Gö lich, Simone; Mehli z, Pa ick; Schillinge , Thomas
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In e se demand acking in anspo a ion ne wo ks
Ma hema ical Me hods o Ope a ions Resea ch
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ORIGINAL ARTICLE
In e se demand acking in anspo a ion ne wo ks
Simone Gö lich1·Pa ick Mehli z2·Thomas Schillinge 1
Recei ed: 14 July 2023 / Re ised: 9 Augus 2024 / Accep ed: 15 Augus 2024 /
Published online: 5 Oc obe 2024
© The Au ho (s) 2024
Abs ac
This pape deals wi h he econs uc ion o he desi ed demand in an op imal con ol
p oblem, s a ed o e a ee-shaped anspo a ion ne wo k which is go e ned by a lin-
ea hype bolic conse a ion law. As desi ed demands ypically unde go luc ua ions
due o seasonali y o unexpec ed e en s making sho - e m adjus men s necessa y,
such an app oach can exempla y be used o o ecas ing om pas da a. We sugges
o model his p oblem as a so-called in e se op imal con ol p oblem, i.e., a hie a -
chical op imiza ion p oblem whose inne p oblem is he op imal con ol p oblem and
whose ou e p oblem is he econs uc ion p oblem. In o de o gua an ee he exis-
ence o solu ions in he unc ion space amewo k, he hype bolic conse a ion law
is in e p e ed in weak sense allowing o con ol unc ions in Lebesgue spaces. Fo
he compu a ional ea men o he model, we ans e he hie a chical p oblem in o a
nonsmoo h single-le el one by plugging he uniquely de e mined solu ion o he inne
op imal con ol p oblem in o he ou e econs uc ion p oblem be o e applying ech-
niques om nonsmoo h op imiza ion. Some nume ical expe imen s a e p esen ed o
isualize a ious ea u es o he model including di e en ypes o noise in he demand
and s a egies o how o obse e he ne wo k in o de o ob ain good econs uc ions
o he desi ed demand.
Keywo ds In e se op imal con ol ·Linea hype bolic conse a ion laws ·
T anspo a ion ne wo ks
Ma hema ics Subjec Classi ica ion 49J20 ·65M32 ·90C33 ·90C35
BSimone Gö lich
[email p o ec ed]
Pa ick Mehli z
mehli z@uni-ma bu g.de
Thomas Schillinge
[email p o ec ed]
1School o Business In o ma ics and Ma hema ics, Uni e si y o Mannheim, 68159 Mannheim,
Ge many
2Depa men o Ma hema ics and Compu e Science, Philipps-Uni e si ä Ma bu g, 35032 Ma bu g,
Ge many
123
636 S. Gö lich e al.
1 In oduc ion
Flow p oblems o e ene gy and supply ne wo ks model a b oad ange o in e es ing
applica ions, see (B essan e al. 2014) o a su ey. In his pape , we in es iga e ans-
po a ion ne wo ks o ee shape whe e he low on edges is modeled, o simplici y,
ia (linea ) hype bolic conse a ion laws, as ypically used o elec ic ansmission
lines (Gö lich e al. 2016), hea ing ne wo ks (Rein e al. 2020), o ne wo ks o gas
pipelines (Banda e al. 2006; Guga e al. 2018). A con ol unc ion is used o model he
in low a some sou ce e ex, and he aim o op imiza ion is o choose his unc ion
in such a way ha ce ain desi ed demands a he sinks o he ne wo k a e acked
as close as possible. As men ioned in some ecen con ibu ions, see (Gö lich e al.
2019; Gö lich and Schillinge 2022a,b), hese desi able demands a e subjec o pe -
u ba ions, noise, o o he sou ces o s ochas ici y. In he a o emen ioned pape s, his
issue has been aced by modeling he p oblem as a s ochas ic op imal con ol p oblem
which is in luenced by andomness ia app op ia ely chosen s ochas ic p ocesses.
In his pape , we a e conce ned wi h ela ed phenomena. Le us conside he ol-
lowing p ac ically ele an si ua ion. The e exis s a company (C2) which appoin s a
second company (C1) o deli e a ce ain amoun o elec ici y/hea /gas a he demand
e ices o e ime by inse ing he eques ed p oduc a he sou ce o he ne wo k o e
ime. In his ega d, C1 has o sol e he op imal con ol p oblem men ioned abo e. We
now en ich he conside ed si ua ion by assuming ha he e is a ne wo k ope a o (NO),
di e en om C1 and C2, which pa ially obse es he low along he ne wo k and,
depending on his, cha ges C1 and C2 o pay some ax o employing he ne wo k. As
ou lined abo e, he desi ed demands eques ed by C2 a e subjec o s ochas ic in lu-
ences and, addi ionally, may a y due o a seasonal beha io . F om pas da a, NO now
wan s o o ecas he desi ed demand o C2 and he associa ed ac ions o C1, exem-
pla y o ixing axes o plan u u e income. Typically, NO is no awa e o he desi ed
demand as he only obse es he ac ual ne wo k low along some bu , mos likely, no
all edges o he ne wo k (as i migh be expensi e o equip he o e all ne wo k wi h
senso s o o un hem on each edge o e all ime). Fu he mo e, he o ecas ing model
should be capable o ecognizing seasonal beha io o he desi ed demands as i is
exempla ily p esen ed o an elec ici y ma ke in Coskun and Ko n (2021).
In o de o model his si ua ion, we conside i om he iewpoin o in e se op imal
con ol, i.e., we aim o iden i y pa ame e s in an op imal con ol p oblem (and no
only in a dynamical sys em). He e, he op imal con ol p oblem o in e es is he
a o emen ioned ne wo k low p oblem, and he appea ing desi ed demand plays he
ole o his pa ame e . We assume ha we a e gi en obse ed (bu , mos likely, noisy)
pai s o op imal in low and op imal ne wo k low, and aim o econs uc he desi ed
demands which a e modeled as a con ex combina ion o gi en ansa z unc ions. I is,
hus, ou goal o ind he associa ed weigh pa ame e s which cha ac e ize a sui able
s anda d (pe iodically eme ging) choice o he desi ed demand. As we a e in e es ed
in he obus ness o ou app oach, we conside addi ional pe u ba ions in he model
and s udy di e en ypes o empo al es ic ions in he obse a ion o he ne wo k o
e alua e whe he hese a e su icien o good o ecas ing.
Na u ally, he model o in e es is a hie a chical op imiza ion p oblem wi h wo
decision le els. Coming back o ou exempla y si ua ion om abo e, a he ou e
123
In e se demand acking in anspo a ion ne wo ks 637
(o uppe -le el) p oblem, he NO is in posi ion o pa ially obse e he ne wo k and
chooses ce ain weigh s, which hen gi e a angible desi ed demand. A he inne (o
lowe -le el) p oblem, C1 now can sol e he ne wo k low p oblem. Along hose pa s
o he ne wo k, which a e obse ed by NO, he la e can compa e he pas da a and he
eal- ime da a ob ained om he inne p oblem o his pa icula choice o he weigh
pa ame e s. No ing ha his decision o de leads o a well-posed p oblem, NO aims
o choose he weigh pa ame e s in such a way ha pas da a and eal- ime da a ma ch
as good as possible. As ou model has wo decision le els, i is a so-called bile el
op imiza ion p oblem.
Fo mo e han 50 yea s, bile el op imiza ion is a majo ield o esea ch in ma h-
ema ical p og amming due o nume ous unde lying applica ions e.g. in da a science,
economy, inance, machine lea ning, o na u al sciences, see (Ba d 1998; Dempe
2002; Shimizu e al. 1997) o an in oduc ion and Dempe (2020) o a ecen su ey
which p esen s an o e iew o con ibu ions in his a ea. Recen ly, bile el op imiza-
ion u ned ou o be o pa icula in e es in he con ex o anspo a ion o ene gy
ne wo ks, see e.g. Dempe e al. (2015). This also includes he apidly g owing ield
o hie a chical con ol, see e.g. Mehli z and Wachsmu h (2020) o an o e iew, and,
pa icula ly, so-called in e se op imal con ol al eady men ioned ea lie , see (Hinze
e al. 2009; T öl zsch 2010; T ou man 1996; Vin e 2010) o an in oduc ion o he
opic o op imal con ol. In e se con ol possesses se e al in e es ing applica ions e.g.
in he con ex o human locomo ion, see (Alb ech e al. 2012; Alb ech and Ulb ich
2017; Alb ech e al. 2010; Mombau e al. 2010). The heo y on in e se op imal
con ol including o dina y and pa ial di e en ial equa ions add esses he exis ence
o solu ions, op imali y condi ions, and solu ion algo i hms, see e.g. (Dempe e al.
2019; F iedemann e al. 2023; Ha de and Wachsmu h 2019;Ha ze al.2012; Holle
e al. 2018; Su yan e al. 2016) and is de eloping as . In abs ac bile el op imiza ion,
wo decision make s, a leade and a ollowe , need o choose a iables in o de o
minimize hei associa ed cos unc ion which also depends on he a iables o he
o he decision make , espec i ely. Mo e p ecisely, he leade chooses his a iables
i s which a e handed o e o he ollowe who now can sol e his op imiza ion p ob-
lem (which is pa ame ic in he leade ’s a iable) o global op imali y. The solu ions
a e hen gi en o he leade , who now can e alua e his objec i e. O en, one assumes
ha leade and ollowe coope a e in o de o op imize he leade ’s objec i e, and his
p ocedu e is e e ed o as he op imis ic app oach o he p oblem, see (Zemkoho 2016)
o an o e iew o o he app oaches a oiding ill-posedness in bile el op imiza ion.
The leade ’s and ollowe ’s p oblem a e o en e e ed o as uppe - and lowe -le el
p oblem, espec i ely. As he ollowe has o de e mine globally op imal solu ions o
his p oblem by na u e o bile el op imiza ion, one ypically equi es ha he lowe -
le el p oblem is con ex in he ollowe ’s a iable in o de o ci cum en issues ela ed
o noncon ex global op imiza ion a he lowe -le el s age.
We s a ou in es iga ions by modeling he p oblem o in e es as an in e se con-
ol p oblem in Sec .2. The e o e, we i s s udy he exis ence o solu ions o linea
hype bolic conse a ion laws in a unc ion space which is sui able o op imal con ol
be o e se ing up he lowe - and uppe -le el p oblem consecu i ely. Fu he mo e, we
demons a e ha he esul ing op imiza ion p oblem possesses an op imal solu ion in
he unc ion space se ing we a e in es iga ing. In Sec . 3, we add ess he compu a-
123
638 S. Gö lich e al.
ional ea men o he model. Sec ion3.1 desc ibes ou app oach o he nume ical
solu ion o he p oblem. As i is analy ically possible o compu e he ne wo k low
associa ed wi h he inpu , we a e in posi ion o dis ill a s a e- educed e sion o he
pa ame ic op imal con ol p oblem. The associa ed solu ion ope a o , which, a leas
in poin wise ashion, can simila ly be compu ed analy ically due o he nice s uc u e
o he p oblem, u ns ou o be a nonsmoo h single- alued mapping. Plugging he
la e in o he supe o dina e econs uc ion p oblem and pe o ming a sui able dis-
c e iza ion, we end up wi h a nonsmoo h op imiza ion which we sol e wi h he aid
o MATLAB’s pa e nsea ch sol e in de aul mode. The gene al se -up o ou
compu a ional expe imen s is ca ed ou in Sec . 3.2. Nume ical esul s a e p esen ed
in Sec .3.3 in o de o isualize he e ec i eness and se e al di e en ea u es o
he app oach. Pa icula ocus is laid on he obus ness o he model wi h espec o
addi ional unce ain ies, es ic ed obse a ion op ions, and he p esence o addi ional
in low cons ain s. Some concluding ema ks close he pape in Sec . 4.
2 The model p oblem
In his sec ion, we se up he model o ou in e es . Fi s , we discuss he pa icula
shape o he lowe -le el pa ame ic op imal con ol p oblem in Sec .2.1. The e o e,
we i s p esen he unde lying ne wo k dynamics and discuss egula i y ea u es o
associa ed solu ions. Second, he lowe -le el objec i e unc ion is cons uc ed, and
sol abili y o he o e all lowe -le el p oblem is discussed. In Sec . 2.2, we de i e
he supe o dina e uppe -le el p oblem and demons a e ha i possesses an op imal
solu ion in he unc ion space se ing.
2.1 The lowe -le el p oblem
In his subsec ion, we a e conce ned wi h he de i a ion and analysis o he lowe -
le el op imal con ol p oblem. To s a , we s a e he lowe -le el dynamics and discuss
exis ence and uniqueness o solu ions associa ed wi h his sys em. A e wa ds, we
se up he (pa ame ic) lowe -le el p oblem, show ha , o each se o pa ame e s,
i possesses a unique solu ion, and in es iga e p ope ies o he associa ed solu ion
ope a o .
2.1.1 Se ing up he ne wo k and ne wo k dynamics
We conside a di ec ed g aph G=(V,E)which is a ee (in he sense ha whene e
he di ec ed edges a e in e p e ed as undi ec ed, hen he esul ing g aph would be
ee o cycles). Le us use he no a ion V:= { 0,...,
n}and no e ha |E|=nby
na u e o ees. Some mo e de ails on Gand he no a ion we a e going o exploi a e
discussed below.
•The uniquely de e mined sou ce e ex o he ne wo k Gis 0∈V. Fu he mo e,
we assume ha 0is a lea o G, i.e., he e is only one edge which lea es 0, and
he e ex a i s end will be deno ed by 1.
123
In e se demand acking in anspo a ion ne wo ks 639
Fig. 1 An exempla y ne wo k wi h VD={ 4,
5,
6},VI={ 1,
2,
3},E+(2)={(4), (5)},and
ED={(4), (5), (6)}
•In VD⊂V, we collec all e ices which possess no ou going edges. These a e
he demand e ices.
•All emaining in e media e (o inne ) e ices o he ne wo k a e collec ed in he
se VI:= V (VD∪{ 0}).
•Fo i∈V { 0}, we iden i y he uniquely de e mined edge which ends a iby
(i).
•The se E+(i)is used o deno e he se o all edges s a ing a e ex i. Fu he mo e,
we use ED:= {(i)∈E| i∈VD} o deno e he se o edges ha end a a demand
e ex. Clea ly, |ED|=|VD|.
We isualize he abo e no a ion in Fig.1. Fo he heo y o his pape , i is no
manda o y ha he e ex 0possesses jus one ou going edge. One can in e p e 0as
an ups eam supe sou ce. Besides, his addi ional assump ion simpli ies he no a ion
because we can abs ain om he in oduc ion o dis ibu ion pa ame e s a he in low
e ex la e on.
A he sou ce 0, he injec ion o low o e ime T:= (0,T), whe e T>0is he
inal ime, is modeled by he con ol a iable u:T→Rwhich has o be chosen om
an app op ia e unc ion space.
The low o e (i)a ime ∈Ta he spa ial coo dina e x∈will be deno ed by
z(i)( ,x). He e, we assume ha := (0,ω)is a bounded eal in e al. The densi y
has o obey he linea hype bolic conse a ion law
z(i)
( ,x)+λ(i)z(i)
x( ,x)=0,a.e. on T×, (i)∈E,(2.1a)
z(i)(0,x)=0,a.e. on , (i)∈E,(2.1b)
λ(1)z(1)( ,0)=u( ), a.e. on T,(2.1c)
λ(k)z(k)( ,0)=αi,kλ(i)z(i)( ,ω), a.e. on T,
i∈VI,(k)∈E+(i). (2.1d)
Pa icula ly, he lux unc ions o he conse a ion law a e o linea s uc u e. Fo each
i∈{1,...,n},λ(i)>0 is a gi en cons an . Abo e, o each i∈VIand (k)∈E+(i),
αi,k>0 is a cons an such ha (k)∈E+(i)αi,k=1 holds, i.e., he coe icien s αi,k
model how he low spli s a e ex iin o he lows along he edges om E+(i).
This way, (2.1d) conse es he low. We no e ha he heo y can be ex ended o mo e
gene al si ua ions. Exempla y, s anda d linea damping e ms o ype μ(i)z(i)( ,x)
123
640 S. Gö lich e al.
can be inco po a ed in (2.1a) o eal cons an s μ(i)>0 o each (i)∈Ewi hou
any p oblem. Unde addi ional assump ions, he coe icien s λ(i)and μ(i)may also
depend on ime. Wi hou loss o gene ali y one could choose ω:= 1. Howe e , in
o de o clea ly dis inguish be ween empo al and spa ial a iables in no a ion, we
s ick o he seemingly mo e gene al si ua ion whe e ω>0 is a bi a y. Fu he mo e,
he indings in his pape ex end o connec ed ne wo ks wi hou cycles, bu apa om
a mo e di icul no a ion, which also allows o e ices whe e lows a e me ged, we
do no belie e ha such a model comes along wi h a signi ican ly di e en heo y. We,
hus, concen a e on ee-shaped ne wo ks.
2.1.2 Discussion o he hype bolic conse a ion law
Le us i s e iew a classical exis ence esul o he linea hype bolic conse a ion
law (2.1). The e o e, we de ine a sui able con ol space by
C1
00(T):= u∈C1(T)|u(0)=0,u(0)=0.
We equip C1
00(T)wi h he classical C1-no m, and no e ha his space is a closed
subspace o C1(T). The p oo o he ollowing esul , which is based on he me hod o
cha ac e is ics, can be dis illed om B essan (B essan 2000, Sec ion 3.1, Theo ems 3.4
and 3.6) unde he condi ion ha we only conside posi i e eloci ies on he ne wo k
and, hus, all wa es a e mo ing wi h posi i e speed.
P oposi ion 2.1 Fo each u ∈C1
00(T), he hype bolic conse a ion law (2.1) pos-
sesses a unique solu ion z := (z(1),...,z(n))∈C1(T×, Rn). The la e is
explici ly gi en by
∀( ,x)∈T×:z(1)( ,x)=1
λ(1)u( −x/λ(1)) −x/λ(1)>0,
0 −x/λ(1)≤0(2.2)
on edge (1), and o each i ∈{1,...,n}such ha i∈VIand (k)∈E+(i), we ind
∀( ,x)∈T×:z(k)( ,x)=αi,kλ(i)
λ(k)z(i)( −x/λ(k),ω) −x/λ(k)>0,
0 −x/λ(k)≤0.
(2.3)
Addi ionally, he e is a cons an κ>0, no depending on u, such ha zC1(T×,Rn)≤
κuC1(T).
Le us no e ha o mula (2.3) can be used ecu si ely o de e mine he solu ion
along all edges o he ne wo k. Indeed, based on (2.2), he solu ion along all a cs om
E+(1)can be compu ed. Nex , using (2.3), i is possible o de e mine he low along
all edges s a ing in hose e ices which a e he end e ex o some edge in E+(1).
Repea ing his p ocedu e, one can i e a e h ough he whole ne wo k.
123
In e se demand acking in anspo a ion ne wo ks 641
Clea ly, P oposi ion 2.1 jus i ies o in oduce a map om C1
00(T) o C1(T×, Rn)
which assigns o each con ol unc ion om C1
00(T) he associa ed uniquely de e -
mined solu ion o (2.1). This mapping is a linea ope a o which is con inuous by
P oposi ion 2.1.
Since we a e in e es ed in he op imal con ol o he sys em (2.1), wo king wi h
he con ol space C1
00(T)induces some inhe en di icul ies. Fi s , his space is non-
e lexi e, i.e., o show he exis ence o op imal solu ions o op imiza ion p oblems
o e (2.1) and he supe o dina e in e se op imal con ol p oblem, which we s a e in
Sec .2.2, would be challenging. Second, he dual o his space, which na u ally a ises
when using he adjoin app oach o he de i a ion o op imali y condi ions, is la ge
and di icul o handle nume ically. I is, hus, a easonable ask o econside (2.1)
om he iewpoin o con ol unc ions u∈L2(T). Besides, his choice allows o dis-
con inuous con ols which can be exploi ed o model swi ches in he in low. Obse e
ha (2.1) does no need o possess a classical solu ion in he sense o P oposi ion 2.1
anymo e whene e he con ol unc ion is no con inuously di e en iable. To p oceed,
we ollow (Keime 2014, Sec ion 2.2), see (Guga e al. 2015, Sec ion 2) as well, o
in oduce a sui able weak o mula ion o (2.1) as s a ed below. Fi s , o he s a e z(1),
we demand
Tτ
z(1)( ,x)(ϕ ( ,x)+λ(1)ϕx( ,x))dxd
=−Tτ
u( )ϕ( ,0)d ∀ϕ∈Wτ(2.4)
o all τ∈T, whe e Tτ:= (0,τ)and
Wτ:= ϕ∈C1(Tτ×)
ϕ(·,ω)=0on
Tτ
ϕ(τ,·)=0on
is he space o es unc ions. Simila ly as abo e, we demand
Tτ
z(k)( ,x)(ϕ ( ,x)+λ(k)ϕx( ,x))dxd
=−αi,kλ(i)Tτ
z(i)( ,ω)ϕ( ,0)d ∀ϕ∈Wτ(2.5)
o all τ∈T, i∈VI, and (k)∈E+(i). A unc ion z∈C(, L2(T,Rn)) sa is ying
hese equi emen s is e e ed o as a weak solu ion o he hype bolic conse a ion
law (2.1). Recall ha he unc ion space C(, L2(T,Rn)) comp ises all unc ions
z:T×→Rnsuch ha , o each x∈,z(·,x)belongs o L2(T,Rn), and x→
z(·,x)∈L2(T,Rn)is con inuous. Le us emphasize ha he bounda y condi ions
(2.1b), (2.1c) a e inco po a ed in his al e na i e o mula ion o he dynamics also in
weak sense only (by de ini ion o he space Wτ) since poin wise conside a ions a e
meaningless in Lebesgue spaces.
123
642 S. Gö lich e al.
The ollowing esul shows ha he (classical) solu ion cha ac e ized in P oposi ion
2.1 (wi h con ols chosen om C1
00(T)) also p o ides he uniquely de e mined weak
solu ion o he hype bolic conse a ion law (2.1) i he con ol is chosen om L2(T).
P oposi ion 2.2 Fo each u ∈L2(T), he unc ion z := (z(1),...,z(n))∈
C(, L2(T,Rn)) cha ac e ized ia (2.2), (2.3) is he uniquely de e mined weak solu-
ion o he hype bolic conse a ion law (2.1). Addi ionally, he e is a cons an κ>0,
no depending on u, such ha zC(,L2(T,Rn)) ≤κuL2(T).
P oo Le us s a o show ha z(1)gi en in (2.2) sa is ies (2.4) o each τ∈Tand
gi en u∈L2(T). The e o e, we in oduce a unc ion ¯u∈L2((−ω/λ(1),T)) by
∀ ∈(−ω/λ(1),T):¯u( ):= u( ) >0,
0 ≤0.
Using a coo dina e ans o ma ion wi h espec o he new domain
τ:= {(s,x)∈R2|x∈, s∈(−x/λ(1),τ −x/λ(1))},
we ind, o each ϕ∈Wτand ¯ϕ(s,x):= ϕ(s+x/λ(1),x) o all (s,x)∈τ, he
iden i ies
Tτ
z(1)( ,x)(ϕ ( ,x)+λ(1)ϕx( ,x))dxd
=1
λ(1)Tτ
¯u( −x/λ(1))(ϕ ( ,x)+λ(1)ϕx( ,x))dxd
=τ
¯u(s)¯ϕx(s,x)d(s,x)
=τ
−ω/λ(1)min(ω,λ(1)(τ−s))
max(0,−λ(1)s)
¯u(s)¯ϕx(s,x)dxds
=τ
0
u(s)min(ω,λ(1)(τ−s))
0
¯ϕx(s,x)dxds
=τ
0
u(s)( ¯ϕ(s,min(ω, λ(1)(τ −s))) −¯ϕ(s,0))ds
=−Tτ
u(s)ϕ(s,0)ds.
Abo e, we used he ac ha he de e minan o he Jacobian associa ed wi h he chosen
coo dina e ans o m is 1, he undamen al heo em o calculus, and
¯ϕ(s,min(ω, λ(1)(τ −s))) =ϕ(τ,λ(1)(τ −s)) =0λ(1)(τ −s)<ω,
ϕ(s+ω/λ(1),ω)=0λ(1)(τ −s)≥ω,
123
In e se demand acking in anspo a ion ne wo ks 649
Le us demons a e ha he econs uc ion p oblem (UL) possesses an op imal
solu ion.
P oposi ion 2.9 The op imiza ion p oblem (UL) possesses a globally op imal solu ion.
P oo We no e ha (UL) can be ans e ed in o a ini e-dimensional op imiza ion
p oblem by plugging he lowe -le el solu ion ope a o in o he objec i e unc ion.
I is ob ious ha a poin β∈(Rm)|VD|is a global minimize o he esul ing con ol-
educed p oblem i and only i (β, (β)) is a global minimize o (UL). By con inui y
o , see P oposi ion 2.7, and con inui y o Cas well as D, he objec i e unc ion o he
educed p oblem is hen con inuous, while i s easible se (m)|VD|is nonemp y and
compac . Thus, he educed p oblem possesses a global minimize ¯
β∈(Rm)|VD|by
he Weie s aß heo em, and his yields ha (¯
β,( ¯
β))sol es (UL) o global op imali y.
Al hough being globally Lipschi z con inuous, see P oposi ion 2.7, he lowe -le el
solu ion ope a o , which, a leas in disc e ized o m, see Appendix A, can be
ep esen ed as he composi ion o a linea , con inuous ope a o and he p ojec ion
on o Uad, is likely o be nonsmoo h apa om he special si ua ion whe e no con ol
cons ain s a e p esen , see Rema k 2.8. Elimina ing he con ol a iable uin (UL)
by plugging in o he objec i e unc ion, hus, leads o a ini e-dimensional bu
noncon ex, nonsmoo h op imiza ion p oblem wi h polyhed al cons ain s. Whene e
Uad =L2(T)holds, is linea , see Rema k 2.8 again, and (UL) is ac ually a con ex
op imiza ion p oblem. In his pa icula si ua ion, nume ical me hods which iden i y
s a iona y poin s o (UL) may al eady compu e global minimize s o he p oblem.
This is a a e p ope y in hie a chical op imiza ion whe e he mul ile el s uc u e is,
ypically, a sou ce o noncon exi y and nonsmoo hness, and his p oblem we also ace
in he gene al se ing whe e con ol cons ain s a e p esen .
3 Nume ical solu ion and compu a ional esul s
In his sec ion, we i s desc ibe how (UL) can be sol ed in nume ical p ac ice. Second,
esul s o some compu a ional expe imen s a e p esen ed.
3.1 Nume ical solu ion o he p oblem
Fo he ne wo k disc e iza ion, we choose a ime g id ( j)J
j=1o J∈Ndisc e iza ion
poin s such ha j:= (j−1) o all j∈{1,...,J}, whe e >0isagi en
empo al s epsize, and a spa ial disc e iza ion o each edge (i), ep esen ed by he
in e al (0,ω),as(x(i)
q)L(i)
q=1, whe e L(i)∈Nis he numbe o disc e iza ion poin s,
x(i)
q:= (q−1)x(i) o all q∈{1,...,L(i)}, and x(i)>0 is he spa ial s epsize
o edge (i). The anspo ed quan i ies z(i)
j,qa ime jand posi ion x(i)
qgi en by he
PDE in (2.1a) a e calcula ed using a le -sided upwind scheme, i.e.,
z(i)
j,q=z(i)
j−1,q−
x(i)λ(i)z(i)
j−1,q−z(i)
j−1,q−1,j∈{2,...,J},q∈{2,...,L(i)}.
123
650 S. Gö lich e al.
We also no e ha (2.1b) ansla es in o z(i)
1,q=0 o all q∈{1,...,L(i)}.A he
junc ions, acco ding o (2.1c) and (2.1d), we equi e
z(1)
j,1=uj
λ(1),z(k)
j,1=αi,k
λ(i)
λ(k)z(i)
j,L(i),
i∈VI,(k)∈E+(i), j∈{1,...,J}
whe e uj:= u( j) o all j∈{1,...,J}.Fo
x(i)λ(i)=1, he upwind scheme shows
no di usion. The e o e, we se x(i):= λ(i) which leads o di e en spa ial g ids
on he di e en edges whene e he espec i e coe icien s λ(i)a e no he same.
We use his disc e iza ion o a ini e di e ences app oxima ion o he lowe -le el
p oblem (LL(β)). We de ine S(i),L(i)∈RJ×J o be he (disc e e) ealiza ion o S(i)
ω
such ha J
ν=1S(i),L(i)
j,ν uνapp oxima es he in luence o he disc e ized in low on he
densi y z(i)
j,L(i)a ime jand spa ial poin ω. Fu he , we deno e he disc e e e sions
o he demand p o iles D(i)
1,...,D(i)
m o edge (i)∈EDby ˜
D(i)
1,..., ˜
D(i)
m∈RJ.
Fo ou compu a ions, we will exploi ha he columns o S(i),L(i)a e o hogonal
o each o he . This is he case since, due o he special s uc u e o he PDEs, he e is
a one- o-one co espondence be ween he in low in o he sys em and he ou low ou
o he sys em. The e o e, in he disc e ized se ing, he e is a unique ime poin o he
in low ha de e mines he ou low a he co esponding ou low ime. This p ope y
en o ces he ma ix S(i),L(i) o be nonze o on i s subdiagonal. Consequen ly, S(i),L(i)
is o hogonal.
Fo a gi en con ex combina ion o base demands by he ec o βand using (2.10),
we ob ain he op imal in low in he disc e ized se ing in he absence o con ol con-
s ain s when sol ing he linea sys em Au −Bβ=0 whe e Ais gi en by
A:=
(i)∈EDS(i),L(i)S(i),L(i)+σIJ,
whe e IJ∈RJ×Jis he iden i y ma ix, and
B:= Q(i)(i)∈ED.
Abo e, o each (i)∈ED,Q(i)∈RJ×mis gi en by
Q(i):= S(i),L(i)˜
D(i)
1... S(i),L(i)˜
D(i)
m.
We no e ha he disc e ized lowe -le el p oblem is equi alen o
min
u{1
2uAu −(Bβ)u|ua≤u≤ub},
whe e ua,j:= ua( j)and ub,j:= ub( j) o all j=1,...,J. We ob ain he solu ion
o his p oblem by p ojec ing he solu ion o he linea equa ion Au −Bβ=0 on o
123
In e se demand acking in anspo a ion ne wo ks 651
Fig. 2 The ne wo k conside ed in Sec .3.3 wi h VD={ 7,
8,
9,
10,
11},VI={ 1,
2,
3,
4,
5,
6},
and ED={(7), (8), (9), (10), (11)}
he easible box, since Ais a diagonal posi i e de ini e ma ix by o hogonali y o
S(i),L(i),(i)∈ED, see Appendix A o de ails.
Fo he uppe -le el p oblem (UL), we apply he same disc e iza ion echnique wi h
di e en s epsizes, see Sec . 3.3, and conside , i no speci ied di e en ly, he obse -
a ion ope a o Cin which we only obse e he densi ies a he demand e ices om
VD, co esponding o he las disc e iza ion poin s o he edges in ED,aswellasa
he i s disc e iza ion poin o edge (1), moni o ing he in low a 0. Addi ionally, D
is he ze o ope a o in ou expe imen s. Fu he de ails and some nume ical examples
a e explained in Sec .3.3 whe e i is also desc ibed how Cand Dcan be adjus ed.
Inse ing he disc e ized solu ion ope a o o he lowe -le el p oblem in o he
objec i e unc ion o he disc e ized uppe -le el p oblem esul s in a nonsmoo h op i-
miza ion p oblem wi h a ine cons ain s, and we sol e he la e using MATLAB’s
pa e nsea ch sol e in de aul mode. We wan o emphasize ha he pe o -
mance o his op imiza ion ou ine hea ily depends on he ini ial poin ha is handed
o e o he sol e . This, howe e , is no su p ising as he conside ed nonsmoo h p ob-
lem o in e es is noncon ex and, hus, likely o possess se e al local minimize s
and s a iona y poin s which a e di e en om i s global minimize s. As he model
is designed o econs uc ce ain e e ence pa ame e s om noisy da a, we ini ialize
pa e nsea ch wi h a pe u bed e sion o hese e e ence pa ame e s o ace his
p oblem. We no e ha , in he absence o lowe -le el con ol cons ain s, he esul -
ing single-le el p oblem is a simple con ex quad a ic p oblem which can be sol ed,
exempla y, wi h he aid o MATLAB’s quadp og ou ine, and he a o emen ioned
issues do no occu .
3.2 Gene al se -up o expe imen s
We conside he ee-shaped ne wo k p esen ed in Fig.2in which each edge has a
leng h o ω=1.
The eloci ies a e chosen iden ically o all edges, we use λ(i)=10, i=1,...,11.
The s epsizes a e gi en by =1
60 ,x=1
6 o he backwa d calcula ion and
123
652 S. Gö lich e al.
=1
70 ,x=1
7 o he o wa d calcula ion, which a e chosen di e en ly o
a oid an in e se c ime, see (Col on and K ess 2013, page 154), o he uncons ained
examples, and =1
20 ,x=1
2(backwa d calcula ion), =1
30 ,x=1
3( o wa d
calcula ion) when applying cons ain s o he in low in o de keep easonable unning
imes. No e ha x(i)=xis exploi ed, i=1,...,11. In bo h cases, he Cou an –
F ied ichs–Lewy condi ion holds ue wi h equali y, i.e.,
xλ(i)=1, i=1,...,11,
o a oid di usion in he nume ical scheme. The dis ibu ion pa ame e s a e se o
α1,2=0.65,α
2,4=0.7,α
4,7=0.5,α
6,10 =0.4,
α1,3=0.35,α
2,5=0.3,α
4,8=0.5,α
6,11 =0.6.
We conside he e olu ion o he demand wi hin one week, i.e., T=168 whe e one
ime uni ep esen s one hou and assume ou unde lying base demand le els which
a e isualized in Fig.3and chosen as
•a ime cons an le el o he demand:
D1( )=4,
•a daily a ying le el a which we a ain he highes le el in he mo ning:
D2( )=2+sin (π( −2)/12),
•a daily a ying le el a which we a ain he highes le el in he a e noon:
D3( )=2+sin (π( −10)/12),
•a le el ha illus a es he lowe demand du ing he weekend:
D4( )=1[0,120]( ).
These choices can simila ly be ound o example o he elec ici y ma ke in
Coskun and Ko n (2021) and desc ibe he iden i ied wo-peak pa e n o demand in
he in aday ma ke (D2,D3) as well as he phenomenon e e ed o as he weekend
e ec (D4). Fo he p o o ypical demand p o iles, we make use o D(i)
:= ˆ
d(i)D,
i∈{7,...,11},∈{1,2,3,4}, whe e
ˆ
d(7)=0.2275,ˆ
d(8)=0.2275,ˆ
d(9)=0.195,ˆ
d(10)=0.14,ˆ
d(11)=0.21.
This choice p opo ionally accoun s o he di e en dis ibu ion pa ame e s in he
ne wo k. The his o ical obse a ions a e basically gene a ed using he ini ial weigh s
(β1,β
2,β
3,β
4)=(0.2,0.15,0.2,0.45). (3.1)
In e e y ime s ep and o e e y demand e ex, he base demand le els a e pe u bed
by andom a iables
Z(i)
1∼N(0,1), Z(i)
2∼N(0,1/4), Z(i)
3∼N(0,1/4), Z(i)
4∼N(0,1/4),
such ha he his o ically desi ed demands a e gi en by ealiza ions o
D(i)
d=
4
=1
βˆ
d(i)D+Z(i)
,i∈{7,...,11}.(3.2)
123
In e se demand acking in anspo a ion ne wo ks 653
Fig. 3 Illus a ion o he ou base demand le els
The his o ically obse ed pai s (zo,uo)a e compu ed as solu ions o he associa ed
p oblem (2.12).
3.3 Documen a ion o expe imen s
In he ollowing, we in es iga e di e en a ian s o he bile el op imiza ion p oblem
(UL). The s anda d e sion is p esen ed in Sec .3.3.1, and se ings wi h addi ional
pe u ba ions in he his o ical obse a ions a e shown in Sec s.3.3.2 and 3.3.3.A
ime- es ic ed obse a ion ope a o Cis in es iga ed in Sec . 3.3.4. All subsec ions
a e cons uc ed in a simila way. Fi s , we p esen exempla y his o ical demand obse -
a ions, hen we p o ide a compa ison o he in- and ou lows o he means o he
his o ical obse a ions and he ini ially chosen βas well as o he econs uc ed βin a
amewo k wi hou an in low cons ain , which can also be conside ed as a amewo k
wi h a high cons ain ha does no eally a ec he in low. These illus a ions a e
p esen ed o he in low e ex and he demand e ex 7( he beha io a all o he
demand e ices is simila ). We can e i y ha , on he one hand, he op imal in lows
a e calcula ed co ec ly and, on he o he hand, see whe he he econs uc ion o he
weigh s βwas success ul. The second aspec is u he unde lined by a able p esen -
ing he means and a iances o βo a Mon e Ca lo simula ion o N=40 uns o
123
654 S. Gö lich e al.
di e en numbe s o his o ical obse a ions p. Second, we epea he in es iga ions o
each subcase based on a medium in low cons ain ub≡2 and a low in low cons ain
ub≡1.5, whe e we also ensu e nonnega i e in lows, i.e., ua≡0, he la e being
non es ic i e as he desi ed demand a he e ices in VDis nonnega i e.
3.3.1 S anda d model wi hou addi ional adjus men s
In his scena io, no u he pe u ba ions o model changes a e included, and we con-
side he amewo k p esen ed in he p e ious sec ions. Th ee examples o his o ical
obse a ions a e gi en in Fig.4which show he sinusoidal beha io o demand, as well
as hed op o >120 du ing he weekend. Fu he mo e, we de ec he s ochas ic
noise in he demands, howe e , s ill e i y ha he demands show a e y simila s uc-
u e. The compa ison o he in low and ou low o demand e ex 7a e p esen ed
in Fig.5, whe e he blue cu e shows he mean alues o he p=6 his o ical obse -
a ions, he yellow do ed line ep esen s he cu e o he ue βgi en in (3.1), and
he ed line he in- o ou low o he econs uc ed β. All conside a ions we e made
wi hou cons aining he in low con ol. I can be concluded ha all h ee cu es ma ch
e y well, which means ha , on he one hand, he in low is calcula ed app op ia ely
and, on he o he hand, also he weigh s o he base demands a e eob ained e y well.
The ou low beha io a he demand e ices 8,...,
11 shows simila pa e ns and
is ( o b e i y o p esen a ion) no illus a ed. A he beginning and he end o he
conside ed ime ho izon, some cu es in Fig.5decay o ze o o show a jump. This can
be explained by he ac ha a ound ime =0, i akes some ime un il (s a ing om
an emp y sys em) he i s inse ed quan i y eaches he demand e ex. The e o e, he
ou lows a e ze o in he e y beginning o he ime pe iod. Con e sely, he in low o
imes close o T=168 anishes, since hese quan i ies do no each he demand nodes
wi hin he conside ed ime ho izon. The inc ease a T=168 in he ou low igu e
can be explained by conside ing T=168 o be Monday al eady, whe e he demand
is la ge again. Simila a i ac s show up in some o he igu es in his sec ion due o
analogous easons.
Table 1shows he means and a iances o he econs uc ed weigh s o he base
demands o di e en numbe s o pe u bed his o ical obse a ions in a Mon e Ca lo
simula ion o N=40 uns and unde lines he esul s om Fig. 5quan i a i ely. As
i can be expec ed o la ge numbe s o his o ical obse a ions, he means app oach
he alues in (3.1) and he a iances in he uns dec ease in he numbe o his o ical
obse a ions p.
Accoun ing o a po en ial cons ain on he in low, we compa e a scena io whe e
he in low is limi ed o 2 (medium cons ain ) and 1.5 (low cons ain ). We epea he
idea o Fig.5in Fig.6emphasizing ha , excep o he cons ain , all o he quan i ies
emain unchanged. Howe e , he demand illus a ion seems o be less luc ua ing
which can be explained by he coa se disc e iza ion g id ha is used o he cons ained
op imiza ion. In he medium cons ain case, we obse e ha he in- and ou low ollow
he uncons ained case bu a e unca ed a he e y highes peaks and o he wise ollow
he a e aged demand well. Rega ding he econs uc ion o he weigh s o he base
demand le els when zooming in, one can s ill obse e a qui e good ma ch in he in- and
ou lows o he op imized and ini ial choices o β. Table 2unde lines his obse a ion,
123
In e se demand acking in anspo a ion ne wo ks 655
Fig. 4 Th ee o he pe u bed his o ically obse ed demands o demand e ex 7in he case o Sec .3.3.1
Fig. 5 A compa ison be ween he mean ealiza ion o he p=6 his o ical in- and ou lows wi h he in- and
ou low o he econs uc ed βin he case o Sec . 3.3.1 and he ini ial β
Table 1 Means and a iances o he eob ained weigh s o he base demands o di e en choices o he
numbe o pe u bed his o ical obse a ions pin he se ing o Sec . 3.3.1
mean a iance
p=1p=6p=20 p=200 p=1p=6p=20 p=200
β10.2003 0.2000 0.1999 0.2000 2.16e−06 0.25e−06 0.16e−06 0.10e−07
β20.1496 0.1499 0.1500 0.1500 6.65e−06 0.42e−06 0.29e−06 0.21e−07
β30.2001 0.2001 0.2002 0.2000 2.82e−06 0.87e−06 0.31e−06 0.18e−07
β40.4503 0.4500 0.4499 0.4500 4.16e−06 0.57e−06 0.34e−06 0.21e−07
123
656 S. Gö lich e al.
Fig. 6 A compa ison be ween he mean ealiza ion o he p=6 his o ical in- and ou lows wi h he in-
and ou low o he econs uc ed βin he case o Sec . 3.3.1 and he ini ial βwi h wo di e en in low
cons ain s
Table 2 Means and a iances o he eob ained weigh s o he base demands o di e en choices o he
numbe o pe u bed his o ical obse a ions pin he se ing o Sec .3.3.1 wi h addi ional in low cons ain s
mean a iance
medium low medium low
p=6p=200 p=6p=200 p=6p=200 p=6p=200
β10.1960 0.1972 0.2460 0.2474 2.87e−06 0.11e−06 0.83e−04 0.11e−05
β20.1549 0.1536 0.0997 0.0988 5.93e−06 0.22e−06 0.91e−04 0.10e−05
β30.1997 0.1997 0.1297 0.1284 6.77e−06 0.35e−06 1.10e−04 0.21e−05
β40.4493 0.4495 0.5245 0.5254 6.24e−06 0.30e−06 1.11e−04 0.15e−05
bu shows a small de ia ion especially in he pa ame e s β1and β2compa ed o he
un es ic ed case. Fo he low cons ain , he in low is cu om Monday o F iday and
in some peak imes also du ing he weekend, so ha mos o he ime demand canno
be sa is ied on a e age. Then he econs uc ion ask is also no success ul, and we
can obse e a isible misma ch in he g een ci cles (associa ed o he op imal ou low
o he ini ial β) and pu ple diamonds ( ep esen ing he ou low o he econs uc ed
β) du ing he weekend. Re e ing again o Table 2, one can see ha he e is a la ge
de ia ion in he econs uc ed alues o β, whe e he e y low alues o β2and β3
a e pa icula ly s iking. This e ec can be explained by he ac ha D2and D3a e
he sinusoidal componen s o demand and ha he obse a ions a e smoo hed and
unca ed a he majo i y o ime.
123
In e se demand acking in anspo a ion ne wo ks 657
Fig. 7 Th ee o he pe u bed his o ically obse ed demands o demand e ex 7in he case o Sec .3.3.2
3.3.2 Resul s wi h addi ional noise in he weigh s ˇ
In addi ion o he in es iga ion o Sec .3.3.1, we in oduce a s uc u al and unce ain
de ia ion in he choice o β, when gene a ing he his o ically desi ed demand in (3.2).
We assume ha he unce ain y mainly comes in o play o β4such ha o any
his o ical obse a ion, he weigh s o he demand le els a e chosen as
(β1,β
2,β
3,β
4)=0.2
1+˜
Z,0.15
1+˜
Z,0.2
1+˜
Z,0.45 +˜
Z
1+˜
Z(3.3)
o a uni o mly dis ibu ed andom a iable ˜
Z∼U([−0.05,0.05]).
The esul s o some his o ical obse a ions a e p esen ed below in Fig.7. The e is
no only noise in he demands bu also s uc u ally di e en beha io due o di e -
en ealiza ions o ˜
Zin he weigh s o he demands. The e o e, he yellow cu e o
his o ic da a 3 seems o be lowe (co esponding o a la ge alue o ˜
Z) han he blue
cu e (co esponding o a smalle alue o ˜
Z). Figu e8shows he di e en in- and
ou lows which a e supplemen ed by Table 3showing he means and he a iances
o a Mon e Ca lo simula ion o he econs uc ed weigh s o he base demands o
di e en numbe s o pe u bed his o ical obse a ions. We obse e ha in Fig.8, he
expec ed ou low and in low ma ch qui e well, bu conside ing Table 3, i can be seen
ha he econs uc ion is mo e di icul han in he s anda d se ing. Fo small p, he
econs uc ed βde ia es mo e signi ican ly om he ini ial choice. Fo a la ge numbe
o obse a ions p, he da a indica es ha he pe o mances a e imp o ed and lead o
good econs uc ed alues o β.
Also in his scena io, we in es iga e a cons ain on he in low con ol on a medium
le el o 2 and a low cons ain o 1.5. Simila o Sec .3.3.1, he econs uc ion wo ks a
123
658 S. Gö lich e al.
Fig. 8 A compa ison be ween he mean ealiza ion o he p=6 his o ical in- and ou lows wi h he in- and
ou low o he econs uc ed βin he case o Sec . 3.3.2 and he ini ial β
Table 3 Means and a iances o he eob ained weigh s o he base demands o di e en choices o he
numbe o pe u bed his o ical obse a ions pin he se ing o Sec . 3.3.2
mean a iance
p=1p=6p=20 p=200 p=1p=6p=20 p=200
β10.2012 0.2010 0.2001 0.2001 0.36e−04 0.08e−04 0.19e−05 0.01e−05
β20.1497 0.1509 0.1500 0.1501 0.26e−04 0.04e−04 0.14e−05 0.01e−05
β30.2013 0.2013 0.2003 0.2001 0.40e−04 0.06e−04 0.19e−05 0.01e−05
β40.4477 0.4468 0.4496 0.4497 2.76e−04 0.49e−04 1.37e−05 0.08e−05
leas sa is ac o ily in he medium cons ain case, whe eas i ails in he low cons ain
case. Ne e heless, in bo h cases, he a e age ou low ma ches he op imal ou low o
he econs uc ed β, see Fig.9. Table 4shows o p∈{6,200} he mean and he a i-
ance as he adap ed e sion o Table 3wi h medium and low in low cons ain , whe e
he a iances a e simila bu sligh ly highe han in he uncons ained amewo k. The
obse ed e ec s a e compa able o hose ob ained o he cons ained bu unpe u bed
egime in Table 2.
3.3.3 Resul s wi h changed base demand le el D4
This sec ion is based on he in es iga ions in Sec .3.3.1. Ins ead o pe u bing β,we
assume ha he e is a s uc u al de ia ion in he base demand le els. Pa icula ly,
we assume ha in he gene a ion o he obse a ions, we adjus he base demand
D4 o D4( )=3
21[0,120]( ), which means ha he e is la ge sha e o demand on
weekdays. Fu he mo e, we omi he no maliza ion es ic ion o he weigh s, i.e., we
me ely assume β≥0, ∈{1,...,4}, and d op he cons ain 4
=1β=1, since
he inc ease in he base demand le el should now be cap u ed by a la ge weigh on
123
In e se demand acking in anspo a ion ne wo ks 665
a ine cons ain s, see he classical pape (Dempe and Ba d 1992) o a ela ed idea
and G e e and Ou a a (2024) o a mode n iew.
A Special quad a ic p oblems wi h box cons ain s
Le us ix ec o s θ, d∈Rnas well as a∈(R∪{−∞})nand b∈(R∪{∞})nsuch
ha all en ies o θa e posi i e while a≤ bholds componen wise. Fo := diag(θ),
we aim o sol e
min
{1
2 −
d | ∈Vad}(QP)
whe e Vad ⊂Rnis he box gi en by
Vad := { ∈Rn| a≤ ≤ b}.
Fi s , we obse e ha he objec i e unc ion in (QP) is uni o mly con ex while he
easible se is nonemp y, closed, and con ex. Hence, (QP) possesses a uniquely de e -
mined global minimize ¯ ∈Vad. The la e can be cha ac e ized in e ms o he
necessa y and su icien op imali y condi ion
∀ ∈Vad :(¯ − d)( −¯ ) ≥0.(A.1)
We no e ha is a posi i e de ini e diagonal ma ix. Hence, i is easonable o se
˜ := −1 d.
No e ha ˜ i=θ−1
i d,iholds o all i=1,...,n. We will now show ha
¯ =max( a,min(˜ , b)) (A.2)
holds ue, i.e., ha ¯ is he p ojec ion o ˜ on o he box Vad. No e ha max and min
ha e o be in e p e ed componen wise in (A.2). We in oduce index se s Ia,I0,Ib⊂
{1,...,n}by means o
Ia:= {i∈{1,...,n}|˜ i<
a,i},
I0:= {i∈{1,...,n}| a,i≤˜ i≤ b,i},
Ia:= {i∈{1,...,n}| b,i<˜ i}.
Clea ly, hese se s o m a disjoin pa i ion o {1,...,n}, and (A.2) can be ew i en
as
∀i∈{1,...,n}: ¯ i=⎧
⎪
⎨
⎪
⎩
a,ii∈Ia,
˜ ii∈I0,
b,ii∈Ib.
123
666 S. Gö lich e al.
Pick ∈Vad a bi a ily. Taking oge he all o he abo e indings, we end up wi h
(¯ − d)( −¯ )
=
i∈Ia
(θi a,i− d,i)(
i− a,i)
≥0
+
i∈Ib
(θi b,i− d,i)(
i− b,i)
≤0
+
i∈I0
(θi˜ i− d,i)
=0
( i−˜ i)
≥
i∈Ia
(θi˜ i− d,i)
=0
( i− a,i)+
i∈Ib
(θi˜ i− d,i)
=0
( i− b,i)=0,
and his shows ha ¯ cons uc ed as in (A.2) is, indeed, a solu ion o (A.1) and, hus,
he uniquely de e mined global minimize o (QP).
Acknowledgemen s The au ho s wish o hank he wo anonymous e iewe s whose aluable commen s
and sugges ions helped o imp o e he o e all quali y o his pape . Fu he mo e, one o he e iewe s
ecommended an inspec ion o he PhD hesis (Keime 2014) which is g a e ully acknowledged. Simone
Gö lich was suppo ed by he Deu sche Fo schungsgemeinscha (DFG) wi hin he p ojec s GO1920/10-1
and GO1920/11-1.
Funding Open Access unding enabled and o ganized by P ojek DEAL.
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