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103Methods
31
whereXisanon-empty,closedandconvexsetoffeasiblesolutions,and
P[ę]isaprojectionofęontoX.Thestationarityisreachedwhenx∗1P[x∗−
∇xF(x)|x1x∗].
Inourapproach,weassumedthat:
(
I
0if
ę(k)≤η(k),
P[ę(k)]1
4
1if
ę(k)≥1−η(k),
I
l
ę(k)
otherwise.
(1.45)
Thischoicewasmotivatedbythebinarysteeringtechniqueproposedin[11].
Followingthetemperatureschedule(1.45),weproposetoiterativelyshrinkthe
bound-constraints[η(k),1−η(k)]accordingtotheexponentiatedrule:
η(k)1
1
2(1−exp{−
k
δ}),
(1.46)
whereδ>1isaconstantthatcontrolsaspeedofconvergencetowardsabinary
solution.
SimilarlyasintheMFA,thedescenttowardsabinarysolutioncannotbetoo
steepbecausetheiterativeprocesscanstuckinlocalminima.Theparameterδ
shouldbethereforecarefullyselected.Ourexperimentsshowedthattherecon-
structionisnotverysensitivetoδ∈[10,500].
Ingeneral,gradientalgorithmsareregardedasslowmethods,however,their
convergenceratestronglydependsonthesteplengthI(k)∈(0,1]in(1.44).
TherearemanyrulesforchoosingI(k),butwerestrictourconsiderationsonly
totheinexactlinesearchthatisknownasthe”Armijorulealongaprojection
arc”[2].Thesteplengthisoptimalif:
I(k)1;mk,
wheremkisthefirstnonnegativeintegermforwhich:
F(x(k+1))−F(x(k))≤σ∇F(x(k))T(x(k+1)−x(k)),
(1.47)
(1.48)
forσ∈(0,1).Theconvergenceofthismethodwasproved,e.g.in[9,30].
Inouralgorithm,weselected;10.5,σ10.001,andthequadraticobjective
function,i.e.:
F(x)1||b−Ax||2
2+;||Lx||2
2,
(1.49)
whereL1I−W/4isdefinedby(1.15)and(1.16).Ourproblembringsthere-
foretotheNLIP(1.9).
