. 41
Computational aspects
In the 90’s: cutting plane methods (Balas-Ceria-Cornuejols, Math.
Prog. 1993).
Aim: separate x from ∪j Pj or equivalently from P = conv(∪j Pj )
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
Let us consider the problem:
(PC )
inf f (x)
s.t. x ∈ C
where f : X → IR ∪ {+∞} is quasiconvex lower semicontinuous
C ⊂ int(dom f ) is a locally finite union of closed convex sets,
C = ∪lf
α∈A Cα .
– p. 42
Locally finite union
A subset C of X is said to be a locally finite union of closed sets if
there exists a (possibly infinite) family {Cα : α ∈ A} of convex subsets
of X such that
C = ∪α∈A Cα
for any x ∈ C, there exist ρ > 0 and a finite subset Ax of A such that
B(x, ρ) ∩ C = B(x, ρ) ∩ [∪α∈Ax Cα ]
and
∀ α ∈ Ax ,
x ∈ Cα .
Notation: C = ∪lf
α∈A Cα
– p. 43
Local mapping
For any subset C of X, locally finite union of closed sets
C = ∪lf
α∈A Cα , a family M = {(ρx , Ax ) : x ∈ C} with ρx > 0 and
Ax satisfying
B(x, ρx ) ∩ C = B(x, ρx ) ∩ [∪α∈Ax Cα ]
and
∀ α ∈ Ax ,
x ∈ Cα .
is called a local mapping of C.
– p. 44
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
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 each gi : X → IR quasiconvex
coercive and
(H)
∀ x ∈ X, ∃ ε > 0 and Jx finite ⊂ J such that
{j ∈ J : gj (u) = g(u)} ⊂ Jx , ∀, u ∈ B(x, ρ).
– p. 45
Existence for lfuc
Theorem 34
Let f : X → IR ∪ {+∞} be a quasiconvex function, continuous on dom f .
Assume that
• for every λ > inf X f , int(Sλ ) 6= ∅
subset of C
• for any α ∈ A, Cα ∩ B̄(0, n) is weakly compact for any n ∈ IN
and the following coercivity condition holds
there exist ρ > 0 such that ∀ x ∈ Cα \ B(0, ρ),
∃ yx ∈ Cα ∩ B(0, kxk) such that f (yx ) < f (x).
If there exists a local mapping M = {(ρx , Ax ) : x ∈ C} of C such that the
set {x ∈ C : card(Ax ) > 1} is included in a weakly compact subset of C,
then problem (PC ) admits a local solution.
– p. 46
Existence for disjunctive program
min
(DP )
s.t.
f (x)
8
< h (x) ≤ 0 i = 1, . . . , l,
i
: minj∈J gj (x) ≤ 0
– p. 47
Existence for disjunctive program
Theorem 38
Let f : X → IR ∪ {+∞} be a quasiconvex function, continuous on dom f .
Assume that
• (H) holds and for every λ > inf X f , int(Sλ ) 6= ∅
• for any j, gj is lsc quasiconvex and coercive
• for any i, hi is lsc quasiconvex
• for any j and any n ∈ IN, the subset
S0 (gj ) ∩ [∩i S0 (hj )] ∩ B(0, n) is weakly compact
If there exists a local mapping M = {(ρx , Ax ) : x ∈ C} of C such that the
set {x ∈ C : card(Ax ) > 1} is included in a weakly compact subset of C,
then problem (DP ) admits a local solution.
– p. 47
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
Appendix 5 - Normally compactness
A subset C is said to be sequentially normally compact at x ∈ C if
for any sequence (xk )k ⊂ C converging to x and any sequence
(x∗k )k , x∗k ∈ N F (C, xk ) weakly converging to 0, one has kx∗k k → 0.
Examples:
X finite dimensional space
C epi-Lipschitz at x
C convex with nonempty interior
A function f is said to be sequentially normally subcompact at
x ∈ C if the sublevel set Sf (x) is sequentially normally compact at x.
Examples:
X finite dimensional space
f is locally Lipschitz around x and 0 6∈ ∂ L f (x)
f is quasiconvex with int(Sf (x) ) 6= ∅
– p. 49
Appendix 2 - Limiting Normal Cone
Let C be any nonempty subset of X and x ∈ C̄. The
limiting normal cone to C at x, denoted by N L (C, x) is defined by
N L (C, x) = lim sup N F (C, x′ )
x′ →x
where the Frèchet normal cone N F (C, x′ ) is defined by
∗
′
hx , u − x i
F
′
∗
∗
≤0 .
N (C, x ) = x ∈ X : lim sup
′
u→x′ , u∈C ku − x k
The limiting normal cone is closed (but in general non convex)
– p. 50
Appendix - Limiting subdifferential
One can define the Limiting subdifferential (in the sense of
Mordukhovich), and its asymptotic associated form, of a function
f : X → IR ∪ {+∞} by
∂ L f (x) = Limsup f ∂ F f (y)
y →x
∗
∗
∗
L
= x ∈ X : (x , −1) ∈ N (epi f, (x, f (x)))
∂ ∞ f (x) = Limsup
f
λ∂ F f (y)
y →x
λց0
∗
= x ∈X
∗
∗
L
: (x , 0) ∈ N (epi f, (x, f (x))) .
– p. 51