3.4 Optimality System Proper Orthogonal Decomposition
3.4.3 Existence of regular Lagrange multipliers
Again we are confronted with the problem that the convex cone representing the inequal-ity constraints has no inner points. We therefore proceed in two steps: First we show that the gradient of the equality constraints is surjective to derive a variaitonal inequal-ity. In the second step, we construct Lagrange multipliers for the inequality constraints by projecting the gradient of these contraints on the corresponding natural active sets.
We start with a generalization of Thm. 1.20; the central steps of the proof can be adapted:
Theorem 3.11. (Existence of Lagrange multipliers)
LetZbe a Banach space,(Vi)i∈Ibe a familiy of Hilbert spaces,V =Q
Vi,(Wj)i∈J be a family of Banach spaces,W =Q
Wj,X=V×W,f :X→Rbe a continuously Fréchet differentiable objective function, e :X → Z0 be a continuously Fréchet differentiable equality constraints operator,g:V →V be a linear and invertible mapping,g˜:X →V be given by ˜g(x) = g(v) and (Ki)i∈I be a family of closed convex cones Ki ⊆ V˜i
representing a relationv≤K 0via −v∈K on the product coneK =Q
Ki with
∀i∈I :∀v∈V˜i : v∈Vi ⇐⇒ ∀v0 ∈Ki :hv0,viVi ≥0, (3.51a)
∀i∈I :∀v∈V˜i : v∈/Ki =⇒ −v∈Ki. (3.51b) For bounds ga, gb ∈V with∀i∈I :gai6=gbi we define the cylinders
Ci={v∈V˜ |gbi−gi(v)∈Ki &gi(v)−gai∈Ki} (i∈I) and set C = T
Ci. Let V be a Hilbert space endowed with a scalar product h·,·iV which satisfies for allv∈V,ˆv∈C the conformity condition
∀˜v∈C:hv,˜v−viˆ V ≥0 ⇐⇒ ∀i∈I :∀˜v ∈Ci:hvi,v˜i−vˆiiVi ≥0. (3.51c) Letx¯∈X be a solution to the optimization problem
minx∈Xf(x) subject to e(x) = 0, ga≤K g(v)≤K gb (3.52) and assume thate0(¯x) :X→Z0 is surjective.
Then there exist multipliersκ¯ ∈V andz¯∈Z such that for each family (γi)i∈I ⊆Rof positive scalars γi >0 the following nonlinear system of equations is satisfied:
0 =f0(¯x) +e0(¯x)?z¯+ ˜g0(¯x)?κ¯ (3.53a) κ¯i =χai
−gi−?
∂f
∂v(¯x) + ∂e
∂v(¯x)?z¯
+γi(gi(¯v)−gai)
+χbi
−gi−?
∂f
∂v(¯x) +∂e
∂v(¯x)?z¯
+γi(gi(¯v)−gbi)
(3.53b) where the cut-off functionsχai, χbi:Xi→Xi are defined as
χai(x) =
0 x∈Ki
x x∈/ Ki , χbi(x) =
0 −x∈Ki
x −x∈/ Ki .
Proof. K is additive: Let v,v˜∈K and i∈I, thenvi,˜vi ∈Ki and we get by (3.51a):
∀v0 ∈Ki :hv0, vi+ ˜viiXi =hv0, viiXi+hv0,v˜iiXi ≥0 =⇒ vi+ ˜vi∈Ki
and hence v+ ˜v ∈ K. Therefore, C = {v ∈ V | g(v)−ga ∈ K & gb−g(v) ∈ K} is convex: Let v,˜v∈C and α∈(0,1), then
g(αv+ (1−α)˜v)−ga=αg(v) + (1−α)g(˜v)−ga
=α(g(v)−ga) + (1−α)(g(˜v)−ga)∈K and gb−g(αv+ (1−α)˜v) =α(gb−g(v)) + (1−α)(gb−g(˜v))∈K.
Sincee0(¯x)is surjective, there exists a Lagrange multiplierz¯∈Z refering to the equality constrainte(x) = 0such that the Lagrange functionL(x, z) =f(x)+he(x), ziZ0,Zsatisfies
∀w˜ ∈W :
∂L
∂w(¯x,z),¯ w˜
W0,W
= ∂f
∂w(¯x) + ∂e
∂w(¯x)?z,¯ w˜
W0,W
= 0, (3.54)
∀˜v∈C:
∂L
∂v(¯x,z),¯ v˜−¯v
V0,V
= ∂f
∂v(¯x) + ∂e
∂v(¯x)?z,¯ v˜−v¯
V0,V
≥0. (3.55) No the transformation arguments given on pp. 31, 32 are adapted; notice thatκ¯ has the
same form asλ¯a−λ¯b.
In contrast to the full-order optimal control problem (1.46) whereg(u, w) =εw+ISuis chosen or the reduced-order optimal control problem with fixed POD basis (3.31) where we define g(u, w) = εw+ I(ψ)S(ψ)u, the OS-POD problem is governed by nonlinear constraints g(u, w,ψ) = εw+ I(ψ)S(ψ)u, so the existence result Thm. 3.9 for regular Lagrange multipliers corresponding to the inequality constraints cannot be applied di-rectly. Therefore, we use the transformation of the penalty variable introduced in (2.16) to replace the nonlinear implicit constraints by explicit ones: We definegas the identity on U×W and interprete control-state pairs (u, ω) ∈L2(Θ,Rm)×L2(Θ,Rn) as families (vi)i∈I with the index setI = Θ× {1, ...,m} ×Θ× {1, ...,n}. We choose the transformed objective functionf(x) = J(y, u,1ε(ω−I(ψ)y),ψ)wherex= (y, u, ω, y,ψ,λ)belongs to the spaceX =H1(Θ,R`)×U×W×Y×V`×R`. We select the standard scalar product onX induced from the scalar products of its six factor spaces and sethv1,v2iVi = v1·v2 on the spacesVi =R. Choosing Ki ={v∈R |v≥0},i∈I, the conditions (3.51a) and (3.51b) are satisfied.
Further, the conformity condition (3.51c) between the multiplication scalar products h·,·iVi and theL2 scalar producth·,·iV holds true: “⇐” is obvious; for “⇒” assume that there are component indices i ∈ {1, ...,m}, j ∈ {1, ...,n} and a time set Θ˜ ⊆ Θ with Lebesgue measure λ( ˜Θ)>0 as well as test functions (˜u,ω)˜ ∈U ×W such that
ui(t)(˜ui(t)−uˆi(t))<0 or ωj(t)(˜ωj(t)−ωˆj(t))<0
for some controls u ∈U, uˆ ∈ [ua, ub] and penalties ω ∈ W, ωˆ ∈ [ˆya,yˆb]. May the first case hold true. We define the test control
˜
so(˜u0,ω)ˆ ∈C violates the variational inequality on the left-hand side of (3.51c). We re-mark that we read conditions onI componentwise almost everywhere inΘ; the Lagrange multipliers construced pointwise inΘby different representants(¯u,ω)¯ just differ on a set Θ˜ ⊆Θ of Lebesgue measure zero and hence are well-defined inL2.
Since the OS-POD problem with transformed penalty w = 1ε(ω−I(ψ)y) is equivalent to the original one (3.44), the existence of optimal OS-POD solutions is granted by the existence theorem, Thm. 3.9. It remains to show that the linearized equality constraints operator e0(¯x) : X → Z0 given in (3.44) is surjective. We hereby follow the arguments presented in [82]:
Theorem 3.12. (Existence of OS-POD multipliers) Consider the transformed OS-POD problem
minx∈X
Proof. For given ζ = (p0,p0◦, p0, p0◦,µ0,µ0◦) ∈Z0 we construct ξ = (y, u, ω, y,ψ,λ) ∈X
where the operatorD:V`× {1, ..., `} → V` is defined via D(ψ, l) = (δlkψl)`k=1 and the operatorR˜l:Y ×R→ Lb(X,X) is given by R˜l(y, λ)φ=R
Θhy(t), φiXy(t) dt−λφ.
1. Letu∈U andω ∈W be arbitrary elements. We choose ξ2 =u and ξ3=ω.
2. According to the well-posedness result, Thm. 1.14, there is somey∈Y satisfying E(y, ξ2) =p0−f & E◦(y) =p0◦−y◦.
We chooseξ4=y.
3. Let l ∈ {1, ..., `}. We complete {ψ¯l} to an orthonormal system (vn)n∈N of X with v1 = ¯ψl. For alln∈N letwn= ˜Rl(¯y,λ¯l)vn. Since ker ˜Rl(¯y,λ¯l) =span{ψ¯l},(wn)n≥2
is a maximal generating system of imR˜l(¯y,λ¯l). Therefore there exists a coefficient vector (αn)n∈N⊆Rsuch that
µ0l− Z
Θ
hψ¯l, ξ4(t)iXy(t) +¯ hψ¯l,y(t)i¯ Xξ4(t) dt=α1ψ¯l+X
n≥2
αnwn.
We chooseξ5l= µ2◦ψ¯l+P∞
n=2αnvn and ξ6l=−α1. Then R˜l(¯y,λ¯l)ξ5l−ξ6lψ¯l+
Z
Θ
hψ¯l, ξ4(t)iXy(t) +¯ hψ¯l,y(t)i¯ Xξ4(t) dt=µ0l, 2hξ5l,ψ¯li=µ0◦l is satisfied according to (3.58e), (3.58f).
4. We selecty∈H1(Θ,R`) as the solution to E( ¯ψ,y, ξ2) = p0−f( ¯ψ) +
`
X
l=1
f(D(ξ5, l)) +
`
X
l=1
B(D(ξ5, l))¯u
−
`
X
l=1
(M(D(ξ5, l),ψ) + M( ¯¯ ψ, D(ξ5, l))) ˙¯y
−
`
X
l=1
(A(D(ξ5, l),ψ) + A( ¯¯ ψ, D(ξ5, l)))¯y;
choosingξ1 = ycompletes the proof.
3.4.4. Optimality conditions Consider the Lagrange function
L(x, z,κ) = ˜J(x) +he(x), ziZ0,Z+h˜g(x),κiZ˜. (3.59) We derive a system of optimality conditions for the OS-POD problem (3.44) of the form
∂L
∂u,∂L
∂ω
(¯x,z,¯ κ¯) =
σuu¯− B?p¯−B( ¯ψ)T¯p + ¯κu
σw(¯ω−I( ¯ψ)¯y) +ε2κ¯w
= 0 (3.60)
by the Lagrange multiplier technique; the multipliers p¯ = p(¯y,µ,¯ ψ)¯ ∈ L2(Θ, V) and
¯
p = p(¯y,w,¯ ψ)¯ ∈ H1(Θ,R`) will satisfy adjoint equations, µ¯ ∈ V` solves a nonlinear equation of the form(R(¯y)−¯λl)¯µl=Gl(¯y,¯p,u,¯ w,¯ ψ)¯ and κ¯u∈U×U,κ¯w ∈W ×W are projections of the gradients (3.60) on the admissible domain.
The Lagrange function has the representation L(x, z,κ) = ˜J(x) +he(x), ziZ0,Z+h˜g(x),κiZ˜
Let x¯ ∈ X be an optimal OS-POD solution. Thm. 3.12 guarantees the existence of multipliers z¯∈ Z and κ¯ ∈Z˜ and we get the following first-order optimality conditions in variational form:
Interpreting the variational argument y˜ as a test function, p¯can be considered as a
weak solution to the following backwards partial differential equation:
Again we successfully isolated the variational argument y˜ and hence get p¯ as the solution to the following system of ordinary differential equations:
−M( ¯ψ) ˙¯p(t)+A( ¯ψ)¯p(t)−σw
ε2 IT( ¯ψ)(¯ω(t)−I( ¯ψ)¯y(t))+σQ(M( ¯ψ)¯y(T)−yQ( ¯ψ;t)) = 0 M( ¯ψ)¯p(T) +σΩ(M( ¯ψ)¯y(T)−yΩ( ¯ψ)) = 0 M( ¯ψ)(¯p(0)−p¯◦) = 0.
3. ∂L∂u(¯x,z,¯ κ¯)˜u= 0 for allu˜∈U gives us the gradient condition
By isolation and variation ofu,˜ ω˜ we receive
σuu(t)¯ − B?p(t)¯ −BT( ¯ψ)¯p(t) + ¯κu(t) = 0 (3.61c)
We replace the terms depending on D( ˜ψ, l) by a more explicit representation: Let
then the first-order optimality condition forµ¯l reads as
iXR˜l(¯y,λ¯l)¯µl+ 2µ◦liXψ¯l+Gl(¯x) = 0; (3.61e) we hereby eliminated the multiplierp◦ by the optimality condition ¯p◦= ¯p(0).
5. ∂L∂λ(¯x,z,¯ κ¯)˜λ= 0 for all λ˜ ∈R` finally implies
−
`
X
l=1
λ˜lhψ¯l,µ¯liX = 0 :
for alll∈ {1, ..., `},ψ¯l is orthogonal toµ¯l. Especially, one getshR˜l(¯y,¯λl)¯µl,ψ¯liX = 0.
Testing (3.61e) withψ¯l allows to eliminate the multiplierµ◦:
¯
µ◦l =−1
2hGl(¯x),ψ¯liX0,X. (3.61f) Altogether, we obtain the following system of first-order optimality condintions:
Theorem 3.13. (OS-POD optimality conditions) Consider the OS-POD problem
minx∈XJ(x) s.t. e(x) = 0 & h(x)≤0. (3.62) Assume that x¯= (¯y,u,¯ w,¯ y,¯ ψ,¯ λ)¯ ∈X is a solution to (3.56).
Then there exist multipliers z¯= (¯p,p¯◦,p,¯ ¯p◦,µ,¯ µ¯◦)∈Z andκ¯ = ( ¯κu,κ¯w)∈Z˜ with
−p(t) +˙¯ Ap(t) +¯
`
X
l=1
hy(t),¯ ψ¯liXiXµ¯l+
`
X
l=1
h¯y(t),µ¯liXiXψ¯l= 0 (3.63a)
¯
p(T) = 0
−M( ¯ψ) ˙¯p(t) + A( ¯ψ)¯p(t) + IT( ¯ψ) ¯κw(t) +σQ(M( ¯ψ)¯y(t)−yQ( ¯ψ;t)) = 0 (3.63b) M( ¯ψ)¯p(T) +σΩ(M( ¯ψ)¯y(T)−yΩ( ¯ψ)) = 0
σuu(t)¯ − B?p(t)¯ −BT( ¯ψ)¯p(t) + ¯κu(t) = 0 (3.63c) σww(t) +¯ ε¯κw(t) = 0 (3.63d) (R(¯y)−λ¯l)¯µl− hGl(¯x),ψ¯liX0,Xψ¯l+ i−1X Gl(¯x) = 0 (3.63e) κ¯u−max(0,B?p¯+ BT( ¯ψ)¯p−σuub)−min(0,B?p¯+ BT( ¯ψ)¯p−σuua) = 0 (3.63f)
κ¯w−max
0,σw
ε (I( ¯ψ)¯y−yb)
−min
0,σw
ε (I( ¯ψ)¯y−ya)
= 0. (3.63g)
Remark 3.14. We obtain the optimality system (3.63) by application of the Lagrange calculus on the transformed problem in the penalization variableωand backtransforma-tionω=ω(w). The optimality conditions coincide with those we get by directly applying the formal Lagrange technique to the original problem with penaltyw. ♦