Structure of risk-averse stochastic linear bilevel programs · 3/27/2019 · Linear bilevel...

Structure of risk-averse stochastic linear

bilevel programs

Computational Management Science Conference Chemnitz

Johanna Burtscheidt, Matthias Claus, Stephan Dempe

March 27, 2019

University of Duisburg-Essen, TU Bergakademie Freiberg

Linear bilevel programming

” minx

c>x + q>y

∣∣∣∣ x ∈ X , y ∈ Argminyd>y | Ay ≤ Tx + z

Problem: no uniqueness in the lower level

→ Optimistic formulation

c>x + min

∣∣∣ y ∈ Argminyd>y | Ay ≤ Tx + z

∣∣∣∣ x ∈ X

0 1 2 3 4 5 60

−∇f −∇F

Challenges:

→ no convexity

→ no differentiability

→ finding ε-optimal solutions

→ is NP-hard [Deng 98]

” minx

c>x + q>y

Problem: no uniqueness in the lower level → Optimistic formulation

c>x + min

∣∣∣∣ x ∈ X

0 1 2 3 4 5 60

−∇f −∇F

Challenges:

→ no convexity

” minx

c>x + q>y

c>x + min

∣∣∣∣ x ∈ X

0 1 2 3 4 5 60

−∇f −∇F

Challenges:

→ no convexity

” minx

c>x + q>y

c>x + min

∣∣∣∣ x ∈ X

0 1 2 3 4 5 60

−∇f −∇F

Challenges:

→ no convexity

Stochastic linear bilevel programming

” minx

c>x+miny

∣∣∣ y ∈ Argminyd>y | Ay ≤ Tx+Z(ω)

∣∣∣∣ x ∈ X

Model: upper level nonanticipativity

→ Induced random variable

f (x ,Z(·)) = c>x + miny

∣∣∣ y ∈ Argminyd>y | Ay ≤ Tx + Z(·)

→ Risk aversion can be modeled by a mapping

R : L0(Ω,F ,P)→ R

• EEη[·] = E[max· − η, 0]

• SDρ[·] = E[·] + ρEEE[·][·]

• CVaRα

• VaRα [Ivanov 14]

QR : Rn → R, QR(x) := R[f (x ,Z(·))]

R[c>x+min

] ∣∣∣∣ x ∈ X

f (x ,Z(·)) = c>x + miny

R : L0(Ω,F ,P)→ R

• EEη[·] = E[max· − η, 0]

• SDρ[·] = E[·] + ρEEE[·][·]

• CVaRα

QR : Rn → R, QR(x) := R[f (x ,Z(·))]

R[c>x+min

] ∣∣∣∣ x ∈ X

f (x ,Z(·)) = c>x + miny

R : L0(Ω,F ,P)→ R

• EEη[·] = E[max· − η, 0]

• SDρ[·] = E[·] + ρEEE[·][·]

• CVaRα

QR : Rn → R, QR(x) := R[f (x ,Z(·))]

R[c>x+min

] ∣∣∣∣ x ∈ X

f (x ,Z(·)) = c>x + miny

R : L0(Ω,F ,P)→ R

• EEη[·] = E[max· − η, 0]

• SDρ[·] = E[·] + ρEEE[·][·]

• CVaRα

QR : Rn → R, QR(x) := R[f (x ,Z(·))]

Basic properties

Assume dom f 6= ∅, then

f (x , z) = c>x + minq>y

is real-valued and Lipschitz continuous on the polyhedron

F := (x , z) ∈ Rn × Rs | ∃y : Ay ≤ Tx + z.

dom f 6= ∅ holds if and only if there exists (x , z) ∈ Rn × Rs such that

1. y | Ay ≤ Tx + z is nonempty,

2. there is some u ∈ Rs satisfying A>u = d and u ≤ 0 and

3. y 7→ q>y is bounded from below on Argminyd>y | Ay ≤ Tx + z.

Basic properties

is real-valued and Lipschitz continuous on the polyhedron

F := (x , z) ∈ Rn × Rs | ∃y : Ay ≤ Tx + z.

dom f 6= ∅ holds if and only if there exists (x , z) ∈ Rn × Rs such that

1. y | Ay ≤ Tx + z is nonempty,

2. there is some u ∈ Rs satisfying A>u = d and u ≤ 0 and

3. y 7→ q>y is bounded from below on Argminyd>y | Ay ≤ Tx + z.

Basic properties

QR(x) = R[f (x ,Z(·))]

Notation:

• µZ := P Z−1

• Mps =

µ ∈ P(Rs) |

∫Rs ‖z‖p µ(dz) <∞

Proposition

Assume dom f 6= ∅ and µZ ∈M1s . Then the mappings QE, QEEη , QSDρ and

QCVaRα are real-valued and Lipschitz continuous on

FZ = x ∈ Rn | (x , z) ∈ F ∀z ∈ supp µZ.

Basic properties

QR(x) = R[f (x ,Z(·))]

Notation:

• µZ := P Z−1

• Mps =

µ ∈ P(Rs) |

∫Rs ‖z‖p µ(dz) <∞

Proposition

Assume dom f 6= ∅ and µZ ∈M1s . Then the mappings QE, QEEη , QSDρ and

QCVaRα are real-valued and Lipschitz continuous on

FZ = x ∈ Rn | (x , z) ∈ F ∀z ∈ supp µZ.

Differentiability

Modified lower level:

∣∣∣ y ∈ Argminyd>y | Ay = Tx + z , y ≥ 0

→ WLOG the rows of A can be assumed to be linearly independent.

Base matrices:

A := AB ∈ Rs×s | AB is a regular submatrix of A

A∗ := AB ∈ A | d>N − d>B A−1B AN ≥ 0

f (x , z) = c>x + minAB

q>B A−1

B (Tx + z) | A−1B (Tx + z) ≥ 0, AB ∈ A∗

holds for any (x , z) ∈ F .

Differentiability

Modified lower level:

∣∣∣ y ∈ Argminyd>y | Ay = Tx + z , y ≥ 0

→ WLOG the rows of A can be assumed to be linearly independent.

Base matrices:

A := AB ∈ Rs×s | AB is a regular submatrix of A

A∗ := AB ∈ A | d>N − d>B A−1B AN ≥ 0

q>B A−1

B (Tx + z) | A−1B (Tx + z) ≥ 0, AB ∈ A∗

holds for any (x , z) ∈ F .

Differentiability

q>B A−1

B (Tx + z) | A−1B (Tx + z) ≥ 0, AB ∈ A∗

Regions of stability:

R(AB) := (x , z) ∈ F | A−1B (Tx + z) ≥ 0, c>x + q>B A−1

B (Tx + z) = f (x , z)

Observations:

•⋃

AB∈A∗R(AB) = F

• If x0 ∈ int FZ and z0 ∈ supp µZ are such that (x0, z0) ∈ int R(AB) for

some AB ∈ A∗, then f (·, z0) is continuously differentiable at x0 and

∇x f (x0, z0) = c> + q>B A−1B T .

Assume dom f 6= ∅ and x0 ∈ int FZ . Then z0 ∈ supp µZ and

(x0, z0) /∈ int R(AB) for all AB ∈ A∗ implies z0 ∈ Nx0 , where Nx0 ⊂ Rs is

contained in a finite union of hyperplanes.

Differentiability

q>B A−1

B (Tx + z) | A−1B (Tx + z) ≥ 0, AB ∈ A∗

B (Tx + z) = f (x , z)

Observations:

•⋃

AB∈A∗R(AB) = F

∇x f (x0, z0) = c> + q>B A−1B T .

Differentiability

q>B A−1

B (Tx + z) | A−1B (Tx + z) ≥ 0, AB ∈ A∗

B (Tx + z) = f (x , z)

Observations:

•⋃

AB∈A∗R(AB) = F

∇x f (x0, z0) = c> + q>B A−1B T .

Differentiability

q>B A−1

B (Tx + z) | A−1B (Tx + z) ≥ 0, AB ∈ A∗

B (Tx + z) = f (x , z)

Observations:

•⋃

AB∈A∗R(AB) = F

∇x f (x0, z0) = c> + q>B A−1B T .

Differentiability

B (Tx + z) = f (x , z)

Nx0 = Fx0 ∪⋃

AB∈A∗Zx0 (AB) ∪

⋃AB ,AB′∈A

q>B A−1B6= q>

B′A−1B′

Vx0 (AB ,AB′)

• Fx0 := z ∈ Rs | (x0, z) ∈ F \ int F

• Zx0 (AB) := z ∈ Rs | A−1B (Tx0 + z) ≥ 0, ∃i : (A−1

B (Tx0 + z))i = 0

• Vx0 (AB ,AB′) := z ∈ Rs | (q>B A−1B − q>B′A

−1B′ )(z + Tx0) = 0

Observation: There is some ε(x0) > 0 such that Nx ⊆ Nx0 ∀x ∈ Bε(x0)(x0).

Differentiability

B (Tx + z) = f (x , z)

Nx0 = Fx0 ∪⋃

AB∈A∗Zx0 (AB) ∪

⋃AB ,AB′∈A

q>B A−1B6= q>

B′A−1B′

Vx0 (AB ,AB′)

• Fx0 := z ∈ Rs | (x0, z) ∈ F \ int F

• Zx0 (AB) := z ∈ Rs | A−1B (Tx0 + z) ≥ 0, ∃i : (A−1

B (Tx0 + z))i = 0

• Vx0 (AB ,AB′) := z ∈ Rs | (q>B A−1B − q>B′A

−1B′ )(z + Tx0) = 0

Observation: There is some ε(x0) > 0 such that Nx ⊆ Nx0 ∀x ∈ Bε(x0)(x0).7

Differentiability

Theorem

Assume dom f 6= ∅, µZ ∈M1s and let x0 ∈ int FZ be such that µZ [Nx0 ] = 0.

Then QE is differentiable at x0.

Derivative:

Q ′E(x0) =

∫supp µZ\Nx0

∇x f (x0, z) µZ (dz) = c> +∑∆∈D

µZ [W(x0,∆)]∆

D := q>B A−1B T | AB ∈ A∗

W(x0,∆) := z ∈ supp µZ\Nx0 | ∃AB ∈ A∗ : (x0, z) ∈ int R(AB), ∆ = q>B A−1B T

Differentiability

Theorem

Assume dom f 6= ∅, µZ ∈M1s and let x0 ∈ int FZ be such that µZ [Nx0 ] = 0.

Then QE is differentiable at x0.

Derivative:

Q ′E(x0) =

∫supp µZ\Nx0

∇x f (x0, z) µZ (dz) = c> +∑∆∈D

µZ [W(x0,∆)]∆

D := q>B A−1B T | AB ∈ A∗

W(x0,∆) := z ∈ supp µZ\Nx0 | ∃AB ∈ A∗ : (x0, z) ∈ int R(AB), ∆ = q>B A−1B T

Differentiability

W(x ,∆) = z ∈ supp µZ \ Nx | ∃AB ∈ A∗ : (x , z) ∈ int R(AB), ∆ = q>B A−1B T

→ There is a neighborhood of x0 on which µZ [Nx ] = 0 and QE is differentiable with

Q′E(x) = c> +∑

∆∈DµZ [W(x ,∆)]∆.

→ µZ [Nx ] = 0 implies µZ [W(x ,∆)] = µZ [W(x ,∆)], where

W(x ,∆) = z ∈ supp µZ | ∃AB ∈ A∗ : (x , z) ∈ cl int R(AB), ∆ = q>B A−1B T.

→ Fix ∆ ∈ D, then the mapping W(·,∆) : int FZ ⇒ Rs is outer semicontinuous

→ at x0, i.e. lim supx→x0W(x ,∆) ⊆ W(x0,∆).

→ This implies that µZ [W(·,∆)] : int FZ → [0, 1] is upper semicontinuous at x0.

→ By construction,∑

∆′∈D µZ [W(x ,∆′)] = 1 holds for any x ∈ int FZ . Thus,

µZ [W(·,∆)] = 1−∑

∆′∈D: ∆ 6=∆′µZ [W(x ,∆′)]

→ is lower semicontinuous at x0.

Differentiability

Q′E(x) = c> +∑

∆∈DµZ [W(x ,∆)]∆.

µZ [W(·,∆)] = 1−∑

∆′∈D: ∆ 6=∆′µZ [W(x ,∆′)]

Differentiability

Q′E(x) = c> +∑

∆∈DµZ [W(x ,∆)]∆.

µZ [W(·,∆)] = 1−∑

∆′∈D: ∆ 6=∆′µZ [W(x ,∆′)]

Differentiability

Q′E(x) = c> +∑

∆∈DµZ [W(x ,∆)]∆.

µZ [W(·,∆)] = 1−∑

∆′∈D: ∆ 6=∆′µZ [W(x ,∆′)]

Differentiability

Q′E(x) = c> +∑

∆∈DµZ [W(x ,∆)]∆.

µZ [W(·,∆)] = 1−∑

∆′∈D: ∆ 6=∆′µZ [W(x ,∆′)]

Differentiability

Theorem

Assume dom f 6= ∅, µZ ∈M1s and let x0 ∈ int FZ be such that µZ [Nx0 ] = 0. Then

QE is continuously differentiable at x0.

Theorem

Assume dom f 6= ∅, µZ ∈M1s and let x0 ∈ int FZ and η ∈ R be such that

µZ [Nx0 ∪ L(x0, η)] = 0, where

L(x0, η) :=⋃

AB∈A∗: q>B

A−1B6=0

z ∈ Rs | q>B A−1B (Tx0 + z) = η.

Then QEEη (·) = E[maxf (·,Z)− η, 0] is continuously differentiable at x0.

Theorem

Assume dom f 6= ∅, µZ ∈M1s and let x0 ∈ int FZ be such that QE(x0) 6= 0 and

µZ [Nx0 ∪ L(x0,QE(x0))] = 0. Then

QSDρ (·) = QE(·) + ρE[maxf (·,Z)− QE(·), 0]

is continuously differentiable at x0.

Differentiability

Theorem

L(x0, η) :=⋃

AB∈A∗: q>B

A−1B6=0

z ∈ Rs | q>B A−1B (Tx0 + z) = η.

Theorem

µZ [Nx0 ∪ L(x0,QE(x0))] = 0. Then

is continuously differentiable at x0.

Differentiability

Theorem

L(x0, η) :=⋃

AB∈A∗: q>B

A−1B6=0

z ∈ Rs | q>B A−1B (Tx0 + z) = η.

Theorem

µZ [Nx0 ∪ L(x0,QE(x0))] = 0. Then

is continuously differentiable at x0.10

Differentiability

Corollary

Assume dom f 6= ∅ and let µZ ∈M1s be absolutely continuous w.r.t. the

Lebesgue measure. Then QE and QEEη are continuously differentiable on

int FZ . Furthermore, QSDρ is continuously differentiable at any x0 ∈ int FZ

satisfying QE(x0) 6= 0.

Consequence: Assuming X ⊂ int FZ first-order necessary optimality conditions

for the stochastic linear bilevel problem can be formulated in terms of

directional derivatives.

Differentiability

Corollary

Assume dom f 6= ∅ and let µZ ∈M1s be absolutely continuous w.r.t. the

Lebesgue measure. Then QE and QEEη are continuously differentiable on

int FZ . Furthermore, QSDρ is continuously differentiable at any x0 ∈ int FZ

satisfying QE(x0) 6= 0.

Consequence: Assuming X ⊂ int FZ first-order necessary optimality conditions

for the stochastic linear bilevel problem can be formulated in terms of

directional derivatives.

Convex risk measures

Assume that R : L0(Ω,F ,P)→ R is law-invariant. Then R[f (x ,Z)] is

determined by

(δx ⊗ µZ ) f −1 ∈ P(R).

In addition, assume that (Ω,F ,P) is atomless (WLOG) and that there is some

p ≥ 1 such that R is real-valued on Lp(Ω,F ,P).

Then for any ν ∈Mp1 there exists some Xν ∈ Lp(Ω,F ,P) such that

ν = P X−1ν .

Define QR : Rn ×Mps → R by

QR(x , µ) = R[X (x , µ)],

for some X (x , µ) ∈ Lp(Ω,F ,P) with P X (x , µ)−1 = (δx ⊗ µ) f −1.

Finally, assume that R is convex and nondecreasing on Lp(Ω,F ,P).

determined by

(δx ⊗ µZ ) f −1 ∈ P(R).

ν = P X−1ν .

QR(x , µ) = R[X (x , µ)],

determined by

(δx ⊗ µZ ) f −1 ∈ P(R).

ν = P X−1ν .

QR(x , µ) = R[X (x , µ)],

determined by

(δx ⊗ µZ ) f −1 ∈ P(R).

ν = P X−1ν .

QR(x , µ) = R[X (x , µ)],

determined by

(δx ⊗ µZ ) f −1 ∈ P(R).

ν = P X−1ν .

QR(x , µ) = R[X (x , µ)],

Qualitative stability

(Pµ) minxQR(x , µ) | x ∈ X

Localized optimal values and solution set mappings: For T ⊆ Rn consider

ϕT (µ) := minxQR(x , µ) | x ∈ X ∩ cl T

ψT (µ) := ArgminxQR(x , µ) | x ∈ X ∩ cl T

CLM sets: Given µ ∈Mps and an open set T ⊆ Rn, ψT (µ) is called a complete

local minimizing set of (Pµ) w.r.t. T if ∅ 6= ψT (µ) ⊆ T . [Robinson 87]

→ Global minimizers, isolated local minimizers

Theorem

Let M⊆Mps be locally uniformly ‖ · ‖ps -integrating. Assume dom f 6= ∅ and

X × Rs ⊆ F . Then

(a) QR|X×M is weakly continuous.

(b) ϕRn |M is weakly upper semicontinuous.

In addition, assume that µ0 ∈M is such that ψT (µ0) is a CLM set of Pµ0

w.r.t. some open bounded set T ⊂ Rn. Then the following statements hold

(c) ϕT |M is weakly continuous at µ0.

(d) ψT |M is weakly Berge upper semicontinuous at µ0.

(e) There exists some weakly open neighborhood U of µ0 such that ψT (µ) is

a CLM set for (Pµ) w.r.t. T for any µ ∈ U ∩M.

M⊆Mps is called locally uniformly ‖ · ‖ps -integrating if for any µ ∈M

and any ε > 0 there exists some weak open neighborhood N of µ such

lima→∞

supν∈N∩M

∫Rs\Ba(0)

‖t‖ps ν(dt) ≤ ε.

• For any κ, ε > 0, the set

µ ∈ P(Rs) |∫Rs

‖t‖p+εs µ(dt) ≤ κ

is locally uniformly ‖ · ‖ps -integrating.

• For any compact set Ξ ⊂ Rs , the set

µ ∈ P(Rs) | µ[Ξ] = 1

is locally uniformly‖ · ‖ps -integrating.

• Any singleton µ ⊂ Mps is locally uniformly‖ · ‖ps -integrating.

lima→∞

supν∈N∩M

∫Rs\Ba(0)

µ ∈ P(Rs) |∫Rs

µ ∈ P(Rs) | µ[Ξ] = 1

lima→∞

supν∈N∩M

∫Rs\Ba(0)

µ ∈ P(Rs) |∫Rs

µ ∈ P(Rs) | µ[Ξ] = 1

lima→∞

supν∈N∩M

∫Rs\Ba(0)

µ ∈ P(Rs) |∫Rs

µ ∈ P(Rs) | µ[Ξ] = 1

Existence

Corollary

Assume dom f 6= ∅ and µZ ∈Mps . Then QR is real-valued and continuous

on FZ . In particular, the stochastic linear bilevel problem

minxQR(x) | x ∈ X

is solvable if X is a nonempty compact subset of FZ .

Preprint:

Johanna Burtscheidt, Matthias Claus, Stephan Dempe:

Risk-Averse Models in Bilevel Stochastic Linear Programming

arXiv:1901.11349

Existence

Corollary

Assume dom f 6= ∅ and µZ ∈Mps . Then QR is real-valued and continuous

on FZ . In particular, the stochastic linear bilevel problem

minxQR(x) | x ∈ X

is solvable if X is a nonempty compact subset of FZ .

Preprint:

Johanna Burtscheidt, Matthias Claus, Stephan Dempe:

Risk-Averse Models in Bilevel Stochastic Linear Programming

arXiv:1901.11349

Structure of risk-averse stochastic linear bilevel programs · 3/27/2019 · Linear bilevel...

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