Unique Lower-Level Response

It turns out that the continuity of the feasible set and optimal value function highly depend on the continuity of the optimal solution set in the lower level. In general, we cannot expect ψ(ρ) being continuous. However, in an upcoming chapter we study a regularized problem for which this property holds. Therefore, we include the claims in which the lower-level response is assumed to be unique for each choice ofρby the upper level. We add a remark to discuss the conditions under which ψ(ρ) behaves continuously with respect to ρ. By a unique lower-level response we mean that for each upper-level choice of ρ, the solution of the lower-level response (f, x) is unique with respect to f and x. It turns out that a unique lower-level response is a sufficient condition for ˜F() and v() to be continuous. Theorem 8.3.2 (F˜() continuous). Let the lower-level response be unique for each

0≤ρ≤, thenF˜() is continuous with respect to. In fact,F˜() is Lipschitz-continuous relative to its domain under Assumption 2.

Proof. First, we prove the claim of the theorem under Assumption 2. We proved earlier that ˜F() is a closed mapping. Let 0_, 1 _{≥ 0 be arbitrary. Then for any (ρ}0_{, y}0_{) =}

(ρ0_{, f}0_{, x}0_)∈F˜₍0_{) we have that there exists (ρ}1_{, y}1_)∈F˜₍1_{) so that}

k(ρ1_{, y}1_)−(ρ0_{, y}0_{)k ≤Ckρ}0_−ρ1_k

≤CNk0₋1_k,

in which the latter inequality follows from Proposition 6.3.1. The first inequality follows from the optimality conditions of the objective function over a polyhedron for a strictly convex function (Theorem 9.1.2).

Let us consider the case which relaxes Assumption 2. Since we have that ˜F() is upper semicontinuous (Lemma 8.3.2) we solely prove that ˜F() is lower semicontinuous relative to. Recall that ψ(ρ) is continuous relative toρ, and thus lower semicontinuous.

So, given ρ0 _{for any y}0 _{= (f}0_{, x}0_{) ∈ ψ(ρ}0_{) and sequence ρ}l _{→ ρ}0_{, exists a sequence}

yl _{so that y}l _{∈ ψ(ρ}l_{) for l large and ky}l_−y0_{k → 0. We emphasize that this sequence is}

unique. DefineE() ={0≤ρ≤}. This multifunction is continuous at each .

Suppose ˜F() is not lower semicontinuous at0 _{≥0. Then exists (ρ}0_{, y}0_)∈F˜(0_{) and}

l _→0_{, then for all (ρ}l_{, y}l_{) so that (ρ}l_{, y}l_)∈F˜(n_{), forl large,k(ρ}l_{, y}l_)−(ρ0_{, y}0_{)k> δ for}

someδ >0.

However, by continuity of E() for each 0 _and l _→ 0 _{exists ρ}l _{∈ E(}l_{), for l large,}

andkρl_−ρ0_{k →0. By continuity ofψ(ρ), existsy}l_∈ψ(ρl_{) so that ky}l_−y0_{k →0. Hence,}

˜

F() is lower semicontinuous.

The result states that the feasible set cannot explode or implode after perturbing . Consequently, the optimal value function cannot become sufficiently smaller or larger by perturbing. If ψ(ρ) is continuous with respect to perturbations inρ (i.e. any ρ≥0 has a unique lower level response (f, x)), the optimal value functionv() is continuous relative to.

Theorem 8.3.3 (v() continuous). If ψ(ρ) is continuous with respect to perturbations in ρ, v() is continuous at all ≥0. In fact, if ψ(ρ) is Lipschitz continuous with respect toρ thenv() is Lipschitz continuous relative to .

Proof. First, we prove this statement for the case in which the latency functions are linear. Suppose (¯ρ0_,y¯0_{), (¯ρ}1_,y¯1_{) are the optimal solutions corresponding to v(}0_{), v(}1₎

respectively. Under the assumptions of the theorem, for all (ρ0_{, y}0_{) = (ρ}0_{, f}0_{, x}0_)∈F˜(0₎

nuity of ˜F(), see Theorem 8.3.2):

v(1_)−v(0_{) =s(¯x}1_)−s(¯x0₎

≤s(x1_)−s(¯x0₎

≤Φkx1−x¯0k ≤ΦNk1−0k.

The second inequality follows from the Lipschitz continuity of the (convex) objective func- tion. x1_∈F˜(1_{) is chosen so that it is closest tox}0_{with respect to the∞-norm. Similarly,}

we find thatv(0_)−v(1_)≤ΦNk1₋0_{k. These two results lead to the claim.}

Let us relax Assumption 2. We know by Lemma 8.3.2 thatv() is lower semicontinuous. Left is to prove that v() is upper semicontinuous. We know that ˜F() is continuous, in particular lower semicontinuous.

Since ˜F(0_{) is lower semicontinuous, for any (ρ}0_{, y}0_)∈F˜₍0_{) and sequence}l _→0_there

exists a sequence (ρl_{, y}l_{) so that (ρ}l_{, y}l_)∈F˜(l_{), for llarge, and k(ρ}l_{, y}l_)−(ρ0_{, y}0_{)k →0.}

Take (ρ0_{, y}0_{) ∈S˜(}0_{), i.e. s(y}0_{) = v(}0_{). Then we have for the mentioned sequence,}

forllarge,

v(l)≤s(yl)≤s(y0) +δ=v(0) +δ,

for someδ >0, by continuity ofs(y). v() is thus upper semicontinuous.

The result of this theorem is in line with our earlier results. By the assumptions of the theorem v() behaves piecewise continuous relative to≥0.

Remark 8.3.1 (Conditions under which lower-level response is unique). Full column rank of

"

Λ Γ

#

guarantees continuity of the optimal solution set ψ(ρ) of the lower problem. Then, any0≤ρ≤leads to a unique solution in(f, x) and the optimal solution set is continuous. The feasible set F() and optimal value function v() are now both continuous as well.

The question remains whether this condition is natural. We answer this question neg- atively. Borchers et al. (2015) showed a range of basic examples for which the condition was not satisfied.

The difficulty of the problem ( ˜Q()) lies in the non-unique lower-level response for a given 0 ≤ ρ ≤ choice of the leader. It is well known that these problems are difficult to solve (see Dempe (2003)). Therefore, we propose a regularized approach in the next chapter to approximate the Best-case BRUE flow distribution.

8.4

Synthesis

We reformulated the Best-case BRUE problem as bilevel optimization problem. This reformulation allowed us to evaluate continuity of (Q()) under more general link cost functions. It turns out that the same difficulties arise as under linear latency functions. However, if the lower-level response is unique for each 0≤ρ≤the feasible set and optimal value function behave continuously under perturbations in. The next chapter therefore proposes a perturbed lower-level problem that forces a unique lower-level response for each ρ.

Chapter 9

A Regularized Approach

In previous chapter we saw that if x uniquely determines f, the feasible set and opti- mal value function behave continuously for any link cost function under Assumption 1. However, f is generally not unique for each x. In this chapter we propose an approach to handle the non-unique lower-level response by perturbing the objective function of the lower-level problem. Now, we find a regularized lower-level problem which we use for algorithm development.

9.1

The Regularized Bilevel Problem

Let us consider, as suggested by e.g. Dempe (2000), the regularized lower-level problem

(qα_{(ρ)) corresponding to (q(ρ)). We define the regularized problem, given 0≤ρ≤and}

α∈R+, as:

min

(f,x)∈F0

˜

z(f, x, ρ, α) :=z(f, x, ρ) +αkfk2 s.t. (f, x)∈ F0. (qα(ρ))

Trivially, ifα= 0 problem (qα_{(ρ)) reduces to (q(ρ)). For any positiveα we have that the}

objective function ˜z(f, x, ρ, α) is strictly convex since the Hessian matrix of this function with respect to variables (f, x) becomes:

∇2_(f,x)z˜(f, x, ρ, α) = " 2αI|P| 0 0 ∇xl(x) # ,

in whichI|P| is the identity matrix of size |P|. This matrix is positive definite by positive

definiteness of 2αI|P|and∇xl(x). From Lemma 8.2.2 it follows that the solution of (qα(ρ))

is unique both inf and x. However, we cannot guarantee that the solution of (qα_{(ρ)) is a}

solution to (q(ρ)). Therefore, we choose α >0 close to zero to assure that the solution is close to the solution of the BRUE lower-level problem (q(ρ)). We say that we approximate a BRUE flow distribution.

Let ψα_{(ρ) denote the optimal solution set corresponding to problem (q}α_{(ρ)) given}

0≤ρ≤. The regularized bilevel problem ( ˜Q()), givenα >0, becomes: min(ρ,f,x)s(x)

s.t. 0≤ρ≤ (f, x)∈ψα_(ρ)

( ˜Qα₍₎₎

As mentioned, the lower level has a unique response for any α > 0. Furthermore, let us introduce the corresponding feasible set

˜

Fα_{() =}n _{(ρ, f, x) 0≤ρ≤,(f, x)∈ψ}α_(ρ) o_,

and optimal value function

vα() := min

(ρ,f,x)∈F˜α₍₎s(x). Lastly, the optimal solution set of problem (qα_(ρ)):

˜

Sα() := arg min

(ρ,f,x)∈F˜α₍₎

vα().

We prove the continuity properties of the regularized lower problem (qα_{(ρ)) for fixedα >0.}

We emphasize that any continuity property for the feasible set, optimal value function and optimal solution set of problem ( ˜Q()) holds in particular for the feasible set, optimal value function and optimal solution set respectively of problem ( ˜Qα_()).

Theorem 9.1.1 (ψα_{(ρ) continuous). For anyα >0, ψ}α_{(ρ) is continuous relative toρ.}

Proof. The same proof as in Theorem 6.5.1 holds.

It turns out that the optimal solution set ψα_{(ρ) behaves Lipschitz continuously with}

respect to changes inρ if Assumption 2 holds and is locally 1

2-H¨older continuous without

any additional assumption. Consider the case in which Assumption 2 holds. In other words, the latency functions of the edges in the network are affine linear.

Theorem 9.1.2 (ψα_{(ρ)Lipschitz continuous). Let Assumption 2 hold. Supposeα >0}

to be fixed. Supposey(ρ) = (f(ρ), x(ρ))∈ψα_{(ρ) for ρ≥0, then there exists C >0so that}

for allρ0_{, ρ}1_≥0,

ky(ρ1)−y(ρ0)k ≤Ckρ1−ρ0k holds.

Proof.If we consider the case in which the latency functions are affine linear, the objective function ˜z(f, x, ρ, α) becomes quadratic for any α ≥ 0. Let us consider this regularized

lower problem for anyα >0,ρ≥0: min (ρ,f,x)z˜(f, x, ρ, α) = 1 2x T_Ax+bT_x−ρT_{f +αkfk}2 _{s.t.(f, x)∈ F} 0 (˜qα(ρ))

Obviously, the matrix

"

2αI|P| 0

0 A

#

is diagonal and positive definite. Consequently, the solution (f, x) of (˜qα_{(ρ)) is unique.}

For eachρ≥0, the KKT conditions of (˜qα_{(ρ)) become:}

Ax+b+β = 0 Λf =d

ΓT_{β+ 2αf−ρ−Λ}T_{λ−γ = 0 Γf −x= 0}

γT_{f = 0 γ, f ≥0}

(9.1.1)

with corresponding multipliers (β, λ, γ). By Assumption 2, (f, x) solves (˜qα_{(ρ)) if and}

only (f, x) satisfies (9.1.1) with the corresponding Lagrange multipliers. As in the proof of Lemma 6.5.1 we can choose I ⊆ {1, . . . ,|P|} and consider solution (ρ, f, x, β, λ, γ) of (9.1.1) so that (f, γ) satisfies:

fp = 0, p∈ I;

γp = 0, p /∈ I;

(9.1.2) and define stability set Ω(I) as

Ω(I) ={ ρ solution of (9.1.1) satisfies (9.1.2) givenI }.

We can apply the same proof as in Lemma 6.5.1 and Theorem 6.5.2 to prove Lipschitz continuity of the optimal solution setψα_(ρ).

In general, we cannot expect ψα_{(ρ) to be Lipschitz continuous. For higher order}

polynomial link cost functions under Assumption 1,ψα_{(ρ) is locally} 1

2-H¨older continuous

with respect to perturbations in ρ.

For the upcoming result we require the optimality conditions of nonlinear program- ming. We can apply the next theorem since a constraint qualification holds for the con- straints in F0. We introduce the Lagrange function corresponding to problem (qα(ρ)),

α >0, forρ≥0:

L(f, x, ρ, β, λ, γ) = ˜z(f, x, ρ, α) + (d−Λf)Tλ+ (Γf −x)Tβ−fTγ in which (β, λ, γ) is the vector of KKT multipliers.

dron). Let ρ0 _{≥ 0. If (f}0_{, x}0_{) satisfies the first-order stationarity conditions (note that}

∇=∇_(f,x))

∇_L(f0_{, x}0_{, ρ}0_{, β}0_{, λ}0_{, γ}0_{) = 0, and γ}0T_f0_{= 0}

with multipliers (β0_{, λ}0_{, γ}0_),γ0_{≥0 in combination with}

τT∇2_L(f0, x0, ρ0, β0, λ0, γ0)τ >0, for all τ ∈C(f0, x0) where τ 6= 0,

then there exists a neighborhood U(f0_{, x}0_{) ⊆}

R|E|×R|P| of (f0, x0) and ζρ0 >0 so that

for all(f, x)∈ F₀∩U(f0_{, x}0_):

˜

z(f, x, ρ0, α)−z˜(f0, x0, ρ0, α)≥ζ_ρ0k(f, x)−(f0, x0)k2. (9.1.3)

Here, C(f0_{, x}0_{) is the set of feasible directions at (f}0_{, x}0_):

C(f, x) :=            τ = (τf_{, τ}x_)∈ R|P|+|E| ∇z˜(f, x, ρ0_{, α)}T_{τ ≤0} Γτf _−τx _{= 0} Λτf _{= 0} τpf ≥0 for all p∈ P :fp= 0           

Proof. (The proof is based on Faigle et al. (2013) and Still and Streng (1996)). For notational purposes, we drop ρ and α in the notation of our analysis. Suppose a flow (f0_{, x}0_{) satisfies the assumptions of the theorem with KKT multipliers (β}0_{, λ}0_{, γ}0_{). Define}

the compact set of feasible directions at (f0_{, x}0_{) as}

K(f0_{, x}0_{) =}n _{τ ∈C(f}0_{, x}0_{) kτk= 1} o_. Putζ as ζ := min τ∈K(f0_,x0₎ 1 2τ T_∇2_L(f0_{, x}0_{, β}0_{, λ}0_{, γ}0_{)τ. (9.1.4)}

The minimization problem in (9.1.4) is well-defined. Indeed, K(f0_{, x}0_{) is compact, the}

objective function is continuous, and

τT∇2_L(f0, x0, β0, λ0, γ0)τ =τT∇2z˜(f0, x0)τ >0

holds for allτ 6= 0. Thus,ζ >0. By any choice ofτ, due to the linear constraints, τ is a feasible direction. That is: for eachτ exists a feasible curve through (f0_{, x}0_{) into feasible}

directionτ. We show that (9.1.3) holds for eachζρ0 < ζ.

Suppose that φ < ζ arbitrarily and suppose (f0_{, x}0_{) is not a second order minimizer}

as in (9.1.3) withζ_ρ0 = φ. Then, exists an infinite sequence of points (fl, xl) → (f0, x0)

satisfying (fl_{, x}l_{) 6= (f}0_{, x}0_{) and ˜z(f}l_{, x}l_)−z˜(f0_{, x}0_{) ≤φk(f}l_{, x}l_)−(f0_{, x}0_)k2_{, l∈}

can write (fl_{, x}l_{) = (f}0_{, x}0_{) +τ}l_tl_,kτl_{k= 1,t}l_{>0. Nowt}l _→0,τl_{→τ and kτk= 1 (for}

a subsequence). For this sequence it holds that ˜

z(fl, xl)−z˜(f0, x0)≤φk(fl, xl)−(f0, x0)k2. We apply the second-order Taylor expansion:

φ(tl)2 ≥z˜(fl, xl)−z˜(f0, x0) =tl " −ρ+ 2αf0 l(x0₎ #T τl+1 2t l2_τlT " 2αI|P| 0 0 ∇xl(x0) # τl 0 = Γfl−xl =tl " Γ −I|E| #T τl 0 = Λfl_−d=tl_ΛT_τf,l 0≥ −f_Pl =tl− ∇ffP0 T τf,l (9.1.5) whereP ={p∈ P |f0 p = 0}.

Multiplying with the respective KKT multipliers and adding, we obtain for (fl_{, x}l_):

φ(tl)2 ≥tl∇L(f0, x0, β0, λ0, γ0)τl+1 2t

l2_τlT_∇2_L(f0_{, x}0_{, β}0_{, λ}0_{, γ}0_)τl _(9.1.6)

By the assumptions of the theorem we find that ∇L(f0_{, x}0_{, β}0_{, λ}0_{, γ}0_{) = 0. Dividing}

(9.1.6) bytl2 _{we obtain:}

ζ > φ≥ 1

2τ

lT_∇2

L(f0, x0, β0, λ0, γ0)τl>0 (9.1.7) Recall that kτl_{k = 1. In case τ ∈ K(f}0_{, x}0_{) we reach a contradiction since we chose ζ}

so that it was the minimum in (9.1.4). We have liml→∞(fl, xl) = (f0, x0), and, hence,

liml→∞tl= 0. We divide the equations in (9.1.5) by tl and letl→ ∞:

0≥ " −ρ+ 2αf0 l(x0₎ #T τ 0 = Γτf _−τx 0 = Λτf 0≥ −τ_Pf ,

which are exactly the (in)equalities defining C(f0_{, x}0_{). So, τ ∈ K(f}0_{, x}0_{) which contra-}

dictsφ < ζ.

We use the second order optimality conditions to prove that the feasible set of the regularized bilevel problem is locally 1

Theorem 9.1.4 (ψα_{(ρ) locally H¨older). Given α > 0, the mapping ψ}α_{(ρ) is locally}

1

2-H¨older continuous relative to its domain, i.e. for all ρ

0_{≥0exists aθ, δ >0 so that for}

allρ1_∈U

δ(ρ0) and y0 ∈ψα(ρ0), y1 ∈ψα(ρ1) the following holds:

ky1_−y0_{k ≤θkρ}1_−ρ0_k1_2.

Proof. Letα >0 andρ0 _{≥0. Sinceψ}α_{(ρ) is continuous (Theorem 9.1.1) we may assume}

that if ρ1 _{is sufficiently close to ρ}0_{, the corresponding optimal solution y}1 _{∈ ψ}α_(ρ1_{) is}

sufficiently close to y0 _{∈ ψ}α_(ρ0_{). Formally, the second order optimality conditions say}

that there exists a γ > 0 so that for all flows in the neighborhood Uγ(ψα(ρ0)) satisfy

(9.1.3). By continuity ofψα_{(ρ): for allτ >0 exists δ >0 so that}

ψα(ρ)⊆Uτ(ψα(ρ0)) for all ρ∈Uδ(ρ0),

and selecting τ = γ. So, denoting y1 _{∈ ψ}α_(ρ1_{) as the optimal solution for ρ}1 _{and y}0 _∈

ψα_(ρ0_{) in which ρ}1 _∈U δ(ρ0) we find: ky1−y0k2 ≤ kz˜(f 1_{, x}1_{, ρ}0_)−z˜(f0_{, x}0_{, ρ}0_)k ζ ≤ kz˜(f 1_{, x}1_{, ρ}0_)−z˜(f1_{, x}1_{, ρ}1_{) +φ}α_(ρ1_)−φα_(ρ1_)k ζ ≤ D 1_kρ1_−ρ0_k+kφα_(ρ1_)−φα_(ρ0_)k ζ .

Then Lipschitz continuity of the lower level optimal value function φα_{(ρ) is sufficient to}

prove the claim of the theorem. Let y1_{, y}0 _{be chosen so that ˜z(y}1_{, ρ}1_{) = φ}α_(ρ1_{) and}

˜

z(y0_{, ρ}0_{) =φ}α_(ρ0_{). Notice thaty}1_{, y}0 _{∈ F}

0. We get:

φα(ρ1)−φα(ρ0)≤z˜(y0, ρ1)−z˜(y0, ρ0)

≤D2kρ1−ρ0k. Analogously, we find that

φα(ρ0)−φα(ρ1)≤z˜(y1, ρ0)−z˜(y1, ρ1)

≤D2_kρ1_−ρ0_k.

The last inequality follows since the functionρT_{f is Lipschitz continuous with respect to}

bothρ and f. Here,D1_{, D}2 _{denote Lipschitz constants. Then,}

ky1−y0k2 ≤ D

1_+D2

ζ kρ

So, ky1−y0k ≤θkρ1−ρ0k12, by defining θ= D1_+D2 ζ 1₂ .

The continuity of the feasible set ˜Fα_{() follows directly. From Theorem 8.3.2 and}

Theorem 8.3.3 we conclude the following.

Corollary 9.1.1 (F˜α_{(), v}α_{()continuous). For anyα >0,F˜}α_(),vα_{()are continuous}

relative to ≥0.

If we assume affine linear latencies, i.e. Assumption 2 holds, then we can strengthen our result.

Corollary 9.1.2 (F˜α_{(), v}α_{()Lipschitz continuous). Let Assumption 2 hold. For any}

α >0, F˜α_{() andv}α_{() are Lipschitz continuous relative to ≥0.}

We proposed a regularized problem to overcome the difficulties of the lower-level prob- lem. The objective function of the lower-level problem is strictly convex in (f, x), assuring a unique result (f, x) ∈ ψ(ρ) for each 0 ≤ ρ ≤ . However, the solution of this lower problem is not necessarily a BRUE flow distribution as in (4.1.1).

In document The boundedly rational user equilibrium : a parametric optimization approach (Page 93-103)