Stephan Dempe
Fakultät für Mathematik und Informatik
TU Bergakademie Freiberg
Akademiestr. 6
09596 Freiberg
Bilevel Optimization Problem:
Existence of optimal solutions and
optimality conditions
November 27, 2013
CIMPA school, November 25 to December 6, 2013,
University of Delhi
Contents
Applications
The model: Formulation and first properties
Transformation to one-level problem and related optimality conditions
First solution algorithms
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
1
Contents
Applications
The model: Formulation and first properties
Transformation to one-level problem and related optimality conditions
First solution algorithms
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
1
Contents
Applications
The model: Formulation and first properties
Transformation to one-level problem and related optimality conditions
First solution algorithms
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
1
Contents
Applications
The model: Formulation and first properties
Transformation to one-level problem and related optimality conditions
First solution algorithms
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
1
Applications
Topics:
1. Traffic control: Optimal tolls
2. Optimal chemical equilibrium
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
2
Applications
Traffic control: Optimal tolls
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
3
Applications
Lower level problem is a multicommodity flow problem, formulated e.g. as
X X
k,l
ck,l
→ min
e xe
(k,l)∈M e∈Ek,l
X
xe +
xk,l
e
∀e ∈ E
=
ue
xk,l
e
=
 k,l
 v
0

−v k,l
xe , xk,l
e
≥
0 ∀(k, l) ∈ M, ∀e ∈ E ,
(k,l)∈M :e∈Ek,l
X
xk,l
e −
e∈Ok,l (j)
X
e∈Ik,l (j)
j=k
j ∈ Vk,l \ {k, l} ∀(k, l) ∈ M
j=l
Upper level problem:
kx − x
bk → min
c
x solves the above problem for fixed c
c∈C
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
4
Applications
Optimal chemical equilibrium
hc, xi → min
p,T,y
(p, T, y) ∈ Y,
x = x(p, T, y) ∈ Ψ(p, T, y)
with
N
P
Ψ(p, T, y) = Argmin {
x,z
ci (p, T )xi +
i=1
S. Dempe
Bilevel Optimization Problem
i=1
G
P
xi ln xzi :
xj , Ax = Ay, x ≥
j=1
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
z=
November 27, 2013
CIMPA school,
University of Delhi
G
P
0}
5
Many other applications can be found in the Annotated Bibliography
(Dempe [9]) a preliminary version of which can be downloaded from
http://www.mathe.tu-freiberg.de/ dempe/Artikel
The bibliography file of this article is outdated.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
6
The model: Formulation and first properties
Topics:
1. The bilevel optimization model
2. Definition of an optimal solution
3. Existence of optimal solutions
4. Optimistic and pessimistic solutions
5. Weak solution of the pessimistic problem
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
7
The model: Formulation and first properties
Lower level problem:
min{f (x, y) : g(x, y) ≤ 0}
(1)
y
f : Rn × Rm → R, g : Rn × Rm → Rp are sufficiently smooth.
Optimal value function:
ϕ(x) := min{f (x, y) : g(x, y) ≤ 0}.
(2)
y
Solution set mapping:
Ψ(x) := {y : g(x, y) ≤ 0, f (x, y) ≤ ϕ(x)}.
(3)
Graph of the solution set mapping:
gph Ψ := {(x, y) : y ∈ Ψ(x)}.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
8
Upper level problem or Bilevel optimization problem (Dempe [8]) in its
original formulation:
0
min0 {F (x, y) : G(x) ≤ 0, y ∈ Ψ(x)}.
x
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
(4)
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
9
The model: Formulation and first properties
Why the brackets in (4)?
Consider the problem
0
min0 {x2 + y : y ∈ Ψ(x)},
x
where
Ψ(x) := Argmin {−xy : 0 ≤ y ≤ 1}.
y
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
10
The model: Formulation and first properties
ϕo (x) = min{F (x, y) : y ∈ Ψ(x)}
(5)
y
and
min{ϕo (x) : G(x) ≤ 0}
x
(6)
Optimistic bilevel optimization Cooperation of the follower or
ϕp (x) = max{F (x, y) : y ∈ Ψ(x)}
(7)
y
and
min{ϕp (x) : G(x) ≤ 0}
x
(8)
Pessimistic bilevel optimization Leader needs to bound the damage resulting
from "bad" selection of the follower.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
11
Definition of an optimal solution
Definition
A feasible point x : G(x) ≤ 0 is a local optimum of problem (6) if ∃ ε > 0:
ϕo (x) ≥ ϕo (x) ∀ x : G(x) ≤ 0, kx − xk ≤ 0.
Definition
A feasible point x : G(x) ≤ 0 is a local optimum of problem (8) if ∃ ε > 0:
ϕp (x) ≥ ϕp (x) ∀ x : G(x) ≤ 0, kx − xk ≤ 0.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
12
Definition of an optimal solution
The optimistic bilevel problem (6):
min{ϕo (x) : G(x) ≤ 0}
x
with
ϕo (x) = min{F (x, y) : y ∈ Ψ(x)}
y
or
min{min{F (x, y) : y ∈ Ψ(x)} : G(x) ≤ 0}
x
y
is often replaced by
min{F (x, y) : G(x) ≤ 0, (x, y) ∈ gph Ψ},
(9)
x,y
F : Rn × Rm → R, G : Rn → Rq sufficiently smooth.
Are they equivalent?
Is a (local) optimal solution of (9) related to a (local) optimal solution of
(6)?
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
13
Definition of an optimal solution
NO !
Consider the problem of minimizing the function F (x, y) = x subject to
x ∈ [−1, 1] and y ∈ Ψ(x) := Argmin {xy : y ∈ [0, 1]}. Then,
y

 [0, 1]
{1}
y(x) ∈

{0}
for x = 0,
for x < 0,
for x > 0.
Hence, the point (x, y) = (0, 0) is a local minimum of the problem
min{x : x ∈ [−1, 1], y ∈ Ψ(x)}
x,y
since, for each feasible point (x, y) with k(x, y) − (x, y)k ≤ 0.5 we have x ≥ 0.
But, the point x does not minimize the function ϕo (x) = x on [−1, 1] locally.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
14
Existence of optimal solutions
Definition
A feasible point (x, y) ∈ gph Ψ, G(x) ≤ 0 is local optimal for (9) if there
exists ε > 0 with
F (x, y) ≥ F (x, y) ∀ (x, y) ∈ gph Ψ, G(x) ≤ 0, k(x, y) − (x, y)k ≤ ε.
It is global optimal if ε can be taken arbitrarily large.
Theorem (Weierstraß)
If X ⊂ Rn is not empty and compact, h : Rn 7→ R is continuous, then
min{h(x) : x ∈ X} and max{h(x) : x ∈ X}
have (global) optimal solutions.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
15
Problem (9):
min{F (x, y) : G(x) ≤ 0, (x, y) ∈ gph Ψ}
x,y
Hence, we have to find properties guaranteeing that
1. the graph gph Ψ is compact.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
16
Properties of set-valued mappings
Definition
A point-to-set mapping Z : Rn ⇒ Rm (mapping points x ∈ Rn to subsets in
Rm ) is called upper semicontinuous at a point x0 ∈ Rn if, for each open set
V ⊃ Z(x0 ) there is an open set U 3 x0 with Z(x) ⊂ V for all x ∈ U.
Definition
A point-to-set mapping Z : Rn ⇒ Rm is closed at z 0 ∈ Rn if for each
k
k
sequence {(xk , y k )}∞
k=1 converging to (x, y) with y ∈ Z(x ) for all k also
y ∈ Z(x).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
17
Properties of set-valued mappings
upper semicontinuous but not lower semicontinuous mapping
Z(x 0)
x0
The point-to-set mapping Z is closed iff its graph is closed.
If Z is closed and its graph is bounded, then its graph is compact.
November
27, 2013
If the graph
of a closed point-to-set
mapping is Fak.
a subset
of a compact set, it
S. Dempe
f. Math. u. Inf.
CIMPA school,
18
is upperof semicontinuous.
University
Delhi
Bilevel Optimization Problem
TU Bergakademie Freiberg
Upper semicontinuity of Ψ
Consider problem (1)
min{f (x, y) : g(x, y) ≤ 0}
y
Feasible set mapping M : Rn ⇒ Rm :
M (x) := {y : g(x, y) ≤ 0}
General assumptions: Let g ∈ C 1 (Rn × Rm , Rp ), G ∈ C(Rn , Rq ),
f ∈ C 1 (Rn × Rm , R), F ∈ C(Rn × Rm , R).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
19
Upper semicontinuity of Ψ
Definition
The Mangasarian-Fromovitz constraint qualification (MFCQ) is satisfied at
(x, y) with y ∈ M (x) if the system
∇y gi (x, y)d < 0 ∀ i ∈ I(x, y) := {j : gj (x, y) = 0}.
Assumption (C) is a compactness assumption:
The set {(x, y) : g(x, y) ≤ 0} is not empty and compact.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
20
Upper semicontinuity of Ψ
Theorem (Necessary optimality condition for problem (1))
y 7→ gi (x, y), i = 1, . . . , p, y 7→ f (x, y) are convex. If (MFCQ) is satisfied at
(x, y) with y ∈ M (x), then y ∈ Ψ(x) iff there exists λ ∈ Λ(x, y) with
Λ(x, y) = {λ : ∇y L(x, y, λ) = 0, λ ≥ 0, λ> g(x, y) = 0}.
Theorem (Bank et al. [2])
Let x with G(x) ≤ 0 be fixed. If for problem (1):
min{f (x, y) : g(x, y) ≤ 0}
y
both assumptions (MFCQ) and (C) are satisfied at each point
(x, y) ∈ gph Ψ then, the mappings Ψ(·) and Λ(·, ·) are upper
semicontinuous at (x, y) and the function ϕ(·) is continuous at x.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
21
Optimistic vs. pessimistic solution
Corollary
Problem (9):
min{F (x, y) : G(x) ≤ 0, (x, y) ∈ gph Ψ}
x,y
has a (global) optimal solution if the assumptions of the last theorem are
satisfied.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
22
Optimistic vs. pessimistic solution
Consider problem (6):
min{ϕo (x) : G(x) ≤ 0}
x
with ϕo (x) = min{F (x, y) : y ∈ Ψ(x)}.
y
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
23
Optimistic vs. pessimistic solution
Definition
A function h : Rn 7→ R is called lower semicontinuous at x0 ∈ Rn if
lim inf
h(x) ≥ h(x0 ).
0
x→x
Theorem
If X ⊂ Rn is not empty and compact, h : Rn 7→ R is lower semicontinuous
on X, then
min{h(x) : x ∈ X}
has a (global) optimal solution.
Hence we need assumptions guaranteeing that ϕo (·) is lower semicontinuous.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
24
Optimistic vs. pessimistic solution
Theorem (Bank et al. [2])
Consider the problem
ξ(x) = min{h(x, y) : y ∈ M (x)}
y
with a continuous function h and an upper semicontinuous feasible set
mapping M . Then, the function ξ(·) is lower semicontinuous.
Corollary
If assumptions (MFCQ) and (C) are satisfied at each point (x, y) ∈ gph Ψ
with G(x) ≤ 0, and the set {x : G(x) ≤ 0} is not empty and compact, the
optimistic bilevel problem (6) has a (global) optimal solution.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
25
Optimistic vs. pessimistic solution
Consider problem (8):
min{ϕp (x) : G(x) ≤ 0}
x
with
ϕp (x) = max{F (x, y) : y ∈ Ψ(x)}.
(10)
y
Definition
Z : Rn ⇒ Rm is called lower semicontinuous at x0 , if for each open set V
with Z(x0 ) ∩ V 6= ∅ there is an open set U 3 x0 such that Z(x) ∩ V 6= ∅ for
all x ∈ U.
Theorem (Bank et al. [2])
If the point-to-set mapping Ψ(·) is lower semicontinuous at x with G(x) ≤ 0,
then the function ϕp (·) is lower semicontinuous.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
26
lower semicontinuous but not upper semicontinuous mapping
Z(x 0)
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
x 0Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
27
Optimistic vs. pessimistic solution
Corollary
The pessimistic bilevel problem has an optimal solution provided
1. the function F : Rn × Rm 7→ R is continuous,
2. the set {(x, y) : G(x) ≤ 0 g(x, y) ≤ 0} =
6 ∅ and compact and
n
m
3. the point-to-set mapping Ψ : R ⇒ R is lower semicontinuous
are satisfied.
For the solution set mapping of the parametric optimization problem (1)
lower semicontinuity is satisfied at x0 if Ψ(x0 )=1.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
28
Weak solution of the pessimistic problem
Remark
A function h : Rp 7→ R is lower semicontinuous iff its epigraph
epi h := {(x, α) : h(x) ≤ α}
is closed.
)
not lower semicontinuos
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
lower semicontinuos
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
29
Weak solution of the pessimistic problem
Definition
The function ϕp (x) defined by
epi ϕp = cl epi ϕp
is called the weak pessimistic optimal value function of the bilevel
optimization problem (4).
Definition
A feasible point x : G(x) ≤ 0 is a weak local pessimistic optimum of problem
(8) if ∃ ε > 0:
ϕp (x) ≥ ϕp (x) ∀ x : G(x) ≤ 0, kx − xk ≤ 0.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
30
Weak solution of the pessimistic problem
Corollary
The pessimistic bilevel problem has a weak pessimistic optimum provided
1. the function F : Rn × Rm 7→ R is continuous,
2. the set {(x, y) : G(x) ≤ 0 g(x, y) ≤ 0} =
6 ∅ and compact and
n
m
3. the point-to-set mapping Ψ : R ⇒ R is upper semicontinuous
are satisfied.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
31
Transformation to one-level problem and related optimality
conditions
Topics:
1. Strongly stable lower level solution
2. KKT-reformulation
3. Transformation using the optimal value function
4. Transformation using primal KKT conditions
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
32
Strongly stable lower level solution
This is the simplest case: Transform the problem (9) into
F (x, y(x)) → min
x
(11)
G(x) ≤ 0.
Necessary: Unique optimal solution in lower level problem
Mangasarian-Fromovitz constraint qualification and (strong) sufficient
optimality condition of second order
Convexity
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
33
Strongly stable lower level solution
Global optimum of nonconvex parametric optimization problems "jumps"
1.5
sin(x)
sin(x)+0.1*x
sin(x)-0.1*x
1
0.5
0
-0.5
-1
-1.5
-6
November 27, 2013
CIMPA school,
University of Delhi
-4
-2
S. Dempe
Bilevel Optimization Problem
0
2
4
6
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
34
Strongly stable lower level solution
Definition (Strong sufficient optimality condition of second order
(SSOSC))
Consider problem (1). The strong sufficient optimality condition of second
order is satisfied at some point (x, y) with g(x, y) ≤ 0 if
for each direction d 6= 0 with ∇gi (x, y)d = 0 for each i : λi > 0 we have
d> ∇2yy L(x, y, λ)d > 0.
Definition (Constant rank constraint qualification (CRCQ))
The constant rank constraint qualification (CRCQ) is satisfied at the point
(x, y) for the problem (1) if there exists an open neighborhood U of (x, y)
such that, for each subset I ⊆ {i : gi (x, y) = 0} the family of gradient
vectors {∇y gi (x, y) : i ∈ I} has constant rank on U .
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
35
Strongly stable lower level solution
Theorem (Ralph and Dempe [24])
Assume convexity, (MFCQ), (CRCQ), and strong sufficient optimality of
second order (SSOSC) for the lower level problem (1).
Then, the optimal solution of the lower level problem (1) is locally unique
and a P C 1 -function.
Definition
A function z : Rn → Rm is a P C 1 -function (locally around a point x ∈ Rn )
if there exist an (open) neighborhood V of x and a finite number of
continuously differentiable functions z k : V → Rm , k = 1, . . . , p:
z(x) ∈ {z 1 (x), . . . , z p (x)} ∀ x ∈ V
and the function z itself is continuous on V .
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
36
Strongly stable lower level solution
Example
(Dempe [8]) Consider the problem
−y
→ min
y
y
≤
1,
y2
≤
3 − x21 − x22 ,
(y − 1.5)2
≥
0.75 − (x1 − 0.5)2 − (x2 − 0.5)2 ,
with two parameters x1 and x2 . Then, y is a continuous selection of three
continuously differentiable functions y 1 = y 1 (x), y 2 = y 2 (x), y 3 = y 3 (x) in an
open neighborhood of the point x0 = (1, 1)> :
 1
1,
x ∈ supp (y, y 1 ),

 y =p
y 2 = 3 − x21 − x22 ,
x ∈ supp (y, y 2 ),
y(x) =

 y 3 = 1.5 − p0.75 − (x − 0.5)2 − (x − 0.5)2 , x ∈ supp (y, y 3 ),
1
November 27, 2013
whereschool,
CIMPA
University of Delhi
S. Dempe
Bilevel Optimization Problem
2
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
37
Strongly stable lower level solution
x
2
1.5
1
Y
{3}
Y
{1}
0.5
Y
{2}
0
0.4
1.4
0.6
0
x1
1.2
0.8
1
1
0.8
1.2
0.6
1.4
0.4
Fig. 1: The sets supp (y, y i ) and the optimal solution in Example 26.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
38
Strongly stable lower level solution
P C 1 -functions are
1. locally Lipschitz continuous
2. directionally differentiable in the classic sense: A function z : Rn → R is
directionally differentiable at x ∈ Rn if
z 0 (x; d) := lim t−1 [z(x + td) − z(x)]
t↓0
exists for all directions d
3. d 7→ z 0 (x; d) is Lipschitz continuous. (Scholtes [27])
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
39
Theorem (Ralph and Dempe [24])
Consider problem (1) at a point x = x and let y be a local optimal solution
of this problem where the assumptions (MFCQ), (SSOSC), and (CRCQ) are
satisfied. Then the directional derivative of the function y(·) at x in
direction r coincides with the unique optimal solution of the convex
quadratic optimization problem QP (λ0 , r)
0.5d> ∇2yy L(x, y, λ0 )d + d> ∇2xy L(x, y, λ0 )r → min
d
0
= 0, if λi > 0,
∇x gi (x, y)r + ∇y gi (x, y)d
≤ 0, if gi (x, y) = λ0i = 0
(12)
for an arbitrary Lagrange multiplier λ0 ∈ Λ(x, y) solving
∇x L(x, y, λ)r →
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
max .
(13)
λ∈Λ(x,y)
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
40
Theorem (Dempe [5])
Assumptions:
(x, y) is a local optimal solution of (1), (4)
(1) is convex, satisfying (MFCQ), (SSOC), and (CRCQ) at (x, y).
Then the following optimization problem has a nonnegative optimal
objective function value:
α
0
∇x F (x, y)r + ∇y F (x, y)y (x; r)
∇Gi (x)r
krk
→ min
α,r
≤
≤
≤
α
α, ∀ i : Gi (x) = 0
1.
(14)
Moreover, if (MFCQ) is satisfied for the upper level problem, problem (14)
can be replaced with
∇x F (x, y)r + ∇y F (x, y)y 0 (x; r) → min
r
∇Gi (x)r
krk
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
≤
≤
(15)
0, ∀ i : Gi (x) = 0
1.
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
41
Strongly stable lower level solution
Theorem (Dempe [5])
Assumptions:
(x, y) is feasible for (1), (4)
problem (1) is convex, (MFCQ), (SSOC), and (CRCQ) are satisfied at (x, y).
If the optimal function value v1 of the problem
∇x F (x, y)r + ∇y F (x, y)y 0 (x; r) → min
∇Gi (x)r ≤ 0, ∀ i : Gi (x) = 0,
krk = 1
(16)
is strongly greater than zero v1 > 0, then (x, y) is a strict local optimal
solution of the bilevel problem, i.e. for arbitrary c ∈ (0, v1 ) there is ε > 0
such that
F (x, y(x)) ≥ F (x, y) + ckx − xk
for all x satisfying kx − xk ≤ ε, G(x) ≤ 0.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
42
KKT-reformulation
Use of KKT conditions of the lower level problem
Assumption: Lower level problem is convex parametric optimization
problem, regular.
F (x, y) → min
x,y,u
G(x) ≤ 0,
∇y L(x, y, u) = 0,
(17)
g(x, y) ≤ 0,
u ≥ 0,
u> g(x, y) = 0.
This problem is a special case of a mathematical program with equilibrium
constraints (MPEC / MPCC).
Both problems are equivalent if global optimal solutions are searched for.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
43
KKT-reformulation
Question: Is the following solution approach possible?
To solve the bilevel optimization problem, transform it into the MPEC
(MPCC), solve the MPEC and interpret the obtained (local or stationary)
solution as a (local or stationary) solution of the bilevel problem.
Hint: The MPEC is a nonconvex optimization problem, usual algorithms
compute (in the best case) local optimal solutions.
Example (Dempe, J. Dutta, 2009)
Consider the linear lower level problem
min{−y : x + y ≤ 1, −x + y ≤ 1}
y
(18)
and the upper level problem
min{(x − 1)2 + (y − 1)2 : (x, y) ∈ gph Ψ}
November
27, 2013 has the unique optimal solution (x, y) = (0.5, 0.5) and no
This problem
S. Dempe
Fak. f. Math. u. Inf.
CIMPA school,
44
University
Delhi
Bilevel Optimization Problem
TU Bergakademie Freiberg
optimalofsolutions.
(19)
local
KKT-reformulation
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
45
KKT-reformulation
Example
Consider the point (x0 , y 0 ) = (0, 1). Then,

{(1, 0)}

{(0, 1)}
Λ(x0 , y 0 ) =

conv {(1, 0), (0, 1)}
if x > 0
if x < 0
if x = 0
where conv A denotes the convex hull of the set A.
Take (x0 , y 0 , u01 , u02 ) = (0, 1, 0, 1)
u2 > 0 implies y = x + 1 ⇒ x ≤ 0 ⇒
(x − 1)2 + (y − 1)2 = (x − 1)2 + x2 ≥ 1
F (0, 1) = 1
Hence, this point is a local optimal solution of the MPEC but does not
correspond to a local optimal solution of the bilevel optimization problem.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
46
KKT-reformulation
Theorem (Dempe, J. Dutta, 2009)
Let the lower level problem (1) be convex, Slater’s constraint qualification
be satisfied at the point x. Then, the point (x, y) is a local optimal solution
of problem (1), (4) if and only if (x, y, u) is a local optimal solution for
problem (17) for all u ∈ Λ(x, y).
Here, Λ(x, y) is the set of regular Lagrange multipliers.
Hence, we need to compute Λ(x, y) and check an infinite number of points
for optimality!
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
47
KKT-reformulation
Corollary (Dempe, J. Dutta, 2009)
If the constant rank constraint qualification is added to the assumptions in
the above Theorem local optimality of (x, y, u) for all vertices u ∈ Λ(x, y)
for problem (17) implies local optimality of (x, y) for the bilevel
optimization problem (1), (4).
This is a finite number of points.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
48
KKT-reformulation
This difficulty is related to nonunique Lagrange multipliers in the lower level
problem at a local optimal solution.
The need to consider more than one feasible point of the MPEC for local
optimality corresponds to nonuniqueness of the Lagrange multiplier for the
lower level problem.
Is LICQ for the lower level problem at a local optimal solution for the bilevel optimization problem
generically satisfied?
NO!
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
49
KKT-reformulation
Example (Dempe, Dutta, 2009)
Consider the lower level problem
ΨL (x) := Argmin {−y − z : x + y ≤ 1, −x + y ≤ 1, 0 ≤ z ≤ 1}
y,z
and the bilevel optimization problem
min{0.5x − y + 3z : (y, z) ∈ ΨL (x)}.
Unique optimal solution of the bilevel problem is
(x, y, z) = (0, 1, 1).
z = 1 for all parameter values. Fix this value and consider the problem only
with variables x, y.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
50
KKT-reformulation
y
lower level objective with minimization direction
optimal solution of bilevel problem
upper level objective with minimization direction
set M
x
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
51
KKT-reformulation
Remark
The (MFCQ) is violated at every feasible solution of problem (17), see
Scheel and Scholtes [26].
Need other regularity conditions, Karush-Kuhn-Tucker conditions will in
general not be satisfied or set of Lagrange multipliers is not bounded.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
52
KKT-reformulation
Use the following sets:
1. IG (x) = {j : Gj (x) = 0},
2. I−0 (x, y, λ) = {i : gi (x, y) < 0, λi = 0},
3. I00 (x, y, λ) = {i : gi (x, y) = 0, λi = 0},
4. I0+ (x, y, λ) = {i : gi (x, y) = 0, λi > 0}.
L(x, y, λ) = ∇y f (x, y) + λ> ∇y g(x, y)
∇x F (x, y) + α> ∇G(x) + β > ∇x g(x, y) + ∇x L(x, y, λ)γ = 0,
(20)
∇y F (x, y) + β > ∇y g(x, y) + ∇y L(x, y, λ)γ = 0,
(21)
>
α ≥ 0, α G(x) = 0,
(22)
∇y gI0+ (x,y,λ) (x, y)γ = 0, βI−0 (x,y,λ) = 0,
(23)
where tI = 0 for a system of inequalities ti ≥ 0 and an index set I means
{ti = 0 ∀ i ∈ I}.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
53
KKT-reformulation
Definition
A feasible solution (x, y, λ) for the problem (17) for which there exists a
vector (α, β, γ) ∈ Rq × Rp × Rm such that
1. conditions (20)-(23) are satisfied is called weakly stationary.
2. conditions (20)-(23) together with
βi ≥ 0 or ∇y gi (x, y)> γ ≥ 0 ∀ i ∈ I00 (x, y, λ)
are satisfied, is an A-stationary solution.
3. equations (20)-(23) together with
βi ∇y gi (x, y)> γ ≥ 0 ∀ i ∈ I00 (x, y, λ)
hold, is C-stationary.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
54
KKT-reformulation
Definition (Continuation)
1. equations (20)-(23) hold together with
(βi > 0 ∧ ∇y gi (x, y)> γ > 0) ∨ βi ∇y gi (x, y)> γ = 0 ∀ i ∈ I00 (x, y, λ),
is a M-stationary solution.
2. conditions (20)-(23) and also
βi ≥ 0 and ∇y gi (x, y)> γ ≥ 0 ∀ i ∈ I00 (x, y, λ)
hold, is a S-stationary solution.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
55
KKT-reformulation
Tightened problem of (17):
F (x, y) → min
November 27, 2013
CIMPA school,
University of Delhi
G(x)
≤
0
0
=
∇y f (x, y) + λ> ∇y g(x, y)
λi
=
0, ∀ i ∈ I−0 (x, y, λ) ∪ I00 (x, y, λ)
gi (x, y)
=
0, ∀ i ∈ I0+ (x, y, λ) ∪ I00 (x, y, λ)
λi
≥
0, i ∈ I0+ (x, y, λ)
gi (x, y)
≤
0, i ∈ I−0 (x, y, λ).
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
(24)
56
KKT-reformulation
Definition
The problem (17) is said to satisfy at the point (x, y, λ) the
1. (MPEC-LICQ ) if the (LICQ) is satisfied for the problem (24).
2. (MPEC-MFCQ) if the (MFCQ) is satisfied for the problem (24).
Remark (Scholtes and Stöhr [29])
The MPEC-LICQ is generically satisfied for problem (17).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
57
KKT-reformulation
Theorem
Let (x, y, λ) be a local optimal solution of problem (17) and assume that the
(MPEC-MFCQ) is satisfied there. Then, (x, y, λ) is a C-stationary point of
(17).
Proof:
F (x, y) → min
November 27, 2013
CIMPA school,
University of Delhi
G(x)
≤
0
L(x, y, λ)
=
0
min{λi , −gi (x, y)}
=
0, ∀ i = 1, . . . , p.
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
58
KKT-reformulation
The generalized derivative in Clarke’s sense of a locally Lipschitz continuous
function z : Rn 7→ R is (by using Rademacher’s theorem)
∂ Cl z(x) = conv { lim ∇x z(xk ) : lim xk = x, ∇x z(xk ) exists ∀k}.
k→∞
∂ Cl si (x, y, λ) =
November 27, 2013
CIMPA school,
University of Delhi
k→∞


−(∇gi (x, y), 0),
(0, ei> ),

conv {−(∇gi (x, y), 0), (0, ei> )},
S. Dempe
Bilevel Optimization Problem
(25)
if i ∈ I0+ (x, y, λ),
if i ∈ I−0 (x, y, λ),
if i ∈ I00 (x, y, λ).
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
59
KKT-reformulation
Theorem (Dempe and Zemkoho [13])
Let (x, y, λ) be a local optimal solution of problem (17) and assume that the
constraint qualification




α> ∇G(x) + β > ∇x g(x, y) + ∇x L(x, y, λ)γ = 0, 





>

β ∇y g(x, y) + ∇y L(x, y, λ)γ = 0,


 α=0
α ≥ 0, α> G(x) = 0,
β = 0 (26)
⇒



γ=0

∇y gI0+ (x,y,λ) (x, y)γ = 0, βI−0 (x,y,λ) = 0,




(βi > 0 ∧ ∇y gi (x, y)γ > 0) ∨ βi ∇y gi (x, y)γ = 0 



∀ i ∈ I00 (x, y, λ)
is satisfied there. Then, there exists (α, β, γ) with k(α, β, γ)k ≤ r for some
r < ∞ such that the point is a M-stationary solution.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
60
KKT-reformulation
Proof:
N̂A (z) := {d : d> (z − z) ≤ o(kz − zk) ∀ z ∈ A}
is the Fréchet normal cone to the set A at z. Here, o(t) is a function
satisfying lim o(t)/t = 0. The Mordukhovich normal cone to A at z is the
t→0
Kuratowski-Painlevé upper limit of the Fréchet normal cone, i.e.
k ∞
k
NAM (z) = {r : ∃ {z k }∞
k=1 ⊆ A, ∃{r }k=1 with lim z = z,
k→∞
lim rk = r, rk ∈ N̂A (z k ) ∀ k}.
k→∞
The Mordukhovich subdifferential of a lower semicontinuous function
f : Rn → R at some point x ∈ dom f is
∂ M f (x) := {z ∗ ∈ Rn : (z ∗ , −1) ∈ Nepif (x, f (x))},
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
61
KKT-reformulation
Theorem (Mordukhovich [19])
Let f : Rp → Rq be Lipschitz continuous around x and g : Rq → R be
Lipschitz continuous around y = f (x) ∈ dom g. Then,
[
∂ M (g ◦ f )(x) ⊆ {∂ M hw, f (x)i : w ∈ ∂ M g(y)}.
Theorem (Mordukhovich [20])
Let the functions f, g : Rn → R be locally Lipschitz continuous around
z ∈ Rn . Then,
∂ M (f + g)(z) ⊆ ∂ M f (z) + ∂ M g(z).
Equality holds if one of the functions is continuously differentiable.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
62
KKT-reformulation
Theorem (Rockafellar and Wets [25])
Let f : Rn → R be a locally Lipschitz continuous function and A ⊆ Rn a
closed set. If z ∈ A is a local minimizer of the function f over A, then
0 ∈ ∂ M f (z) + NAM (z).
Let
Θ = {(a, b) ∈ R2p : a ≥ 0, b ≥ 0, a> b = 0}.
Then, it can be shown, see Flegel and Kanzow [14], that


 ∗
ui = 0,
∀ i : ui > 0 = v i 

M
∀ i : ui = 0 < v i
.
NΘ
(u, v) = (u∗ , v ∗ ) :  vi∗ = 0,


(u∗i < 0 ∧ vi∗ < 0) ∨ u∗i vi∗ = 0, ∀ i : ui = v i = 0
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
63
Transformation using the optimal value function
Replace problem (1), (4) with
F (x, y) → min
x,y,u
G(x) ≤ 0,
(27)
f (x, y) ≤ ϕ(x)
g(x, y) ≤ 0
Problems (1), (4) and (27) are fully equivalent (both w.r.t. local and global
optima).
Problem (27) is a nondifferentiable nonconvex optimization problem.
Regularity conditions as nonsmooth MFCQ are violated at every feasible
point, see e.g. Pilecka [23].
To formulate optimality conditions and to describe solution algorithms for it
we can use nonsmooth analysis.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
64
Transformation using the optimal value function
Definition (J.J. Ye, D.L. Zhu, 1995 [32])
Problem (27) is called partially calm at (x0 , y 0 ) with
y 0 ∈ Ψ(x0 ), G(x0 ) ≤ 0, g(x0 , y 0 ) ≤ 0,
if ∃ λ > 0 and ∃ U of (x0 , y 0 , 0) ∈ Rn × Rm × R:
∀ (x, y, u) ∈ U , feasible for problem
min{F (x, y) : f (x, y) − ϕ(x) + u = 0, G(x) ≤ 0, g(x, y) ≤ 0}
x,y
we have
F (x, y) − F (x0 , y 0 ) + λ|u| ≥ 0.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
65
Transformation using the optimal value function
Theorem (J.J. Ye, D.L. Zhu, 1995 [32])
Problem (27) is partially calm at a local optimal solution (x∗ , y ∗ ) if and
only if there is λ∗ such that (x∗ , y ∗ ) solves the penalized problem
min{F (x, y) + λ(f (x, y) − ϕ(x)) : g(x, y) ≤ 0, G(x) ≤ 0}
x,y
(28)
for all λ ≥ λ∗ .
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
66
Transformation using the optimal value function
Definition (Mordukhovich [21])
A point-to-set mapping Γ : Rn ⇒ Rm is said to be inner semicontinuous at
(z, α) ∈ gph Γ provided that, for each sequence {z k }∞
k=1 converging to z
k
k
there is a sequence {αk }∞
∈
Γ(z
)
converging
to
α.
,
α
k=1
Theorem (Dempe, Dutta, Mordukhovich [10])
Assumptions:
(x, y) is a local optimal solution for problem (27)
F, f, gi , Gj ∈ C 1
(MFCQ) is satisfied for both the problems of the follower and the leader at
(x, y)
partial calmness of (27) at (x, y)
inner semicontinuity of Ψ(·) at x:
A point-to-set mapping S : Rn ⇒ Rm is inner semicontinuous at
(x, y) ∈ gph S, if: ∀ xν → x ∃ yν ∈ S(xν ) converging to y.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
67
Theorem (continuation)
Then, ∃ λ > 0, λis , i = 1, . . . , p, s = 1, . . . , n + 1, µi , i = 1, . . . , p, ηs ,
s = 1, . . . , n + 1, αj , j = 1, . . . , k such, that:
p
p
n+1
P
P
P
∇x F (x, y) +
ηs
λis ∇x gi (x, y)
µi ∇x gi (x, y) − λ
s=1
i=1
i=1
k
P
+
αj ∇Gj (x) = 0,
j=1
∇y F (x, y) + λ∇y f (x, y) +
∇y f (x, y) +
p
P
p
P
µi ∇y gi (x, y) = 0,
i=1
λis ∇y gi (x, y) = 0 ∀ s = 1, . . . , n + 1.
i=1
λis ≥ 0,
λis gi (x, y) = 0 ∀ i = 1, . . . , p, s = 1, . . . , n + 1,
µi gi (x, y) = 0 ∀ i = 1, . . . , p,
n+1
P
ηs ≥ 0 ∀ s = 1, . . . , n + 1,
ηs = 1,
µi ≥ 0,
s=1
αj ≥ 0, αj Gj (x) = 0 ∀ j = 1, . . . , k.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
68
Transformation using the optimal value function
Theorem (Gauvin and Dubeau [15])
Consider problem (1), let the set {(x, y) : g(x, y) ≤ 0} be nonempty and
compact, f, gi ∈ C 1 (Rn × Rm , R) for all i = 1, . . . , p and assume that
(MFCQ) is satisfied for x = x and all y ∈ Y (x). Then, the function ϕ(·) is
locally Lipschitz continuous at x and
[
[
∂ Cl ϕ(x) ⊆ conv
∇x L(x, y, λ).
(29)
y∈Ψ(x) λ∈Λ(x,y)
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
69
First solution algorithms
Topics:
1. Strongly stable lower level solution
2. KKT-reformulation
3. Transformation using the optimal value function
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
70
Strongly stable lower level solution
F(x) := F (x, y(x)) → min
x
G(x) ≤ 0.
(30)
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
71
Strongly stable lower level solution
Descent algorithm for the bilevel problem (Dempe and Schmidt [12]):
Input: Bilevel optimization problem (30).
Output: A Clarke stationary solution.
Step 1: Select x0 satisfying G(x0 ) ≤ 0, set k := 0,
choose ε, δ ∈ (0, 1).
Step 2: Compute a direction rk , krk k ≤ 1, satisfying
Algorithm:
F 0 (xk ; rk ) ≤ sk , ∇Gi (xk )rk ≤ −Gi (xk ) + sk , i = 1, . . . , q,
and sk < 0.
Step 3: Choose a step-size tk such that
F(xk + tk rk ) ≤ F(xk ) + εtk sk , G(xk + tk rk ) ≤ 0.
Step 4: Set xk+1 := xk + tk rk , compute the optimal solution
y k+1 = y(xk+1 ) ∈ Ψ(xk+1 ), set k := k + 1.
Step 5: If a stopping criterion is satisfied stop, else
goto step 2.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
72
Strongly stable lower level solution
L(x, y, λ) = f (x, y) + λ> g(x, y)
(FRR) For each vertex λ0 ∈ Λ(x, y) the matrix
2
2
0
∇yy L(x, y, λ0 ) ∇>
y gJ(λ0 ) (x, y) ∇xy L(x, y, λ )
M :=
∇y gI0 (x, y)
0
∇x gI0 (x, y)
has full row rank m + |I0 |.
Here I0 = {i : gi (x, y) = 0} and J(λ) = {i : λi > 0}.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
73
Strongly stable lower level solution
Theorem (Dempe and Schmidt [12])
Consider problem (30), where y(x) is an optimal solution of the lower level
problem (1) which is a convex parametric optimization problem. Assume
that all functions F, f, G, g are sufficiently smooth and the set
{(x, y) : G(x) ≤ 0, g(x, y) ≤ 0} is nonempty and bounded. Let the
assumptions (FRR), (MFCQ), (CRCQ), and (SSOC) for all
(x, y), y ∈ Ψ(x), G(x) ≤ 0 be satisfied for problem (1) and (MFCQ) is
satisfied with respect to G(x) ≤ 0 for all x. Then, each accumulation point
(x, y) of the sequence of iterates is Clarke stationary.
Proof:
1. (MFCQ), (CRCQ), and (SSOC) imply: optimal solution lower level
problem is P C 1 -function, problem (30) is Lipschitz continuous problem.
2. algorithm realized algorithm of feasible directions (Topkis and Veinott
[31])
3. Step-size rule is Armijo step-size rule, see e.g. Bazaraa et al. [3]
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
74
Strongly stable lower level solution
Algorithm converges to Clarke stationary solution. Need not to be
Bouligand stationary.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
75
Strongly stable lower level solution
Theorem (Ralph and Dempe [24])
Consider problem (1) at a point x = x and let y be a local optimal solution
of this problem where the assumptions (MFCQ), (SSOSC), and (CRCQ) are
satisfied. Then the directional derivative of the function y(·) at x in
direction r coincides with the unique optimal solution of the convex
quadratic optimization problem QP (λ0 , r)
0.5d> ∇2yy L(x, y, λ0 )d + d> ∇2xy L(x, y, λ0 )r → min
d
0
= 0, if λi > 0,
∇x gi (x, y)r + ∇y gi (x, y)d
≤ 0, if gi (x, y) = λ0i = 0
(31)
for an arbitrary Lagrange multiplier λ0 ∈ Λ(x, y) solving
∇x L(x, y, λ)r →
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
max .
(32)
λ∈Λ(x,y)
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
76
Strongly stable lower level solution
Idea of an bundle algorithm:
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
77
Strongly stable lower level solution
∂F(x0 ) = {∇x F (x0 , y(x0 )) + ∇y F (x0 , y(x0 ))∂y(x0 )},
i
i
k
i
(33)
i
max {v(x )d + v(x )(z − x ) + F(x )}
(34)
1 >
d d,
2tk
(35)
1≤i≤k
with d = x − z k .
max {v(xi )d − αik } + F(xk ) +
1≤i≤k
with
αik = F(z k ) − v(xi )(z k − xi ) − F(xi )
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
(36)
78
Strongly stable lower level solution
∂z(x0 ) = conv { lim ∇z(xk ) : lim xk = x0 , ∇z(xk ) exists ∀k}.
k→∞
k→∞
Theorem ([16, 27])
Let z : Rp → R be a P C 1 -function. Then,
∂z(x0 ) = conv {∇z i (x0 ) : x0 ∈ cl int supp (z, z i )}.
Iz0 (x0 ) = {i : x0 ∈ cl int supp (z, z i )}.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
(37)
79
Strongly stable lower level solution
Corollary
The local optimal solution function x(·) is
locally Lipschitz continuous, the generalized Jacobian in the sense of
Clarke is given by
∂y(x0 ) = conv {∇y I (x0 ) : I ∈ Iy0 (x0 )},
directionally differentiable with
y 0 (x0 ; r) ∈ {∇y I (x0 )r : I ∈ Iy0 (x0 )},
B-differentiable.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
80
Strongly stable lower level solution
(C1) ∃λ ∈ EΛ(x0 , y 0 ) such that J(λ) ⊆ I ⊆ I(x0 , y 0 )
(C2) The gradients {∇y gi (x0 , y 0 ) : i ∈ I} are linearly independent.
EΛ(x0 , y 0 ): vertex set of Λ(x0 , y 0 )
y I (·): selection functions of y(·)
I(λ0 ): family of all sets I satisfying both conditions (C1) and (C2) for a
fixed vertex λ0 ∈ Λ(x0 , y 0 )
(SCR) For a vertex (λ0 ) ∈ Λ(x0 , y 0 ), a set I ∈ I(λ0 ) and a direction r strict
complementarity slackness is satisfied in problem QP (λ0 , µ0 , r).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
81
Strongly stable lower level solution
Theorem
Consider the problem (1) at a point (x0 , y 0 ) and let the assumptions
(MFCQ), (SSOC), and (CRCQ) be satisfied there.
[6] Take any vertex (λ0 ) ∈ Λ(x0 , y 0 ) and a set I ∈ I(λ0 ). Let r0 be a
direction such that condition (SCR) is satisfied. Then
∇y I (x0 ) ∈ ∂y(x0 ).
[7] If condition (FRR) is valid then
[
∂y(x0 ) = conv
[
(λ0 )∈EΛ(x0 ,y 0 )
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
∇y I (x0 ).
I∈I(λ0 )
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
82
Strongly stable lower level solution
Schematic step in the inner iteration in the bundle trust region algorithm
(Outrata et al. [22])
Algorithm: Input: Sequences of iterates {z i }ki=1 and trial points {y i }ki=1 , a
regularization parameter tk .
Output: A new sufficiently better iteration point z k+1 or an improved
model.
1. Compute an optimal solution dk of (35). Set y k+1 = z k + dk .
2. If F(y k+1 ) is sufficiently smaller than F(z k ) then either
2.1 enlarge tk and go back to 1. or
2.2 make a serious step: Set z k+1 = y k+1 .
3. If F(y k+1 ) is not sufficiently smaller than F(z k ) then either
3.1 reduce tk and go back to 1. or
3.2 make a null step: Set z k+1 = z k , compute v(y k+1 ) ∈ ∂F(y k+1 ).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
83
Algorithm: Input: A convex function F(y) to be minimized.
Output: A Clarke stationary point.
1. Choose a starting point z 1 ∈ Rm and parameters
T > 0, 0 < m1 < m2 < 1, 0 < m3 < 1, ν > 0, ε > 0 and an upper
bound jmax ≥ 3 for the maximal number of subgradients in the bundle.
2. Compute F(z 1 ) and a subgradient v(z 1 ) ∈ ∂F(z 1 ). Set
k := 1, J1 := {1}.
3. Apply the inner iteration algorithm either to compute a new (trial or
iteration) point z k+1 and a new subgradient v(y k+1 ) or realize that z k
is ε-optimal.
4. If |Jk | = jmax reduce the bundle in Step 5. Else goto Step 6.
5. Choose a subset J ⊂ Jk with |J| ≤ jmax − 2 and
max{i ∈ Jk : αik = 0} ∈ J.
Introduce some additional index k̃ and set
v(y k̃ ) := %k , αk̃k := σk , J := J ∪ k̃.
6. Compute αi,k+1 for all i, set Jk+1 := J ∪ {k + 1}, k := k + 1 and goto
Step 2.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
84
Strongly stable lower level solution
Theorem ([30])
If the function F(·) is bounded below and the sequence {z k }∞
k=1 computed
by the above algorithm remains bounded, then there exists an accumulation
Cl
point z of {z k }∞
k=1 such that 0 ∈ ∂ F(z).
Corollary
If the assumptions (C), (MFCQ), (CRCQ), (SSOC), and (FRR) are
satisfied for the convex lower level problem at all points (x, y), y ∈ Ψ(x),
the sequence of iteration points {z k }∞
k=1 remains bounded, then the bundle
algorithm computes a sequence {z k }∞
k=1 having at least one accumulation
point z ∈ Y with 0 ∈ ∂ Cl F(z).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
85
Strongly stable lower level solution
The Theorem by J. Ye gives us the possibility to use a
combination of penalization approach and bundle algorithm
to solve problem (28), see Mehlitz [17].
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
86
KKT-reformulation
F (x, y) → min
x,y,u
G(x) ≤ 0,
∇y L(x, y, u) = 0,
(38)
g(x, y) ≤ 0, u ≥ 0,
u> g(x, y) = 0.
MFCQ violated at every feasible point
F (x, y) → min
x,y,u
G(x) ≤ 0, ∇y L(x, y, u) = 0,
(39)
g(x, y) ≤ 0, u ≥ 0,
− u> g(x, y) ≤ δ
for δ ↓ 0.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
87
KKT-reformulation
Algorithm converges to stationary solutions of the MPEC (17), see Scholtes
and colleagues [28].
Unfortunately, stationary solutions of (17) are in general not related to
stationary solutions of (4).
Way out: Enlarge the feasible set again:
F (x, y) −→
November 27, 2013
CIMPA school,
University of Delhi
min
x,y,λ
G(x)
≤
0, k∇y L(x, y, λ)k∞ ≤ g(x, y)
≤
0, λ ≥ 0
−λi gi (x, y)
≤
, i = 1, . . . , p
S. Dempe
Bilevel Optimization Problem
(40)
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
88
KKT-reformulation
Theorem (Ayalew Getachew Mersha 2008[18])
A method of feasible directions can be described solving problem (40) for
k ↓ 0 converging to a Bouligand-stationary point of the bilevel optimization
problem (1), (4) provided that (MFCQ) is satisfied for the upper level
constraints and the lower level problem is a convex parametric optimization
problem satisfying (MFCQ), (CRCQ) and (SSOC) at the limit point.
Another possibility: Use the elastic mode SQP (Anitescu [1]).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
89
Transformation using the optimal value function
F (x, y) → min
x,y,u
G(x) ≤ 0,
(41)
f (x, y) ≤ ϕ(x)
g(x, y) ≤ 0
Theorem (Ye and Zhu [32], Pilecka [23])
If the function ϕ(·) is locally Lipschitz continuous and the functions
F, G, f, g are at least differentiable, then the (nonsmooth)
Mangasarian-Fromovitz constraint qualification is violated at every feasible
point of the problem (41).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
90
Transformation using the optimal value function
Case of jointly convex lower level problem:
Let {x : G(x) ≤ 0} ⊆ X, X be polyhedron and |ϕ(x)| < ∞ for all x ∈ X.
Then, ϕ(·) is convex on X.
X : set of all vertices of X.
s
s
s
P
P
P
∀ x ∈ X ∃ µi ≥ 0,
µi = 1 : x =
µi xi , ϕ(x) ≤
µi ϕ(xi ).
i=1
ξ(x) = min
µ
( s
X
i=1
i
µi ϕ(x ) : µ ≥ 0,
i=1
i=1
s
X
µi = 1, x =
i=1
s
X
)
µi x
i
.
(42)
i=1
F (x, y) → min
x,y
November 27, 2013
CIMPA school,
University of Delhi
G(x)
≤
0
f (x, y)
≤
ξ(x)
g(x, y)
≤
0
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
(43)
91
Transformation using the optimal value function
Start Compute the set X of all vertices of the
set X, compute the function ξ(·), t := 1.
Step 1 Solve problem (43). Let (xt , y t ) be a global optimal
solution.
Step 2 If y t ∈ Ψ(xt ), stop, (xt , y t ) is the solution of the
problem (41). Otherwise, set X := X ∪ {xt }, compute
ϕ(xt ), update ξ(·), set t := t + 1 and goto Step 1.
Algorithm:
Theorem
Assume that the above algorithm computes an infinite sequence
{(xt , y t )}∞
t=1 , that the set {(x, y) : G(x) ≤ 0, g(x, y) ≤ 0} is compact and
that ∀ x
e ∈ X there exists a point (e
x, ye) such that g(e
x, ye) < 0. Then,
1. the sequence {(xt , y t )}∞
t=1 has accumulation points (x, y),
2. each accumulation point of this sequence is a globally optimal solution
of problem (41).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
92
Transformation using the optimal value function
Case of linear lower level problem:
>
min {a>
1 x + a2 y : Ax ≤ b, y ∈ Ψ(x)}
(44)
Ψ(x) := Argmin {x> y : By ≤ d}.
(45)
x,y
y
Theorem ([4])
The function ϕ(·) is a piecewise linear concave function over
Q := {x : |ϕ(x)| < ∞}. Moreover, Q is a convex polyhedron.
v i ∈ ∂ϕ(xi ), i = 1, . . . , t
∂ϕ(z) := {α : ϕ(x) ≤ ϕ(z) + α> (x − z) ∀ x ∈ Rn }.
{(x, y) : Ax ≤ b, x> y ≤ ϕ(x), By ≤ d}
⊆ T := {(x, y) : Ax ≤ b, By ≤ d, x> y ≤ ϕ(xi ) + v i> (x − xi ), i = 1, . . . , t}
>
min {a>
1 x + a2 y : (x, y) ∈ T }.
(46)
x,y
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
93
Transformation using the optimal value function
>
>
>
min {a>
1 x + a2 y : Ax ≤ b, x y ≤ min x z, By ≤ d}
x,y
(47)
z∈Z
with Z ⊆ {y : By ≤ d}.
Theorem (Dempe and Franke [11])
Let the feasible point (x, y) be a global optimal solution of (47) for some
Z ⊆ {y : By ≤ d}. Denote the feasible set of the bilevel programming
problem by TB := {(x, y) : Ax ≤ b, y ∈ Ψ(x)}. If (x, y) ∈ TB , then it is also
globally optimal for problem (44).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
94
Transformation using the optimal value function
Algorithm: Global optimality
Step 0 Choose a vector for Z (see following Remark). Set k := 1.
Step 1 Solve problem (47) globally. The optimal solution is (xk , y k ).
Step 2 If (xk , y k ) is feasible for (44), stop. Otherwise, compute an optimal
k
solution z
ofthe lower level problem with the parameter xk . Set
Z := Z ∪ z k , k := k + 1 and go to Step 1.
Remark
One possible way for finding the first vector for Z is solving the surrogate
problem (47) without any additional constraints, i.e. ignoring the lower level
objective function. Concerning Step 2, the lower level problem is solved in
order to compute a supergradient of the optimal value function ϕ(·).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
95
Transformation using the optimal value function
Theorem
Assume that the set M = {(x, y) : Ax ≤ b, By ≤ d} is bounded. Then,
every accumulation point (x, y) of the sequence {(xk , y k )}∞
k=1 produced by
the last Algorithm is globally optimal for (44).
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
96
[1]
M. Anitescu, On using the elastic mode in nonlinear programming
approaches to mathematical programs with complementarity constraints,
SIAM Journal on Optimization 15 (2006), 1203–1236.
[2]
B. Bank, J. Guddat, D. Klatte, B. Kummer, and K. Tammer, Non-linear
parametric optimization., Basel - Boston - Stuttgart: Birkhäuser Verlag, 1983.
[3]
Mokhtar S. Bazaraa, Hanif D. Sherali, and C. M. Shetty, Nonlinear
programming. Theory and algorithms. 3rd ed., Hoboken, NJ: John Wiley
&amp; Sons. xv, 853 p. , 2006 (English).
[4]
K. Beer, Lösung großer linearer Optimierungsaufgaben, Deutscher Verlag der
Wissenschaften, Berlin, 1977.
[5]
S. Dempe, A necessary and a sufficient optimality condition for bilevel
programming problems, Optimization 25 (1992), 341–354.
[6]
, On generalized differentiability of optimal solutions and its
application to an algorithm for solving bilevel optimization problems, Recent
advances in nonsmooth optimization (D.-Z. Du, L. Qi, and R.S. Womersley,
ed.), World Scientific Publishers, Singapore, 1995, pp. 36–56.
[7]
, An implicit function approach to bilevel programming problems,
Multilevel Optimization: Algorithms and Applications (A. Migdalas, P.M.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
96
Pardalos, and P. Värbrand, eds.), Kluwer Academic Publishers, Dordrecht,
1998, pp. 273–294.
[8]
, Foundations of bilevel programming, Kluwer Academie Publishers,
Dordrecht, 2002.
[9]
, Annotated bibliography on bilevel programming and mathematical
programs with equilibrium constraints, Optimization 52 (2003), 333–359.
[10] S. Dempe, J. Dutta, and B.S. Mordukhovich, New necessary optimality
conditions in optimistic bilevel programming, Optimization 56 (2007),
577–604.
[11] S. Dempe and S. Franke, Solution algorithm for an optimistic linear
Stackelberg problem, Computers & Operations Research 41 (2014), 277–281.
[12] S. Dempe and H. Schmidt, On an algorithm solving two-level programming
problems with nonunique lower level solutions, Computational Optimization
and Applications 6 (1996), 227–249.
[13] S. Dempe and A. B. Zemkoho, On the Karush-Kuhn-Tucker reformulation of
the bilevel optimization problem, Nonlinear Analysis: Theory, Methods &
Applications 75 (2012), 1202–1218.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
96
[14] M. L. Flegel and C. Kanzow, A direct proof for M-stationarity under
MPEC-GCQ for mathematical programs with equilibrium constraints,
Optimization with Multivalued Mappings: Theory, Applications and
Algorithms (S. Dempe and V. Kalashnikov, eds.), Optimization and its
Applications, vol. 2, Springer Science+Business Media, LLC, New York,
2006, pp. 111–122.
[15] J. Gauvin and F. Dubeau, Differential properties of the marginal function in
mathematical programming, Mathematical Programming Study 19 (1982),
101–119.
[16] B. Kummer, Newton’s method for non-differentiable functions, Advances in
Mathematical Optimization, Mathematical Research, vol. 45,
Akademie-Verlag, Berlin, 1988.
[17] P. Mehlitz, Elemente nichtglatter Analysis in der Zwei-Ebenen-Optimierung,
Bachelorarbeit, TU Bergakademie Freiberg, 2011.
[18] A. G. Mersha, Solution methods for bilevel programming problems, Ph.D.
thesis, TU Bergakademie Freiberg, 2008.
[19] B.S. Mordukhovich, Generalized differential calculus for nonsmooth and
set-valued mappings, Journal of Mathematical Analysis and Applications 183
(1994), 250–288.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
96
[20]
, Generalized differential calculus for nonsmooth and set-valued
mappings, Journal of Mathematical Analysis and Applications 183 (1994),
250–288.
[21]
, Variational analysis and generalized differentiation, vol. 1: Basic
theory, Springer Verlag, Berlin et al., 2006.
[22] J. Outrata, M. Kočvara, and J. Zowe, Nonsmooth approach to optimization
problems with equilibrium constraints, Kluwer Academic Publishers,
Dordrecht, 1998.
[23] M. Pilecka, Combined reformulation of bilevel programming problems,
Master’s thesis, TU Bergakademie Freiberg, Fakultät für Mathematrik und
Informatik, 2011.
[24] D. Ralph and S. Dempe, Directional derivatives of the solution of a
parametric nonlinear program, Mathematical Programming 70 (1995),
159–172.
[25] R. T. Rockafellar and R. J.-B. Wets, Variational analysis, Springer Verlag,
Berlin, 1998.
[26] H. Scheel and S. Scholtes, Mathematical programs with equilibrium
constraints: stationarity, optimality, and sensitivity, Mathematics of
Operations Research 25 (2000), 1–22.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
96
[27] S. Scholtes, Introduction to piecewise differentiable equations, Tech. report,
Universität Karlsruhe, Institut für Statistik und Mathematische
Wirtschaftstheorie, 1994, No. 53/1994,
http://www.eng.cam.ac.uk/~ss248/publications/index.html.
[28]
, Convergence properties of a regularization scheme for mathematical
programs with complementarity constraints, SIAM J. on Optimization 11
(2001), 918–936.
[29] S. Scholtes and M. Stöhr, How stringent is the linear independence
assumption for mathematical programs with stationarity constraints?,
Mathematics of Operations Research 26 (2001), 851–863.
[30] H. Schramm, Eine kombination von bundle- und trust-region-verfahren zur
lösung nichtdifferenzierbarer optimierungsprobleme, Bayreuther
Mathematische Schriften, Bayreuth, No. 30, 1989.
[31] D.M. Topkis and A. F.jun. Veinott, On the convergence of some feasible
direction algorithms for nonlinear programming., SIAM J. Control 5 (1967),
268–279.
[32] J.J. Ye and D.L. Zhu, Optimality conditions for bilevel programming
problems, Optimization 33 (1995), 9–27.
November 27, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
96
Stephan Dempe
Fakultät für Mathematik und Informatik
TU Bergakademie Freiberg
Akademiestr. 6
09596 Freiberg
Bilevel Optimization Problem:
Addendum
November 29, 2013
CIMPA school, November 25 to December 6, 2013,
University of Delhi
Contents
One special linear bilevel optimization problem
Bilevel optimization problems with vectorvalued objective functions in both
levels
Auxiliary results on multicriterial optimization
Reformulation of the lower level as a nonlinear optimization problem
Properties of the linear multiobjective bilevel optimization problem
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
1
Contents
One special linear bilevel optimization problem
Bilevel optimization problems with vectorvalued objective functions in both
levels
Auxiliary results on multicriterial optimization
Reformulation of the lower level as a nonlinear optimization problem
Properties of the linear multiobjective bilevel optimization problem
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
1
One special linear bilevel optimization problem
min{a> y + bd : y ∈ Ψ(x)}
with
Ψ(b) = Argmax {h> y : g > y ≤ b, 0 ≤ yi ≤ 1 ∀i}
y
(Dempe and Franke [4])
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
2
Lower level: maximization of a linear function on a polymatroid, can be
solved using a greedy algorithm.
Let
h1
h2
hn
≥
≥ ... ≥
g1
g2
gn
j:
j−1
X
gi ≤ b <
i=1
yi (b) =




November 29, 2013
CIMPA school,
University of Delhi
gi
i=1
i≤j−1
1,
b−
j−1
P
gi
i=1



j
X
gj
0
S. Dempe
Bilevel Optimization Problem
i=j
i>j
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
3
One special linear bilevel optimization problem
resulting function
optimal value function lower level
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
4
Semivectorial bilevel optimization problem
min {f (x, y) : g(x, y) ≤ 0},
(1)
y
where f : Rn × Rm → Rq , gi : Rn × Rm → R i = 1, . . . , p.
Ψ(x) denotes the set of optimal solutions of problem (1).
” min ” {F (x, y) : x ∈ X, y ∈ Ψ(x)},
(2)
x
where X ⊆ Rn is a closed, nonempty set and F : Rn × Rm → Rk .
In this case, as long as no feasible point exists which minimizes all objective
functions at the same time, problem (1) will not have a unique solution.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
5
Semivectorial bilevel optimization problem
First, consider a multicriterial optimization problem
” min ”{α(x) : x ∈ M },
(3)
where ∅ =
6 M ⊆ Rn is a closed set and α : Rn → Rq .
Definition
Let y, z ∈ Rq . y z if it holds that
yi ≥ zi
∀i = 1, ..., q
and ∃ q ∈ {1, ..., q} : yq > zq .
Definition
A feasible point x of (3) is called Pareto optimal if there does not exist a
feasible point x ∈ M with α(x) α(x).
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
6
Semivectorial bilevel optimization problem
Theorem (Ehrgott [5])
Let λ ∈ Rq , λi > 0 for i = 1, . . . , q. Let x ∈ M be an optimal solution of the
problem
min {λ> α(x) : x ∈ M }.
(4)
Then, x is Pareto optimal for problem (3).
Theorem (Ehrgott [5])
Let y < α(x) for all x ∈ M (e.g. let yi < min{αi (x) : x ∈ M }, i = 1, . . . , q)
and x ∈ M be an optimal solution of the problem
min {kα(x) − yk : x ∈ M },
(5)
where k · k denotes the Euclidean norm in Rq . Then, x is Pareto optimal for
problem (3).
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
7
Semivectorial bilevel optimization problem
Theorem (Ehrgott [5])
Consider problem (3) with M being a convex set and αi being convex
functions for i = 1, . . . , q. Then, for each Pareto optimal solution x ∈ M
q
P
there exists a vector λ ∈ Rq , λi ≥ 0, i = 1, . . . , q,
λi = 1 such that x is
i=1
an optimal solution of problem (4).
Using parametric linear optimization and the vertex property of optimal
solutions in linear optimization, it can be shown that λ can be found such
that λi > 0 for i = 1, . . . , q if M is a polyhedral set and all functions αi are
linear.
Theorem (Ehrgott [5])
If x ∈ M is a Pareto optimal solution of problem (3), then there exists
y ∈ Rq such that x is an optimal solution of problem (5).
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
8
Semivectorial bilevel optimization problem
”max”
x∈X
s.t.
>
>
>
>
>
c>
1 x, c2 x, ..., ck x + f1 y, f2 y, ..., fk y
A1 x + B 1 y ≤ b1
x≥0
y ∈ Ψ(x)
(6)
with the solution set mapping Ψ(x) of the follower’s problem
>
>
max d>
1 y, d2 y, ..., dp y
y∈Y
(7)
A2 x + B 2 y ≤ b2
y ≥ 0,
for ci ∈ Rn (i = 1, ..., k), fl ∈ Rm (l = 1, ..., k), dj ∈ Rm (j = 1, ..., p),
X ⊆ Rn and Y ⊆ Rm being polyhedral sets and matrices A1 , B1 , A2 , B2 as
well as vectors b1 , b2 of appropriate dimensions.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
9
Semivectorial bilevel optimization problem
f1 . . . fk ,
Let C > = c1 . . . ck , F > =
d1 . . . d p
D> =
be the matrices composed of the objective function coefficients of all the
objective functions of both the leader and the follower.
Ψ(x) =
[
Ψs (x, λ) with Λ := {λ ∈ Rp : λi > 0,
p
X
λi = 1}.
i=1
λ∈Λ
Here
Ψs (x, λ) = Argmax {λ> Dy : A2 x + B2 y ≤ b2 , y ≥ 0}.
(8)
y∈Y
Now, if the optimistic approach is used, problem (6) reduces to
max {Cx + F y : A1 x + B1 y ≤ b1 , x ≥ 0, y ∈ Ψ(x)}
x,y
=
max {Cx + F y : A1 x + B1 y ≤ b1 , x ≥ 0, y ∈ Ψs (x, λ), λ ∈ Λ}.
x,y,λ
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
10
Semivectorial bilevel optimization problem
Remark
Problem (6) has been formulated using so-called coupling constraints
A1 x + B1 y ≤ b1 . To prove their satisfaction, the leader needs the solution
selected by the follower. This implies that the leader can recognize his / her
selection as being feasible only after being told the follower’s selection.
Mathematically, this implies that the feasible solution set
{(x, y) ∈ Rn × Rm : A1 x + B1 y ≤ b1 , x ≥ 0, y ∈ Ψ(x)}
of the leader’s probem is now disconnected [6]. Shifting these constraints to
the lower level problem will modify the feasible set of the leader’s problem
[1, 6]. Hence, those constraints are difficult to handle and are, therefore,
usually avoided.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
11
Semivectorial bilevel optimization problem
Theorem
Consider the parametric multicriterial optimization problem (7) and let
[
Ψ :=
Ψ(x)
x∈Rn
+ ∩X
denote the union over all sets of Pareto optimal solutions for every the
parametric optimization problem
(7). Then, Ψ equals the union of faces of
the set T := {(x, y) ∈ Rn+ ∩ X × Rm : A2 x + B2 y ≤ b2 , y ≥ 0}.
max {Cx + F y : x ∈ X, x ≥ 0, y ∈ Ψ(x)}
x,y
(9)
= max {Cx + F y : x ∈ X, x ≥ 0, (x, y) ∈ gph Ψ},
x,y
where gph Ψ := {(x, y) : y ∈ Ψ(x)} denotes the graph of the point-to-set
mapping Ψ.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
12
Semivectorial bilevel optimization problem
Problem (9) is the problem of maximizing a linear function over a
(nonempty) set. This equals maximizing this function over the convex hull
of the set
max {Cx + F y : x ∈ X, (x, y) ∈ conv gph Ψ}.
(10)
x,y
The feasible set of problem (10) is a polyhedron, the problem itself is a
linear multiobjective optimization problem. Unfortunately, its feasible set is
not known. Nevertheless, the set of Pareto optimal solutions of this problem
equals the union of faces of the set
{(x, y) ∈ Rn × Rm : x ∈ X, (x, y) ∈ conv gph Ψ}. Applying Theorem
5, it is easy to see that the feasible set is also the union of faces of the set
{(x, y) ∈ Rn+ ∩ X × Rm : (x, y) ∈ gph Ψ}.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
13
Semivectorial bilevel optimization problem
To compute one Pareto optimal solution of problem (9) we can use a convex
combination of the upper level objective functions and solve
max {µ> (Cx + F y) : x ∈ X, x ≥ 0, (x, y) ∈ gph Ψ}
(11)
x,y
with µ ∈ M := {µ ∈ Rk : µi > 0, i = 1, . . . , k,
k
P
µ = 1}. This problem can
i=1
e.g. be solved using the k-th best algorithm [3] tailored to problem (11).
The algorithm is based on the following important property.
Lemma
[2] Let K := {(x, y) ∈ X × Rm : A2 x + B2 y ≤ b2 , x ≥ 0, y ≥ 0} be
nonempty and compact. Then, a solution of (11) occurs at an extreme point
of K.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
14
Semivectorial bilevel optimization problem
A k-th best algorithm
Inititialization. Set i := 1 and W = T = W i := ∅. Choose µ ∈ M.
Step 1. Compute a first feasible solution (xi , y i ) by dropping the lower level
objective functions and solving the resulting problem. Set W := {(xi , y i )}.
Step 2.
fixed upper level parameter xi . If
S Solve ithe lower level problem with
i
i i
y ∈
Ψs (x , λ), stop. The solution (x , y ) is globally optimal for (11).
λ∈Λ
Otherwise, go to Step 3.
Step 3. Set T := T ∪ {(xi , y i )}. W := (W ∪ W i ) \ T , increase i. Choose
(xi , y i ) with µ> (Cxi + F y i ) = max µ> (Cx + F y) and go to Step 2.
(x,y)∈W
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
15
Semivectorial bilevel optimization problem
Example
Consider the multicriterial bilevel programming problem
max
x∈R,y∈R2
s.t.
(x − 2y2 , 2x)
2 ≤ x ≤ 5,
max2 (y1 + 2y2 , y1 − y2 )
y∈R
(x, y) ∈ K
with
November 29, 2013
CIMPA school,
University of Delhi
K := (x, y) ∈ R3 :
S. Dempe
Bilevel Optimization Problem
y1 ≤ 6,
x + y1 + y2 ≤ 10,
−x + y2 ≤ 0,
y1 , y2 ≥ 0 } .
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
16
Semivectorial bilevel optimization problem
Since the lower level feasible set is compact provided that it is not empty,
we only have to guarantee the latter property. A quick look shows that K is
not empty for all x ∈ [2, 5]. We can apply the k-th best algorithm.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
17
Semivectorial bilevel optimization problem
Initialization. Set i := 1 and W = T = W i := ∅, choose µ := (0.5, 0.5).
First iteration
Step 1. Solving max µ> (Cx + F y) = max (1.5x − y2 ) leads to the
(x,y)∈K
(x,y)∈K
optimal solution (x1 , y 1 ) = (5, 0, 0) with µ> (Cx1 + F y 1 ) = 7.5.
Step 2. The Pareto set of the lower level with x1 = 5 is
[
Ψs (5, λ) = conv {(5, 0), (0, 5)}
λ∈Λ
which means that y 1 = (0, 0) is not a Pareto optimal point. Hence, the
solution obtained in Step 1 is not globally optimal for the bilevel problem.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
18
Semivectorial bilevel optimization problem
Example
Step 3. Set T := {(5, 0, 0)}. The adjacent extreme points of (x1 , y 1 ) are
added to W , namely W := {(0, 0, 0), (5, 5, 0), (5, 0, 5)}. We choose
(x2 , y 2 ) = (5, 5, 0) with µ> (Cx2 + F y 2 ) = 7.5.
Second iteration
Step 2. Now, solving the lower level problem again for x2 = 5 leads to
[
y 2 = (5, 0) ∈
Ψs (5, λ) = conv {(5, 0), (0, 5)}.
λ∈Λ
The algorithm terminates with the globally optimal solution
(x2 , y 2 ) = (5, 5, 0).
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
19
[1] C. Audet, J. Haddad, and G. Savard, A note on the definition of a linear
bilevel programming problem, Applied Mathematics and Computation 181
(2006), 351–355.
[2] W. Bialas and M. Karwan, On two-level optimization, IEEE Transactions on
Automatic Control 27 (1982), 211–214.
[3]
, Two-level linear programming, Management Science 30 (1984),
1004–1020.
[4] S. Dempe and S. Franke, Bilevel programming: Stationarity and stability,
Pacific Journal of Optimization 9 (2013), 183–199.
[5] M. Ehrgott, Multicriteria optimization, Springer Verlag, Berlin, 2005.
[6] A. G. Mersha and S. Dempe, Linear bilevel programming with upper level
constraints depending on the lower level solution, Applied Mathmatics and
Computation 180 (2006), 247–254.
November 29, 2013
CIMPA school,
University of Delhi
S. Dempe
Bilevel Optimization Problem
Fak. f. Math. u. Inf.
TU Bergakademie Freiberg
19