A smoothing Newton method for absolute value equation associated with second-order cone

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Applied Numerical Mathematics

www.elsevier.com/locate/apnum

A smoothing Newton method for absolute value equation associated with second-order cone

Xin-He Miao

^a

^,

¹

, Jian-Tao Yang

^a

, B. Saheya

^b

^,

²

, Jein-Shan Chen

^c

^,∗,

³

aDepartmentofMathematics,TianjinUniversity,China,Tianjin300072,China

bCollegeofMathematicalScience,InnerMongoliaNormalUniversity,Hohhot010022,InnerMongolia,PRChina cDepartmentofMathematics,NationalTaiwanNormalUniversity,Taipei11677,Taiwan

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received20May2016

Receivedinrevisedform26February2017 Accepted28April2017

Availableonline4May2017

Keywords:

Second-ordercone Absolutevalueequations SmoothingNewtonalgorithm

Inthispaper,weconsider thesmoothingNewtonmethodforsolving atypeofabsolute value equations associated with second order cone (SOCAVE for short), which is a generalizationofthestandardabsolutevalueequationfrequentlydiscussedintheliterature during the past decade. Based on a class of smoothing functions, we reformulate the SOCAVE as a family of parameterized smooth equations, and propose the smoothing Newton algorithm to solve the problem iteratively. Moreover, the algorithm is proved to be locally quadratically convergent under suitable conditions. Preliminary numerical resultsdemonstrate that thealgorithm is effective.Inaddition, two kinds ofnumerical comparisonsare presentedwhichprovidesnumericalevidenceaboutwhythesmoothing Newtonmethodisemployedand alsosuggestsasuitablesmoothingfunctionforfuture numericalimplementations.Finally,wepointoutthatalthoughthemainideaforproving theconvergenceissimilartotheone usedintheliterature,theanalysisisindeedmore subtleandinvolvesmoretechniquesduetothefeatureofsecond-ordercone.

©^{2017IMACS.PublishedbyElsevierB.V.Allrightsreserved.}

1. Introduction

Thestandardabsolutevalueequation(AVE)isintheformof

Ax

+

^B

|

^x

| =

^b

,

(1)

where A

∈ R

^{n×n, B}

∈ R

^{n×n, B}

=

^0,andb

∈ R

^n.Here

|

^x

|

^meansthecomponentwiseabsolutevalue ofvectorx

∈ R

^n.When B

= −

^{I,whereI istheidentitymatrix,theAVE(1)reducestothespecialform:}

Ax

− |

^x

| =

^b

.

ItisknownthattheAVE(1)wasfirstintroducedbyRohnin[38]andrecentlyhasbeeninvestigatedbymanyresearchers, forexample,Caccetta,QuandZhou[1],HuandHuang[14],JiangandZhang[22],KetabchiandMoosaei[23],Mangasarian [25–32],MangasarianandMeyer[34],Prokopyev[35],andRohn[40].

*

Correspondingauthor.

E-mailaddresses:[email protected](X.-H. Miao),[email protected](J.-T. Yang),[email protected](B. Saheya),[email protected] (J.-S. Chen).

1 Theauthor’sworkissupportedbyNationalNaturalScienceFoundationofChina(No. 11471241).

2 Theauthor’sworkissupportedbyNaturalScienceFoundationofInnerMongolia(AwardNumber:2014MS0119).

3 Theauthor’sworkissupportedbyMinistryofScienceandTechnology,Taiwan(No. 104-2115-M-003-011-MY2).

http://dx.doi.org/10.1016/j.apnum.2017.04.012

0168-9274/©^{2017IMACS.PublishedbyElsevierB.V.Allrightsreserved.}

Inparticular,MangasarianandMeyer[34]showthat theAVE(1)isequivalent tothebilinearprogram,thegeneralized LCP(linear complementarityproblem),andthestandardLCP provided1 isnot an eigenvalueof A.Withtheseequivalent reformulations, they also show that the AVE(1) isNP-hard inits general formand provideexistence results.Prokopyev [35]furtherimprovestheaboveequivalencewhichindicatesthattheAVE(1)canbeequivalentlyrecastasLCPwithoutany assumptionon A and B,andalsoprovidesarelationshipwithmixedintegerprogramming.Ingeneral,ifsolvable, theAVE (1)canhaveeitherunique solutionormultiple(e.g.,exponentially many)solutions.Indeed,varioussufficiencyconditions on solvability and non-solvability of the AVE (1) with unique and multiple solutions are discussed in [34,35,39]. Some variantsoftheAVE,liketheabsolutevalue equationassociatedwithsecond-order coneandtheabsolutevalue programs, areinvestigatedin[16]and[41],respectively.

Inthispaper,we targetanother typeofabsolutevalue equationwhich isanaturalextension ofthestandard AVE(1).

More specifically the following absolute value equation associated with second-order cones, abbreviated as SOCAVE, as below:

Ax

+

B

|

x

| =

b

,

(2)

where A

,

B

∈ R

^{n×n andb}

∈ R

^{n arethesame asthosein(1);}

|

^x

|

^{denotesthe absolutevalueof x coming fromthesquare} root of the Jordan product “

◦

^{” of x and x. What is the difference betweenthe standard AVE (1) and the SOCAVE (2)?}

Their mathematical formats look the same. In fact, the main difference is that

|

^x

|

^{in the standard AVE (1) means the} componentwise

|

^xi

|

^{ofeach x}i

∈ R

^,i.e.,

|

^x

| = (|

^x1

|, |

^x2

|, · · · , |

^xn

|)

^T

∈ R

^n;however,

|

^x

|

^{intheSOCAVE(2)denotesthevector} satisfying

√

x²

:= √

x

◦

x associatedwithsecond-orderconeunderJordanproduct.Tounderstanditsmeaning,we needto introducethedefinitionofsecond-ordercone(SOC).Thesecond-orderconein

R

ⁿ

(

n

≥

¹

)

,alsocalledtheLorentz cone,is definedas

K

ⁿ

:= 

(

x₁

,

x₂

) ∈ R × R

ⁿ⁻¹

| 

^x2

 ≤

^x1

 ,

where

 · 

^{denotestheEuclideannorm.Ifn}

=

^1,then

K

^nisthesetofnonnegativereals

R

+.Ingeneral,ageneralsecond- ordercone

K

^{couldbetheCartesianproductofSOCs,i.e.,}

K := K

ⁿ¹

× · · · × K

^nr

.

Forsimplicity,wefocusonthesingleSOC

K

^{nbecausealltheanalysiscanbecarriedovertothesettingofCartesianproduct.}

TheSOCisaspecialcaseofsymmetricconesandcanbeanalyzedunderJordanproduct,see[11].Inparticular,foranytwo vectorsx

= (

^x1

,

x₂

) ∈ R × R

^{n−1and y}

= (

^y1

,

y₂

) ∈ R × R

^{n−1,theJordanproduct ofx and}y associatedwith

K

^{n isdefinedas}

x

◦

^y

:=



x^Ty

y1x2

+

^x1y2

 .

TheJordanproduct,unlikescalarormatrixmultiplication,isnotassociative,whichisamainsourceofcomplicationinthe analysisofoptimizationproblemsinvolvedSOC,see[3,10,12]andreferencesthereinformoredetails.Theidentityelement underthis Jordanproduct is e

= (

¹

,

0

, ...,

0

)

^T

∈ R

^{n. Withthese}definitions, x² means theJordan product ofx with itself, i.e.,x²

:=

^x

◦

^x;and

√

x withx

∈ K

^{n denotestheuniquevectorsuchthat}

√

x

◦ √

x

=

^{x.Inotherwords,thevector}

|

^x

|

^inthe SOCAVE(2)iscomputedby

|

x

| := √

x

◦

x

.

As mentioned earlier,the significance ofthe AVE (1)arises fromthe fact that the AVEis capable to formulatemany optimizationproblems(alsosee[26,30,32,34,35]),suchas,linearprograms,quadraticprograms,bimatrixgames,andsoon.

Moreover,theabsolutevalue equationsisequivalent tothelinearcomplementarityproblem[34].Accordingly,weseethat theSOCAVE(2)plays similarrole invariousoptimizationproblemsinvolvedsecond-ordercones. Forsolvingthestandard AVE(1),therearemanyvariousnumericalmethodsproposedintheliterature(see[1,21,22,25–27,35,43]).AsfortheSOCAVE (2),Hu, Huangand Zhang[16] propose a generalizedNewton method forsolving theSOCAVE (2).It iswell known that smoothing-typealgorithmsisapowerfultoolforsolvingmanyoptimizationproblems,forexample,thelinearandnonlinear complementarityproblems[3,12,19,20,24],thesystemofequalitiesandinequalities[17,42].Inthispaper,weareinterested inasmoothing NewtonmethodforsolvingtheSOCAVE(2).OurnumericalresultsalsosupportthatthesmoothingNewton methodisabetterwaythanthegeneralizedNewtonmethodemployedin[16].Thatiswhyweadoptthisalgorithmasthe maintool todonumericalimplementations.Inaddition,we haveshownthat theproposed smoothingNewtonmethodis locallyquadraticallyconvergentundersuitable condition. Wereport somepreliminary numericalresultstoshowthat the methodisefficient.Moreover,numericalcomparisonsbasedonvariousvalueofp arepresentedaswell.

Toclosethissection,wesaya fewwordsaboutnotationsandtheorganizationofthispaper.Asusual,

R

^{n denotesthe} space ofn-dimensional real columnvectors.

R

+ and

R

++ denotethe nonnegative andpositive reals. For anyx

,

y

∈ R

^n, theEuclideaninner productisdenoted

^x

,

y

 =

^{xTy,andtheEuclideannorm}



^x



^isdenotedas



^x

 = √

^x

,

x



^.Thispaper isorganized asfollows.InSection2,webriefly describesome conceptsandpropertiesonsecond-ordercone.Besides,we reviewJordanproductandthespectraldecompositionforelementsx and y in

R

^{n.InSection3,weintroduceasmoothing} function of the absolute value

|

^x

|

^{, and study the Jacobian matrix of the smoothing function. In Section 4, we propose} a smoothing Newton algorithm forsolving the SOCAVE (2), anddiscuss the convergenceof the proposed method under suitableconditions.InSection5,thepreliminarynumericalresultsandnumericalcomparisonsaregiven.

2. Preliminaries

Inthissection,werecallsomebasicconceptsandbackgroundmaterialsregardingthesecond-ordercone,whichwillbe extensivelyusedinthesubsequentanalysis.Moredetailscanbefoundin[3,10–12,16].First,werecalltheexpressionofthe spectraldecomposition ofx withrespecttoSOC.Forx

= (

^x1

,

x2

) ∈ R × R

^{n−1,thespectral}decompositionofx withrespectto SOCisgivenby

x

= λ

1

(

x

)

u⁽_x¹⁾

+ λ

2

(

x

)

u⁽_x²⁾

,

(3)

where

λ

i

(

x

) =

^x1

+ (−

¹

)

ⁱ



^x2



^fori

=

¹

,

2 and u⁽_xⁱ⁾

=

⎧ ⎪

⎨

⎪ ⎩

1 2

1

, (−

¹

)

^{i x}

T

x22



T

if



^x2

 =

⁰

,

1 2



1

, ( −

¹

)

ⁱ

ω

^T

^T if



^x2

 =

⁰

,

(4)

with

ω ∈ R

^{n−1 beinganyvector satisfying}

 ω _{ =}

1.Thetwo scalars

λ

1

(

x

)

and

λ

2

(

x

)

are calledspectral valuesof x;while thetwovectorsu⁽x¹⁾andu⁽x²⁾arecalledthespectralvectorsofx.Moreover,itisobviousthatthespectraldecompositionof x

∈ R

^nisuniqueifx2

=

^0.

Lemma2.1.Foranyx

= (

^x1

,

x2

) ∈ R × R

^{n−1withthespectral}decompositiongivenasin(3)-(4),thefollowingresultshold.

(a) u⁽_x¹⁾

◦

^u(x²⁾

=

^{0 andu(}xⁱ⁾

◦

^u(xⁱ⁾

=

^u(xⁱ⁾fori

=

¹

,

2;

(b)



^u(x¹⁾



²

= 

^u(x²⁾



²

=

¹₂^and



^x



²

=

¹₂

(λ

²₁

(

x

) + λ

²₂

(

x

))

.

Proof. Thepropertycanbeverifieddirectlyorcanbefoundin[3,11,12,16,10].

2

Inthenextcontent,wetalkabouttheprojectionontosecond-ordercone.Weletx₊betheprojectionofx ontoSOC

K

^n, and x₋ betheprojection of

−

^{x ontothe dualcone}

(K

ⁿ

)

^∗ of

K

ⁿ,wherethedual cone

(K

ⁿ

)

^∗ isdefinedby

(K

ⁿ

)

^∗

:= {

^y

∈ R

ⁿ

|

^x

,

y

 ≥

⁰

, ∀

^x

∈ K

ⁿ

}

^{.Infact,thedualconeof}

K

^{n isitself,i.e.,}

(K

ⁿ

)

^∗

= K

^{n.DuetothespecialstructureofSOC}

K

^n,the explicitformulaofprojectionofx

= (

^x1

,

x2

) ∈ R × R

^{n−1 onto}

K

^{n isobtainedin[3,10–13]asbelow:}

x₊

=

⎧ ⎨

⎩

x if x

∈ K

ⁿ

,

0 if x

∈ − K

ⁿ

,

u otherwise

,

where u

=



_x1+x2



2 x₁+^x2

2



x₂

x2

 .

Similarly,theexpressionofx₋isintheformof

x₋

=

⎧ ⎨

⎩

0 if x

∈ K

ⁿ

,

−

^{x if x}

∈ − K

ⁿ

,

w otherwise

,

where w

=

  −

^x1−₂^x2

x1−x2 2



x2

^x2

 .

Togetherwiththespectraldecompositionofx,itisshownthatx

=

^x+

+

^x−andtheexpressionofx₊hastheform:

x₊

= (λ

1

(

x

))

₊u⁽_x¹⁾

+ (λ

2

(

x

))

₊u⁽_x²⁾

,

and

x₋

= (−λ

1

(

x

))

₊u⁽_x¹⁾

+ (−λ

2

(

x

))

₊u⁽_x²⁾

,

where

( α )

₊

=

^max

{

⁰

, α }

^for

α ∈ R

^.

Next,wetalkabouttheexpressionof

|

^x

|

^{associatedwithSOC.Thereisan}alternativewayviatheso-calledSOC-function toobtaintheexpressionof

|

^x

|

^{,whichcanbefoundin[2,4].More}specifically,foranyx

∈ R

^n,wedefinethe absolutevalue

|

^x

|

^{ofx withrespecttoSOCas}

|

^x

| :=

^x+

+

^x−.Infact,inthesettingofSOC,theform

|

^x

| =

^x+

+

^x−isequivalenttotheform

|

^x

| = √

x

◦

^{x.Combiningtheaboveexpressionofx}+andx₋,itcabbeverifiedthattheexpressionoftheabsolutevalue

|

^x

|

isintheformof

|

x

| = 

(λ

1

(

x

))

₊

+ (−λ

1

(

x

))

₊



u⁽_x¹⁾

+ 

(λ

2

(

x

))

₊

+ (−λ

2

(

x

))

₊



u⁽_x²⁾

= λ

₁

(

x

)

u⁽_x¹⁾

+ λ

₂

(

x

)

u⁽_x²⁾

.

Toendthissection,wepointouttherelationbetweenSOCAVEandSOCLCP(second-orderconelinearcomplementarity problem).In[16],itwasshownthatSOCAVE(2)isequivalenttothefollowingSOCLCP:findx

,

y

∈ R

^{n suchthat}

Mx

+

^{P y}

=

^c

,

and x

∈ K

ⁿ

,

y

∈ K

ⁿ

,

^x

,

y

 =

⁰

,

whereM

,

P

∈ R

^{n×n arematricesandc}

∈ R

^{n.However,theaboveisnotastandardSOCLCPbecausethereexiststheequations} Mx

+

^{P y}

=

^{c therein.Asbelow,weshowthattheSOCAVE(2)canbefurtherconvertedintoastandardSOCLCP.}

Theorem2.1.TheSOCAVE(2)canbereducedtothesecond-orderconelinearcomplementarityproblem(SOCLCP):

v

∈ K

ⁿ

× K

ⁿ

× K

ⁿ

,

w

=

^{Q v}

+

^q

∈ K

ⁿ

× K

ⁿ

× K

ⁿ and

^v

,

w

 =

⁰

,

(5) where

Q

:=

⎡

⎣ −

^{I 2I 0}

A B

−

A 0

−

^{A A}

−

^{B 0}

⎤

⎦ ,

v

:=

⎡

⎣

^2x

|

x⁺

|

0

⎤

⎦

and q

:=

⎡

⎣ −

⁰b b

⎤

⎦ .

(6)

Proof. Bylookinginto(6),wehave

w

=

^{Q v}

+

^q

=

⎡

⎣

Ax

+

^2xB

|

^−x

| −

^b

−

Ax

−

B

|

x

| +

b

⎤

⎦ .

PluggingthisintoSOCLCP(5)impliesthat

Ax

+

B

|

x

| −

b

∈ K

ⁿ and

−

Ax

−

B

|

x

| +

b

∈ K

ⁿ

.

Since

K

^{n ispointed,itfollowsthat Ax}

+

^B

|

^x

| −

^b

=

^{0.Ontheother hand,theaboveargumentis}reversible.Thus,weshow thatSOCAVE(2)isequivalenttosecond-orderconelinearcomplementarityproblem.

2

Remark2.1.From Theorem 2.1,itfollows thatwe can alsosolvethe SOCAVE(2)by employing manyefficientalgorithms forsolving SOCLCP(5).Nonetheless, whenwe apply theNewton methodto solveSOCLCP, itstill needs reformulateit as smoothequationsornonsmoothequations.Thismeansthatweneedtwicereformulationsifwefollowthisway.Inviewof this,inthispaper,wereformulatetheSOCAVE(2)directlyasthesmoothequations,andsolvetheequationsbysmoothing Newtonmethod.

3. SmoothingfunctionsassociatewithSOCAVE

Inthispaper,we employthe smoothingNewtonmethodforsolving theSOCAVE(2).Tothisend, weneed toadopta smoothingfunction. Dueto thenon-differentiability of

| α _|

for

α ∈ R

^{,we consider aclass ofsmoothingfunctionsforthe} absolutevaluefunction

| α _|

.Morespecifically,wedefinethefunction

φ

p

( ·, ·) : R

²

→ R

^as

φ

p

(

a

,

b

) := 

^p

|

^a

|

^p

+ |

^b

|

^p

,

p

>

1

.

(7)

This class of functionsis extracted fromthe so-called generalized Fischer–Burmeister function

φ

p

(

a

,

b

) = 

^p

|

^a

|

^p

+ |

^b

|

^p

− (

a

+

^b

)

,which isheavily studiedinmanyreferences[5–9,15].Forconvenience, westill usethenotation

φ

p even itisno longerexactlythesameasthegeneralizedFischer–Burmeisterfunction.

Lemma3.1.Let

φ

p

: R

²

→ R

^{bedefinedasin(7).Then,thefollowinghold.}

(a)

φ

p

(

a

,

0

) = |

^a

|

^and

φ

p

(

0

,

b

) = |

^b

|

^; (b)

φ

p

(·, ·)

^{isLipschitzcontinuouson}

R

^2; (c)

φ

p

( ·, ·)

^{isstronglysemismoothon}

R

^2;

(d)

φ

p

(

a

,

b

)

iscontinuouslydifferentiableforany

(

a

,

b

) = (

⁰

,

0

) ∈ R

^2with

∂φ

p

(

a

,

b

)

∂

a

=

^sgn

(

a

) |

^a

|

^p−1

(φ

p

(

a

,

b

))

^{p−1 and}

∂φ

p

(

a

,

b

)

∂

b

=

^sgn

(

b

) |

^b

|

^p−1

(φ

p

(

a

,

b

))

^p−1

,

wherethefunctionsgn

( ·)

^{isdefinedbysgn}

( α ) :=

⎧ ⎨

⎩

1 if

α >

0

,

0 if

α ₌

0

,

−

^{1 if}

α <

0

.

Proof. Pleasereferto[5–9,15]foraproof.

2

According to Lemma 3.1, it follows that for any a

∈ R

^anda

→

^{0, we have}

φ

p

(

a

,

b

) → |

^b

|

^{. Therefore, combining the} spectraldecompositionofx andthefunction

φ

p,wedefineavector-valuedsmoothingfunction



p

: R × R

ⁿ

→ R

^{n as}



p

( μ ,

x

) = φ

p

( μ , λ

1

(

x

))

u⁽_x¹⁾

+ φ

p

( μ , λ

2

(

x

))

u⁽_x²⁾

= 

^p

| μ _|

^p

_{+ |λ}

1

(

x

) |

^pu(x¹⁾

+ 

^p

| μ _|

^p

_{+ |λ}

2

(

x

) |

^pu(x²⁾

,

where

μ ∈ R

^{isaparameter,and}

λ

1

(

x

), λ

2

(

x

)

arethespectralvaluesofx.FromLemma 3.1,itiseasytoverifythat μlim→⁰



p

( μ ,

x

) = |λ

1

(

x

)|

u⁽_x¹⁾

+ |λ

2

(

x

)|

u⁽_x²⁾

= |

^x

|.

Inotherwords,thefunction



p

( μ ,

x

)

isauniformlysmoothingfunctionof

|

^x

|

^{associatedwithSOC.Withthisfunction,for} theSOCAVE(2),wefurtherdefineafunction H

( μ ,

x

) : R × R

ⁿ

→ R × R

^{n by}

H

( μ ,

x

) =

 μ

Ax

+

^B



p

( μ ,

x

) −

^b



, ∀ μ ∈ R,

^x

∈ R

ⁿ

.

(8)

Then,weobservethat

H

( μ ,

x

) =

0

⇐⇒ μ =

0 and Ax

+

B



p

( μ ,

x

) −

b

=

0

⇐⇒

^Ax

+

^B

|

^x

| −

^b

=

^{0 and}

μ =

⁰

.

Thisindicatesthat x isasolutiontotheSOCAVE(2)ifandonlyif

( μ ,

x

)

isasolutiontotheequation H

( μ ,

x

) =

^0.Infact, we oftenchoose

μ ∈ R

++.Applying Lemma 3.1again, itisnotdifficulttoshow thatthefunction H

( μ ,

x

)

iscontinuously differentiable on

R

++

× R

^n.Fromdirectcalculation,wecanalsoobtaintheexplicitformulaoftheJacobianmatrixforthe function H asbelow:

H^

( μ ,

x

) =



1 0

B^∂p_∂⁽_μ^μ,x) A

+

B^∂p_∂⁽_x^μ,x)



(9)

forall

( μ ,

x

) ∈ R

++

× R

^nwithx

= (

^x1

,

x2

) ∈ R × R

^n−1,where

∂

p

( μ ,

x

)

∂ μ ⁼

∂φ

p

( μ , λ

1

(

x

))

∂ μ

^u

(1)

x

+ ∂φ

p

( μ , λ

2

(

x

))

∂ μ

^u

(2) x

= μ

^p−1

[φ

p

( μ , λ

1

(

x

))]

^p−1u

(1)

x

+ μ

^p−1

[φ

p

( μ , λ

2

(

x

))]

^p−1u

(2) x and

∂

_p

( μ ,

x

)

∂

x

=

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

sgn(x1)|x1|^p−1

√_p

μ^p+|^x1|^pp−1I if x2

=

⁰

,

⎡

⎣

^{b c x}

T

x22 c_^x_x²

2 aI

+ (

^b

−

^a

)

^x2x

T

x22²

⎤

⎦

if x₂

=

⁰

,

with

a

= φ

p

( μ , λ

2

(

x

)) − φ

p

( μ , λ

1

(

x

)) λ

2

(

x

) − λ

1

(

x

) ,

b

=

¹

2

sgn

(λ

₂

(

x

)) |λ

2

(

x

) |

^p−1

[φ

p

( μ , λ

2

(

x

)) ]

^p−1

+

^sgn

(λ

₁

(

x

)) |λ

1

(

x

) |

^p−1

[φ

p

( μ , λ

1

(

x

)) ]

^p−1



,

(10)

c

=

¹ 2

sgn

(λ

2

(

x

))|λ

2

(

x

)|

^p−1

[φ

p

( μ , λ

2

(

x

))]

^p−1

−

^sgn

(λ

1

(

x

))|λ

1

(

x

)|

^p−1

[φ

p

( μ , λ

1

(

x

))]

^p−1



.

4. SmoothingNewtonmethod

Inthissection,weinvestigatethesmoothingalgorithmbasedonthesmoothingfunction



p

( μ ,

x

)

forsolvingtheSOCAVE (2), and show the convergence properties of the considered algorithm. First, we present the generic framework of the smoothingalgorithm.

Algorithm4.1(AsmoothingNewtonalgorithm).

Step 0 Choose

δ ∈ (

⁰

,

1

)

,

σ ∈ (

⁰

,

1

)

, and

μ

0

∈ R

++, x⁰

∈ R

^{n. Set z0}

:= ( μ

0

,

x⁰

)

, e

:= (

¹

,

0

) ∈ R × R

^{n−1. Choose}

β >

1 satisfyingmin

{

¹

, 

^H

(

z⁰

) 

²

} ≤ β μ

0.Setk

:=

^0.

Step 1 If



^H

(

z^k

)  =

^{0,stop.Otherwise,set}

τ

k

:=

^min

{

¹

, 

^H

(

z^k

) }

^. Step 2 Compute



^zk

= ( μ

k

, 

^xk

) ∈ R × R

^{n by}

H

(

z^k

) +

^H

(

z^k

) 

^zk

=

¹

β τ

_k²e

,

(11)

whereH^

(

z^k

)

denotestheJacobianmatrixof H

(

z^k

)

at

( μ

k

,

x^k

)

givenby(9).

Step 3 Let

α

kbethemaximumofthevalues1

, δ, δ

²

, · · ·

^suchthat



^H

(

z^k

+ α

k



^zk

) ≤



1

− σ (

1

−

¹

β ) α

k





^H

(

z^k

).

⁽¹²⁾

Step 4 Setz^k+1

:=

^zk

+ α

k



^{zk andk}

:=

^k

+

^{1.Goto Step1.}

In orderto explain that Algorithm 4.1 is well defined, we haveto prove that the systemof Newton equation (11)is solvable,andthelinesearch(12)iswell-defined.Tothisend,weneedthenexttwotechnicallemmas.

Lemma4.1.Forany M

,

N

∈ R

^n×n,

σ

min

(

M

) > σ

max

(

N

)

ifandonlyif

σ

min

(

M^TM

) > σ

max

(

N^TN

)

.Inaddition,if

σ

min

(

M^TM

) >

σ

max

(

N^TN

)

,thenM^TM

−

^{NTN ispositivedefinite.Here}

σ

min

(

M

)

denotestheminimumsingularvalueofM,and

σ

max

(

N

)

denotes themaximumsingularvalueofN.

Proof. Theproofisstraightforwardorcanbefoundinusualtextbookofmatrixanalysis,soweomitithere.

2

Lemma4.2.LetA

,

S

∈ R

^{n×nandA besymmetric.Supposethatthe}eigenvaluesofA andS S^Tarearrangedinnon-increasingorder.

Then,foreachk

=

¹

,

2

, · · · ,

^{n,thereexistsa}nonnegativerealnumber

θ

_ksuchthat

λ

min

(

S S^T

) ≤ θ

k

≤ λ

max

(

S S^T

)

and

λ

k

(

S A S^T

) = θ

k

λ

k

(

A

).

Proof. Pleasesee[18,Corollary4.5.11]foraproof.

2

InordertoshowthattheJacobianmatrix H^

( μ ,

x

)

inNewtonequation(11)isnonsingularforany

μ >

0.We needthe followingassumption:

Assumption4.1.FortheSOCAVE(2),itholds

σ

min

(

A

) > σ

max

(

B

)

.

Infact,undertheconditionofAssumption 4.1,the SOCAVE(2)hasauniquesolution,whichisverifiedin[33].

Theorem4.1.LetH bedefinedasin(8).SupposethatAssumption 4.1holds.Then,theJacobianmatrixH^

( μ ,

x

)

inNewtonequa- tions(11)isnonsingularforany

μ >

0.

Proof. From the expression of H^

( μ ,

x

)

given asin (9), we know that H^

( μ ,

x

)

is nonsingular ifand only ifthe matrix A

+

^B∂(_∂^μ_x^{,x) is}nonsingular.Thus, itsufficestoshow thatthematrix A

+

^B∂(_∂^μ_x^{,x) is}nonsingular.Supposenot,i.e.,there existsavector0

=

^v

∈ R

^{n suchthat}



A

+

^B

∂( μ ,

x

)

∂

x



v

=

⁰

.

Thisimpliesthat v^TA^TA v

=

^vT

 ∂( μ ,

x

)

∂

x



T

B^TB

∂( μ ,

x

)

∂

x v

.

(13)

Forconvenience,we denoteC

:=

^∂(_∂^μ_x^{,x).Then, itfollowsthat vTATA v}

=

^{vTCTBTBC v.ByLemma 4.2,thereexistsacon-} stant

ˆθ

suchthat

λ

_min

(

C^TC

) ≤ ˆθ ≤ λ

max

(

C^TC

)

and

λ

max

(

C^TB^TBC

) = ˆθλ

max

(

B^TB

).

Notethatifwecanprovethat0

≤ λ

min

(

C^TC

) ≤ λ

max

(

C^TC

) ≤

^1,wehave

λ

max

(

C^TB^TBC

) ≤ λ

max

(

B^TB

)

.Then,bytheassump- tion that the minimumsingular value of A strictlyexceeds the maximumsingular value of B,andapplying Lemma 4.1, weobtain v^TA^TA v

>

v^TC^TB^TBC v.Thiscontradictstheformula(13),whichshowstheJacobianmatrixH^

( μ ,

x

)

inNewton equations(11)isnonsingularfor

μ >

0.

Thus, asdiscussed above, we only need to prove 0

≤ λ

min

(

C^TC

) ≤ λ

max

(

C^TC

) ≤

^{1.For x}2

=

^{0,we compute that C}

=

sgn(x₁)|^x1|^p−1

√_p

μ^p+|^x1|^pp−1I.Then,itisclearthat 0

< λ(

C^TC

) <

1 for

μ >

0.Forx2

=

^{0,usingthefactthat thematrixMTM isalways} positive semidefiniteforanymatrix M

∈ R

^{m×n,weseethat theinequality}

λ

min

(

C^TC

) ≥

^{0 alwaysholds.Inorderto prove} that

λ

max

(

C^TC

) ≤

^{1,weneedtofurtherprovethatthematrixI}

−

^{CTC ispositive}semidefinite.Toseethis,notethat

I

−

^CTC

=

⎡

⎣

¹

−

^b2

−

^c2

−

^2bc_^x_x^2T_2

−

^2bc_^x_x²_2

(

1

−

^a2

)

I

+ (

^a2

−

^b2

−

^c2

)

_^x2x2T

x2²

⎤

⎦ .

Becauseb²

+

^c2

=

¹ 2

 |λ

2

(

x

) |

^2(p−1)

[φ

p

( μ , λ

2

(

x

)) ]

^2(p−1)

+ |λ

1

(

x

) |

^2(p−1)

[φ

p

( μ , λ

1

(

x

)) ]

^2(p−1)



<

¹

2

·

²

=

^{1 for}

μ >

0,wehave1

−

^b2

−

^c2

>

0.Moreover, theSchurcomplementof1

−

^b2

−

^{c2hastheformof}

(

1

−

^a2

)

I

+ (

^a2

−

^b2

−

^c2

)

^x2x

T 2



^x2



²

−

^4b2c2 1

−

^b2

−

^c2

x2x^T₂



^x2



²

= (

¹

−

^a2

)



I

−

^x2x2T



^x2



²

 +

1

−

^b2

−

^c2

−

^4b2c2 1

−

^b2

−

^c2



x2x₂^T



^x2



²

.

(14)

Ontheotherhand,

|λ

i

(

x

)| < φ

p

( μ , λ

i

(

x

)) (

i

=

¹

,

2

)

for

μ >

0,wehave

φ

_p

( μ , λ

₂

(

x

)) − φ

p

( μ , λ

₁

(

x

))

=

 

 

 

|λ

2

(

x

)|

^p

− |λ

1

(

x

)|

^p



p i=1

 φ

p

( μ , λ

2

(

x

)) 

p−ⁱ



φ

p

( μ , λ

1

(

x

)) 

i−¹

 

 

 

=

 

 

 

( |λ

2

(

x

) | − |λ

1

(

x

) |)



p i=1

|λ

2

(

x

) |

^p−i

|λ

1

(

x

) |

ⁱ⁻¹



p i=¹

 φ

_p

( μ , λ

₂

(

x

)) 

p−i



φ

_p

( μ , λ

₁

(

x

)) 

i−1

 

 

 

< ||λ

2

(

x

)| − |λ

1

(

x

)||

≤ |λ

2

(

x

) − λ

1

(

x

)|.

Thistogetherwith(10)impliesthat1

−

^a2

>

0 forany

μ >

0.Inaddition,forany

μ >

0,weobservethat

(

1

−

^b2

−

^c2

)

²

−

^4b2c2

= (

¹

− (

^b

−

^c

)

²

)(

1

− (

^b

+

^c

)

²

)

=



1

− |λ

1

(

x

) |

^2(p−1)

 φ

p

( μ , λ

1

(

x

)) 

2(p−¹)



·



1

− |λ

2

(

x

) |

^2(p−1)

 φ

p

( μ , λ

2

(

x

)) 

2(p−¹)



>

0

,

where theinequality holds dueto

|λ

i

(

x

) | < φ

p

( μ , λ

i

(

x

))

fori

=

¹

,

2 and

μ >

0. Withall ofthese, we seethat the Schur complementof1

−

^b2

−

^{c2 givenasin(14)isalinearpositive} combinationofthematrices

I

−

_^x_x²₂^x_^T22



and ^x2x

T

x22²,which yields that theSchur complement(14)of1

−

^b2

−

^{c2 is positive}semidefinite. Hence, thematrix I

−

^{CTC isalso positive} semidefinite,whichisequivalenttosaying0

≤ λ

min

(

C^TC

) ≤ λ

max

(

C^TC

) ≤

^{1.Thus,theproofiscomplete.}

2