Risk-averse optimal control of random elliptic variational inequalities

2025-04-15 0 0 1.8MB 31 页 10玖币

侵权投诉

Amal Alphonse∗Caroline Geiersbach†Michael Hinterm¨uller‡Thomas M. Surowiec§

Abstract

We consider a risk-averse optimal control problem governed by an elliptic variational inequality (VI) subject

to random inputs. By deriving KKT-type optimality conditions for a penalised and smoothed problem and

studying convergence of the stationary points with respect to the penalisation parameter, we obtain two forms of

stationarity conditions. The lack of regularity with respect to the uncertain parameters and complexities induced

by the presence of the risk measure give rise to new challenges unique to the stochastic setting. We also propose

a path-following stochastic approximation algorithm using variance reduction techniques and demonstrate the

algorithm on a modiﬁed benchmark problem.

Contents

1 Introduction 2

1.1 Notationandbackgroundmaterial ..................................... 3

1.2 Standingassumptions ............................................ 4

1.3 Example.................................................... 5

2 Analysis of the optimisation problem 5

2.1 AnalysisoftheVI .............................................. 5

2.2 Existenceofanoptimalcontrol....................................... 7

3 A regularised control problem 8

3.1 A penalisation of the obstacle problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Stationarity for the regularised control problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Stationarity conditions 15

4.1 Consistency of the approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Passagetothelimit ............................................. 16

5 Numerical example 23

5.1 Problemformulation............................................. 23

5.2 Path-following stochastic approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Conclusion 26

A Diﬀerentiability of superposition operators 28

B Other results 28

AA and MH were partially supported by the DFG through the DFG SPP 1962 Priority Programme Non-smooth and

Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization within project 10.

∗Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany (alphonse@wias-berlin.de)

†Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany (geiersbach@wias-berlin.de)

‡Weierstrass Institute, Mohrenstrasse 39, 10117 Berlin, Germany (hintermueller@wias-berlin.de)

§Department of Scientiﬁc Computing and Numerical Analysis, Simula Research Laboratory, Kristian Augusts gate 23, 0164, Oslo,

Norway (thomasms@simula.no)

arXiv:2210.03425v1 [math.OC] 7 Oct 2022

1 Introduction

In this work, we consider the following nonsmooth stochastic optimisation problem

min

u∈Uad R[J(S(u))] + (u),(1)

where Uad is a set of controls, Sis the solution map of a random elliptic variational inequality (VI), Jis an objective

function, Ris a so-called risk measure that scalarises the random variable J(S(u)), and is the cost of the control

Here, the state S(u) =: ysatisﬁes, on a pointwise almost sure (a.s.) level, the VI

y(ω)≤ψ(ω) : hA(ω)y(ω)−f(ω)−B(ω)u, y(ω)−vi ≤ 0∀v:v≤ψ(ω),(2)

where ω∈Ω stands for the uncertain parameter taken from a probability space (Ω,F,P), f(ω) is a random source

term, ψ(ω) is a random obstacle, and A(ω) and B(ω) are random operators. The map Ris typically a convex

functional chosen to generate solutions according to given risk preferences, e.g., optimal performance on average,

weight of the tail of J(S(u)), and so on.

Numerous free boundary problems in partial diﬀerential equations such as contact problems in mechanics and

ﬂuid ﬂow through porous media can be modelled as elliptic VIs, and in some cases, coeﬃcients or inputs in the

constitutive equations may be uncertain and are modelled as random. Such random VIs have been studied by

[18,19,11,29,5].

It is important to already note here that (1) typically contains two types of nonsmoothness: on the one hand

from the solution operator S, on the other due to the choice of R, which in many interesting cases is a nonsmooth

risk measure such as the average value at risk (usually written AVaR/CVaR). The problem (1)–(2) is formulated

in the spirit of a “here and now” two-stage stochastic programming problem, where the decision uis made before

the realisation ωis made known. The study of such stochastic mathematical programs with equilibrium constraints

(SMPECs), has been limited to the ﬁnite-dimensional, risk-neutral (i.e., R=E) case; see [40,10,47,49]. For

deterministic elliptic MPECs, there have been many developments in terms of theory and algorithms; see, e.g.,

[4,27,38,24,26,34,25,50,20,53,23,39].

The paper contains two contributions. First, using an adaptive smoothing approach, we derive stationarity

conditions related to the well-known weak and C-stationarity conditions. To the best of our knowledge, this is the

ﬁrst attempt at such a derivation. Secondly, we provide a numerical study by applying a variance-reduced stochastic

approximation method to solve an example of (1). Concerning the theoretical developments, our method for

establishing the stationarity conditions is through a penalty approach similar to [25,45]. As with the deterministic

case, the penalty approach has the advantage that it is directly linked to the convergence analysis of solutions

algorithms for the optimisation problem in a fully continuous, function space setting. The theoretical results will

also highlight a hidden diﬃculty unique to the stochastic setting that contrasts with the deterministic elliptic and

parabolic cases. We also believe that this is an inherent diﬃculity in SMPECs in general, regardless of the dimension

of the underlying decision space.

Our work is related to the problem setting in the recent paper [21] where the authors develop a bundle method

for problems of the form (1) with R=Ethe expectation. The focus in our work is on obtaining stationarity

conditions and a stochastic approximation algorithm for the general risk-averse case. Risk-averse optimisation is a

subject in its own right; cf. [48] and [42] and the references therein. Modeling choices in engineering were explored

in [43] and their application to PDE-constrained optimisation was popularised in [31,32,30]. However, these

papers typically require Sto be Fr´echet diﬀerentiable, which does not hold in general for solution operators of

variational inequalities. It is worth mentioning that typically random VIs are studied in combination with some

quantity of interest such as the expectation or variance as in [5,29]. Another modeling choice could involve ﬁnding

a deterministic solution ysatisfying (2), which leads to an expected residual minimisation problem, or a ysatisfying

the expected-value problem

y≤ψ:hE[A(·)y−f(·)−B(·)u], y −vi ≤ 0∀v:v≤ψ.

These modeling choices are discussed in the survey [46], but will not be pursued in the present work.

With respect to the organisation of the paper, after deﬁning in Section 1.1 some terms and notation, Section 1.2

is dedicated to introducing standing assumptions. An example of (1) with speciﬁc choices for the various terms there

is given in Section 1.3. In Section 2, the framework given above is formalised and it is shown that (2) has a unique

solution (see Lemma 2.1). We will show in Proposition 2.4 that the control-to-state map Smaps into Lq(Ω; V) and

hence the composition in (1) is sensible. Moreover, we show in Proposition 2.7 that an optimal control to (1) exists.

In Section 3, we use a penalty approach on the obstacle problem and show that this penalisation is consistent with

(2) (see Proposition 3.2). Optimality conditions for the control problem associated to the penalisation are given

in Proposition 3.11 and Proposition 3.12 and form the starting point for the derivation of stationarity conditions.

The main results culminate in Section 4, namely Proposition 4.9 and Theorem 4.8, where stationarity conditions of

E-almost weak and C-stationarity type are derived. This is done by taking the limit of the optimality conditions

with respect to the penalisation parameter and is an especially delicate procedure due to the presence of the risk

measure. A numerical example is shown in Section 5. For the experiments, a novel path-following stochastic variance

reduced gradient method is proposed in Algorithm 1. Although, as speciﬁed, we work in a particular setting of

obstacle-type problems, we will explain in the concluding Section 6 how greater generality could also be an option.

1.1 Notation and background material

For exponents t≥1, the Bochner space Lt(Ω; V) := Lt(Ω,F,P;V) is the set of all (equivalence classes of) strongly

measurable functions y: Ω →Vhaving ﬁnite norm, where the norm is given by

kykLt(Ω;V):= ((RΩky(ω)kt

VdP(ω))1/t for t∈[1,∞),

ess supω∈Ωky(ω)kVfor t=∞.

We set Lt(Ω) = Lt(Ω; R) for the space of random variables with ﬁnite t-moments. For a random variable Z: Ω →R,

the expectation is deﬁned by E[Z] := RΩZ(ω) dP(ω).

Recall that separability of Vimplies separability of V∗. Moreover, if Xis a separable Banach space, then

strong and weak measurability of the mapping y: Ω →Xcoincide (cf. [22, Corollary 2, p. 73]1). Hence, we can

call the mapping measurable without distinguishing between the associated concepts. Given Banach spaces X, Y ,

an operator-valued function A: Ω → L(X, Y ) is said to be uniformly measurable in Fif there exists a sequence of

countably-valued operator random variables in L(X, Y ) converging almost everywhere to Ain the uniform operator

topology. A set-valued map with closed images is called measurable if the inverse image2of each open set is a

measurable set, i.e., T−1(E)∈ F for every open set E⊂X.

Set R:= R∪ {∞}. Recall that F:X→Ris proper if its eﬀective domain

dom(F) := {x∈X:F(x)<∞}

satisﬁes dom(F)6=∅.Observe that it is always the case that functions mapping into Rsatisfy F > −∞. The

subdiﬀerential of a convex function F:X→Rat zis the set ∂F (z)⊂X∗deﬁned as

∂F (z) := {g∈X∗:F(z)−F(x)≤ hg, z −xiX∗,X ∀x∈X}.

We list some other notation and conventions that will be frequently used:

•Whenever we write the duality pairing h·,·i without specifying the spaces, we mean the one on V∗, i.e., h·,·iV∗,V .

•For weak convergence, we use the symbol and for strong convergence we use the symbol →.

•For a constant t∈[1,∞), t0will denote its H¨older conjugate, i.e., 1

t+1

t0= 1.

•We write →to mean a continuous embedding and c

−→ for a compact embedding.

•L(X, Y ) is the set of bounded and linear maps from Xinto Y.

•M(Ω; X) denotes the set of all measurable functions from Ω into X.

•Statements that are true with probability one are said to hold almost surely (a.s.).

•A generic positive constant that is independent of all other relevant quantities is denoted by Cand may have a

diﬀerent value at each appearance.

1As noted in [22], this result goes back to Pettis [41] from 1938.

2For a set-valued map T: Ω ⇒Xfrom Ω to a separable Banach space X, the inverse image on a set E⊂Xis

T−1(E) := {ω∈Ω: T(ω)∩E6=∅}.

1.2 Standing assumptions

Let us now describe our problem setup more precisely:

(i) D⊂Rdis a bounded Lipschitz domain for d≤4, and take

H:= L2(D) and V∈ {H1(D), H1

0(D)}.

(ii) Uad ⊂Uis a non-empty, closed and convex set where the control space Uis a Hilbert space.

(iii) (Ω,F,P) is a complete probability space, where Ω represents the sample space, F ⊂ 2Ωis the σ-algebra of

events on the power set of Ω, and P: Ω →[0,1] is a probability measure.

(iv) f∈Lr(Ω; V∗) is a source term and ψ∈Ls(Ω; V) is an obstacle for r, s ∈[2,∞].

(v) R:Lp(Ω) →Rand :U→Rare proper and p∈[1,∞).

To ease the presentation of the results, we make the following assumption on the almost everywhere boundedness

of the operators in play in (2).

Assumption 1.1. The operators A: Ω → L(V, V ∗)and B: Ω → L(U, V ∗)are uniformly measurable and there exist

positive constants Cb, Ca, Ccsuch that for all y, z ∈Vand u∈Uand for a.e. ω,

hA(ω)y, yi ≥ Cakyk2

hA(ω)y, zi ≤ CbkykVkzkV,

hB(ω)u, zi ≤ CckukUkzkV.

(3)

In applications, the operators Aand Bmay be generated by random ﬁelds. There are numerous examples of

random ﬁelds that are compactly supported; while this choice precludes a lognormal random ﬁeld for A, in numerical

simulations truncated Gaussian noise is often employed to generate samples. See, e.g., [37,51,17] for examples of

compactly supported random ﬁelds, including approximations of lognormal ﬁelds as described.

The nature of the feasible set and composite objective function necessitates several assumptions on the objective

functional. These ensure integrability, continuity and later, diﬀerentiability.

Assumption 1.2. Assume that J:V×Ω→Ris a Carath´eodory3function and that there exists C1∈Lp(Ω) and

C2≥0such that

|J(v, ω)| ≤ C1(ω) + C2kvkq/p

where

2≤q < ∞, q ≤min(r, s).(4)

For y: Ω →V, we deﬁne the superposition operator J(y) : Ω →Rby J(y)(ω) := J(y(ω), ω).The necessary and

suﬃcient conditions to obtain continuity of Jare directly related to famous results by Krasnosel’skii; see [33] and

[52, Theorem 19.1]. Thanks to Assumption 1.2, it follows by [16, Theorem 4] that

J:Lq(Ω; V)→Lp(Ω) is continuous.(5)

Remark 1.3. If f∈L∞(Ω; V∗)and ψ∈L∞(Ω; V)(so that r=s=∞), (4)forces us to take qto be ﬁnite. The

case q=∞creates technical diﬃculties that we will address in a future work.

In typical examples4,Uc

−→ V∗is a compact embedding and we would like the operator Bto mimic this compact

embedding. For that purpose, we need the next assumption.

Assumption 1.4. If un u in Uthen B(ω)un→B(ω)uin V∗a.s.

Further assumptions will be introduced as and when required later in the paper.

3That is, J(v, ·) is measurable for ﬁxed vand J(·, ω) is continuous for ﬁxed ω.

4Although Uis often taken to be L2(D) in the literature, other examples of Uone could consider include Rn,L2(∂Ω), H−1/2(∂Ω).

1.3 Example

Take V=H1

0(D) and let a∈L∞(Ω ×D) be a given function such that a0≤a(ω, x)≤a1a.s. and for a.e. x, where

a0>0 and a1> a0are both constants. Deﬁne the operator

A(ω) := −∇ · (a(ω)∇u),

understood in the usual weak sense:

hA(ω)y, zi=ZD

a(ω)∇y∇zdxfor y, z ∈H1

0(Ω).

Set U=L2(D) with the box constraint set

Uad := {u∈L2(D) : ua≤u≤uba.e.},

where ua, ub∈L2(D) are given functions. For B, we take it to be the canonical embedding L2(D)c

−→ H−1(D), i.e.,

B(ω)u≡uas an element of V∗=H−1(Ω). Take f∈L2(Ω; H−1(D)), ψ∈L2(Ω; H1

0(D)) and the exponent q= 2.

Let p= 1 and deﬁne

J(y) := 1

2ky−ydk2

Hand (u) := ν

2kuk2

H(6)

where yd∈L2(D) is a given target state and ν > 0 is the control cost. The risk measure is chosen to be the

conditional value-at-risk, which for β∈[0,1) is deﬁned for a random variable X: Ω →Rby

R[X] = CVaRβ[X] = inf

s∈Rs+1

1−βE[max{X−s, 0}].(7)

This risk measure is ﬁnite, convex, monotone, continuous and subdiﬀerentiable (see [48,§6.2.4]) and turns out to

satisfy every assumption we will make in this paper. CVaR is easily interpretable: given a random variable X,

CVaRβ[X] gives the average of the tail of values Xbeyond the upper β-quantile. The minimisers in (7) correspond

to the β-quantile. CVaRβapproaches the essential supremum as β→1.

Further examples of risk measures can be found in [31,§2.4] and references therein.

2 Analysis of the optimisation problem

We begin by studying various properties of the solution map to the VI (2) and addressing the control problem (1).

The solution mapping u7→ y(ω) in (2) is denoted by Sω:U→Vand its associated superposition operator Sby

S(u)(ω) := Sω(u).(8)

2.1 Analysis of the VI

We make heavy use, in particular, of the standing assumptions Assumption 1.1.

Lemma 2.1. For almost every ω, there exists a unique solution to (2)satisfying the estimate

kSω(u)kV≤C(kf(ω)kV∗+kukU+kψ(ω)kV) (9)

where the constant C > 0depends only on Cb, Caand Cc.

Proof. The conditions (3) ensure the existence and uniqueness of the solution to (2) for each ωby the Lions–

Stampacchia theorem; see [36].

For the estimate we argue as follows. Setting v=ψ(ω) in (2) and splitting with Young’s inequality with , we

obtain

Caky(ω)k2

V≤ hA(ω)y(ω), ψ(ω)i+hf(ω) + B(ω)u, y(ω)−ψ(ω)i

≤Cbky(ω)kVkψ(ω)kV+ (kf(ω)kV∗+CckukU)ky(ω)kV

+ (kf(ω)kV∗+CckukU)kψ(ω)kV

≤Ca

3ky(ω)k2

V+3C2

4Cakψ(ω)k2

V+3

4Ca

(kf(ω)kV∗+CckukU)2+Ca

3ky(ω)k2

2(kf(ω)kV∗+CckukU)2+1

2kψ(ω)k2

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Risk-averseoptimalcontrolofrandomellipticvariationalinequalitiesAmalAlphonse*CarolineGeiersbachMichaelHintermullerThomasM.Surowiec§AbstractWeconsiderarisk-averseoptimalcontrolproblemgovernedbyanellipticvariationalinequality(VI)subjecttorandominputs.ByderivingKKT-typeoptimalityconditionsforapenali...

展开>> 收起<<

Risk-averse optimal control of random elliptic variational inequalities.pdf

共31页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Risk-averse optimal control of random elliptic variational inequalities

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: