Unified smoothing functions for absolute value equation associated with second-order cone

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Applied Numerical Mathematics

www.elsevier.com/locate/apnum

Unified smoothing functions for absolute value equation associated with second-order cone

Chieu Thanh Nguyen

^a

, B. Saheya

^b

^,

¹

, Yu-Lin Chang

^a

, Jein-Shan Chen

^a

^,∗,

²

aDepartmentofMathematics,NationalTaiwanNormalUniversity,Taipei11677,Taiwan

bCollegeofMathematicalScience,InnerMongoliaNormalUniversity,Hohhot010022,InnerMongolia,PRChina

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

Articlehistory:

Received19February2018 Receivedinrevisedform12July2018 Accepted30August2018

Availableonline3September2018

Keywords:

Second-ordercone Absolutevalueequations SmoothingNewtonalgorithm

Inthispaper,weexploreaunifiedwaytoconstructsmoothingfunctionsforsolvingtheab- solutevalueequationassociatedwithsecond-ordercone(SOCAVE).Numericalcomparisons arepresented,whichillustratewhat kindsofsmoothingfunctions workwell alongwith thesmoothingNewtonalgorithm.Inparticular,thenumericalexperimentsshowthatthe wellknownlossfunctionwidelyusedinengineeringcommunityistheworstoneamong theconstructedsmoothingfunctions,whichindicatesthattheotherproposedsmoothing functionscanbeemployedforsolvingengineeringproblems.

©^{2018IMACS.PublishedbyElsevierB.V.Allrightsreserved.}

1. Introduction

Recently,thepaper[36] investigatesafamilyofsmoothingfunctionsalongwithasmoothing-typealgorithmtotacklethe absolutevalueequationassociatedwithsecond-ordercone(SOCAVE)andshowstheefficiencyofsuchapproach.Motivated bythisarticle,wecontinuetoasktwonaturalquestions.(i)Whetherthereareothersuitablesmoothingfunctionsthatcan be employedforsolvingtheSOCAVE?(ii)IsthereaunifiedwaytoconstructsmoothingfunctionsforsolvingtheSOCAVE?

Inthispaper,weprovideaffirmativeanswersforthesetwoqueries.Inordertosmoothlyconveythestoryofhowwefigure outtheanswers,webeginwithrecallingwheretheSOCAVEcomesfrom.

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

.

It isknownthat theAVE(1) wasfirst introducedby Rohnin [40],butwas termedby Mangasarian[34].Duringthe past decade,therehasbeenmanyresearcherspayingattentiontothisequation,forexample,Caccetta,QuandZhou[1],Huand

*

Correspondingauthor.

E-mailaddresses:thanhchieu90@gmail.com(C.T. Nguyen),saheya@imnu.edu.cn(B. Saheya),ylchang@math.ntnu.edu.tw(Y.-L. Chang), jschen@math.ntnu.edu.tw(J.-S. Chen).

1 Theauthor’sworkissupportedbyNaturalScienceFoundationofInnerMongolia(AwardNumber:2017MS0125).

2 Theauthor’sworkissupportedbyMinistryofScienceandTechnology,Taiwan.

https://doi.org/10.1016/j.apnum.2018.08.019

0168-9274/©^{2018IMACS.PublishedbyElsevierB.V.Allrightsreserved.}

Huang[12],JiangandZhang[19],KetabchiandMoosaei[20],Mangasarian[26–33],MangasarianandMeyer[34],Prokopyev [37],andRohn[42].

WeelaboratemoreaboutthedevelopmentsoftheAVE.MangasarianandMeyer[34] showthattheAVE(1) isequivalent to the bilinear program, the generalized LCP (linear complementarity problem), and to the standard LCP provided 1 is not an eigenvalue of A.Withthese equivalent reformulations,they also show that the AVE(1) is NP-hard in its general formandprovide existence results.Prokopyev [37] furtherimproves theabove equivalence whichindicates that theAVE (1) can be equivalently recast asLCP withoutany assumptionon A and B,andalso provides a relationship withmixed integer programming. Ingeneral, if solvable, the AVE(1) can have eitherunique solution ormultiple (e.g.,exponentially many) solutions.Indeed,various sufficiency conditionson solvabilityandnon-solvabilityofthe AVE(1) withunique and multiplesolutions arediscussedin [34,37,41].Some variantsofthe AVE,like theabsolutevalueequation associatedwith second-orderconeandtheabsolutevalueprograms,areinvestigatedin[14] and[45],respectively.

Recently, another type of absolutevalue equation, a natural extension ofthe standard AVE (1), is considered [14,35, 36].Morespecificallythefollowingabsolutevalueequation associatedwithsecond-ordercones,abbreviatedasSOCAVE,is studied:

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,a generalsecond- ordercone

K

couldbetheCartesianproductofSOCs,i.e.,

K := K

ⁿ¹

× · · · × K

^nr

.

Forsimplicity,wefocusonthesingleSOC

K

ⁿbecausealltheanalysiscanbecarriedovertothesettingofCartesianproduct.

TheSOCisaspecialcaseofsymmetricconesandcanbeanalyzedunderJordanproduct,see[9].Inparticular,foranytwo vectorsx

= (

^x1

,

x2

) ∈ R × R

^{n−1and y}

= (

^y1

,

y2

) ∈ R × R

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

K

^{nisdefinedas}

x

◦

y

:=



x^Ty y₁x₂

+

^x1y₂

 .

TheJordanproduct,unlikescalarormatrixmultiplication,isnotassociative,whichisamainsourceofcomplicationinthe analysisofoptimizationproblemsinvolvedSOC,see[5,6,10] 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

.

Asremarkedintheliterature,thesignificanceoftheAVE(1) arisesfromthefactthattheAVEiscapableofformulating manyoptimizationproblemssuchaslinearprograms,quadraticprograms,bimatrixgames,andsoon.Likewise,theSOCAVE (2) plays a similar role in various optimization problemsinvolving second-order cones. There hasbeen manynumerical methodsproposed forsolving thestandardAVE(1) andtheSOCAVE(2).Please referto [36] fora quickreview.Basically, wefollowthesmoothingNewtonalgorithmemployedin[36] todealwiththeSOCAVE(2).Thiskindofalgorithmhasbeen apowerfultoolforsolvingmanyotheroptimizationproblems,includingsymmetricconecomplementarityproblems[21,23, 24],the systemofinequalitiesundertheorder inducedby symmetriccone [17,25,46], andso on.Itis alsoemployed for thestandard AVE(1) in [18,43]. Thenewupshotofthispaperliesondiscoveringmoresuitable smoothingfunctionsand exploringaunifiedwaytoconstructsmoothingfunctions.Ofcourse,thenumericalperformanceamongdifferentsmoothing functionsarecompared.Thesearetotallynewtotheliteratureandarethemaincontributionofthispaper.

Toclosethissection, werecall somebasicconcepts andbackgroundmaterials regardingthesecond-ordercone,which willbe used inthe subsequentanalysis. Moredetails can be foundin [5,6,9,10,14]. First, werecall theexpression ofthe 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}_x^T2

2



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.Itisknownthatthespectralvaluesandspectralvectorspossesthefollowing}properties:

(i) u⁽_x¹⁾

◦

^u(x²⁾

=

^{0 andu(}xⁱ⁾

◦

^u(xⁱ⁾

=

^u(xⁱ⁾fori

=

¹

,

2;

(ii)



^u(x¹⁾



²

= 

^u(x²⁾



²

=

¹₂ ^and



^x



²

=

¹₂

(λ

²₁

(

x

) + λ

²₂

(

x

))

.

Nextistheconceptabouttheprojectionontosecond-ordercone.Letx₊denotetheprojectionofx onto

K

^n,andx₋ ^be theprojectionof

−

^{x ontothedualcone}

(K

ⁿ

)

^∗ of

K

^{n,wherethedualcone}

(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.Duetothespecialstructureof}

K

^{n,theexplicitformulaof} projectionofx

= (

^x1

,

x2

) ∈ R × R

^n−1onto

K

ⁿ isobtainedin[5,6,8–10] 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₋canbewrittenoutas

x₋

=

⎧ ⎨

⎩

0 if x

∈ K

ⁿ

,

−

x if x

∈ − K

ⁿ

,

w otherwise

,

where w

=

⎡

⎣  −

^x1−₂^x2

x1−x2 2



x2

^x2

⎤

⎦ .

Itiseasytoverifythatx

=

^x+

+

^x−and

x₊

= (λ

1

(

x

))

₊u⁽_x¹⁾

+ (λ

2

(

x

))

₊u⁽_x²⁾ x₋

= (−λ

1

(

x

))

₊u⁽_x¹⁾

+ (−λ

2

(

x

))

₊u⁽_x²⁾

,

where

( α )

₊

=

^max

{

⁰

, α }

^for

α ∈ R

^{.As fortheexpression of}

|

^x

|

^{associatedwithSOC.There is an}alternative wayvia the so-calledSOC-functiontoobtaintheexpressionof

|

^x

|

^{,whichcanbefoundin[2,3].Inanycase,itcomesoutthat}

|

^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⁾

.

2. UnifiedsmoothingfunctionsforSOCAVE

AsmentionedinSection1,weemploythesmoothingNewtonmethodforsolvingtheSOCAVE(2),whichneedsasmooth- ingfunctiontoworkwith.Indeed,a familyofsmoothingfunctionswasalreadyconsideredin[36].Inthissection,welook into what kindsofsmoothingfunctionscan be employed towork withthe smoothingNewton algorithmfor solvingthe SOCAVE(2).

Definition2.1.Afunction

φ : R

++

× R → R

^{iscalledasmoothingfunctionof}

|

^t

|

^{ifitsatisfiesthefollowing:}

(i)

φ

iscontinuouslydifferentiableat

( μ ,

t

) ∈ R

++

× R

^; (ii) lim

μ↓⁰

φ ( μ ,

t

) = |

^t

|

^foranyt

∈ R

^.

Givenasmoothingfunction

φ

,wefurtherdefineavector-valuedfunction

 : R

++

× R

ⁿ

→ R

^nas

( μ ,

x

) = φ ( μ , λ

1

(

x

))

u⁽_x¹⁾

+ φ ( μ , λ

2

(

x

))

u⁽_x²⁾ (5)

where

μ ∈ R

++isaparameter,

λ

1

(

x

)

,

λ

2

(

x

)

arethespectralvaluesofx,andu⁽_x¹⁾,u⁽_x²⁾arethespectral vectorsofx.Conse- quently,



isalsosmoothon

R

++

× R

^{n.Moreover,itiseasytoverifythat}

lim

μ→⁰⁺

( μ ,

x

) = |λ

1

(

x

) |

^u(x¹⁾

+ |λ

2

(

x

) |

^u(x²⁾

= |

^x

|

whichmeans eachfunction

( μ ,

x

)

servesasa smoothingfunctionof

|

^x

|

^{associatedwithSOC.Withthis}observation, for theSOCAVE(2),wefurtherdefinethefunction H

( μ ,

x

) : R

++

× R

ⁿ

→ R × R

^{n by}

H

( μ ,

x

) =

 μ

Ax

+

^B

( μ ,

x

) −

^b



, ∀ μ ∈ R

++and x

∈ R

ⁿ

.

(6)

Proposition2.1.Supposethatx

= (

^x1

,

x2

) ∈ R × R

^{n−1hasthespectral}decompositionasin(3)–(4).LetH

: R

++

× R

ⁿ

→ R

^nbe definedasin(6).Then,

(a) H

( μ ,

x

) =

^{0 ifandonlyifxsolvestheSOCAVE(2);}

(b) H iscontinuouslydifferentiableat

( μ ,

x

) ∈ R

++

× R

^{nwiththeJacobianmatrixgivenby} H^

( μ ,

x

) =



1 0

B^∂(_∂^μ_μ^,x) A

+

^B∂(_∂^μ_x^,x)



(7)

where

∂( μ ,

x

)

∂ μ ⁼

∂φ ( μ , λ

1

(

x

))

∂ μ

^u

(1)

x

+ ∂φ ( μ , λ

2

(

x

))

∂ μ

^u

(2) x

,

∂( μ ,

x

)

∂

x

=

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

∂φ(μ,x₁)

∂x1 I if x2

=

0

,

⎡

⎣

^{b c}

x^T₂

^x2 c_^x_x²

2 aI

+ (

^b

−

^a

)

_^x_x^2xT2

2²

⎤

⎦

if x₂

=

⁰

,

with

a

= φ ( μ , λ

2

(

x

)) − φ( μ , λ

1

(

x

)) λ

2

(

x

) − λ

1

(

x

) ,

b

=

¹

2

∂φ ( μ , λ

2

(

x

))

∂

x1

+ ∂φ ( μ , λ

1

(

x

))

∂

x1



,

(8)

c

=

¹ 2

∂φ ( μ , λ

2

(

x

))

∂

x₁

− ∂φ ( μ , λ

1

(

x

))

∂

x₁



.

Proof. (a)First,weobservethat

H

( μ ,

x

) =

0

⇐⇒ μ =

0 and Ax

+

B

( μ ,

x

) −

b

=

0

⇐⇒

^Ax

+

^B

|

^x

| −

^b

=

^{0 and}

μ =

⁰

.

Thisindicatesthatx isasolutiontotheSOCAVE(2) ifandonlyif

( μ ,

x

)

isasolutiontoH

( μ ,

x

) =

^0.

(b)Since

( μ ,

x

)

iscontinuously differentiable on

R

++

× R

^{n,itisclearthat H}

( μ ,

x

)

iscontinuously differentiableon

R

++

× R

^{n.Thus,itremainstocomputetheJacobianmatrixofH}

( μ ,

x

)

.Notethat

( μ ,

x

) = φ( μ , λ

1

(

x

))

u⁽_x¹⁾

+ φ( μ , λ

2

(

x

))

u⁽_x²⁾

=

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

1 2

 φ ( μ , λ

₁

(

x

)) + φ( μ , λ

₂

(

x

))

−φ( μ , λ

1

(

x

))

_^x_x^2T

2

+ φ( μ , λ

2

(

x

))

_^x_x^T2

2



if x2

=

0

,

1

2

 φ ( μ , λ

1

(

x

)) + φ( μ , λ

2

(

x

))

−φ( μ , λ

₁

(

x

)) ω

^T

_{+ φ(} μ , λ

₂

(

x

)) ω

^T



if x2

=

0

=

¹ 2

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

⎡

⎢ ⎢

⎢ ⎢

⎣

φ ( μ , λ

1

(

x

)) + φ( μ , λ

2

(

x

) (−φ( μ , λ

₁

(

x

)) + φ( μ , λ

₂

(

x

)))

_^x_x^¯2

2

.. .

(−φ( μ , λ

₁

(

x

)) + φ( μ , λ

₂

(

x

)))

_x^¯xn

2

⎤

⎥ ⎥

⎥ ⎥

⎦

^{if x2}

=

⁰

,

⎡

⎢ ⎢

⎢ ⎣

φ ( μ , λ

1

(

x

)) + φ( μ , λ

2

(

x

))

0

.. .

0

⎤

⎥ ⎥

⎥ ⎦

^{if x2}

=

⁰

,

wherex2

:= (¯

^x2

, · · · , ¯

xn

) ∈ R

^n−1,

ω _{= (} ω

2

, · · · , ω

n

) ∈ R

^{n−1.Fromchainrule,itistrivialthat}

∂ ( μ ,

x

)

∂ μ ⁼

∂φ ( μ , λ

1

(

x

))

∂ μ

^u

(1)

x

+ ∂φ ( μ , λ

2

(

x

))

∂ μ

^u

(2) x Inordertocompute ^∂(μ,x)

∂x ,forsimplicity,wedenote

( μ ,

x

) :=

¹ 2

⎡

⎢ ⎢

⎢ ⎣ τ

1

( μ ,

x

) τ

2

( μ ,

x

)

.. . τ

n

( μ ,

x

)

⎤

⎥ ⎥

⎥ ⎦ .

Toproceed,wediscusstwocases.

(i)Forx2

=

^0,wecompute

∂ τ

1

( μ ,

x

)

∂

x1

= ∂φ ( μ , λ

1

(

x

))

∂

x1

+ ∂φ ( μ , λ

2

(

x

))

∂

x1

= ∂φ ( μ , λ

1

(

x

))

∂λ

1

(

x

)

∂λ

1

(

x

)

∂

x₁

+ ∂φ ( μ , λ

2

(

x

))

∂λ

2

(

x

)

∂λ

2

(

x

)

∂

x₁

= ∂φ ( μ , λ

1

(

x

))

∂λ

₁

(

x

) + ∂φ ( μ , λ

2

(

x

))

∂λ

₂

(

x

) :=

^2b

and

∂ τ

₁

( μ ,

x

)

∂

x

¯

_i

= ∂φ ( μ , λ

₁

(

x

))

∂

x

¯

_i

+ ∂φ ( μ , λ

₂

(

x

))

∂

x

¯

_i

= ∂φ ( μ , λ

1

(

x

))

∂λ

1

(

x

)

∂λ

1

(

x

)

∂

x

¯

i

+ ∂φ ( μ , λ

2

(

x

))

∂λ

2

(

x

)

∂λ

2

(

x

)

∂

x

¯

i

= − ∂φ ( μ , λ

₁

(

x

))

∂λ

1

(

x

)

¯

x_i



^x2

 + ∂φ ( μ , λ

₂

(

x

))

∂λ

2

(

x

)

¯

x_i



^x2



=

∂φ ( μ , λ

₂

(

x

))

∂λ

2

(

x

) − ∂φ ( μ , λ

₁

(

x

))

∂λ

1

(

x

)



x

¯

_i



^x2



=

∂φ ( μ , λ

2

(

x

))

∂

x1

− ∂φ ( μ , λ

1

(

x

))

∂

x1



x

¯

i



x2

 :=

2c x

¯

i



x2

 ,

i

=

2

,· · · ,

n

.

Moreover,

∂ τ

i

( μ ,

x

)

∂

x₁

=

∂φ ( μ , λ

2

(

x

))

∂

x₁

− ∂φ ( μ , λ

1

(

x

))

∂

x₁



x

¯

_i



^x2

 =

^2c

¯

x_i



^x2

 ,

i

=

²

, · · · ,

ⁿ

.

Similarly,wehave

∂ τ

2

( μ ,

x

)

∂

x

¯

2

=

∂φ ( μ , λ

2

(

x

))

∂

x

¯

2

− ∂φ ( μ , λ

1

(

x

))

∂ ¯

x2



x

¯

2



x2

 + (φ( μ , λ

2

(

x

)) − φ( μ , λ

1

(

x

)))

∂



x¯2

^x2



∂

x

¯

2

=

^2b

¯

^x2

· ¯

^x2



^x2



²

+ (φ( μ , λ

2

(

x

)) − φ( μ , λ

1

(

x

)))

1



^x2

 −

^x

¯

2

· ¯

^x2



^x2



³



=

^2a

+

²

(

b

−

^a

) ¯

x2

· ¯

x2



^x2



²

,

wherea meansa

:= φ ( μ , λ

2

(

x

)) − φ( μ , λ

1

(

x

))

λ

2

(

x

) − λ

1

(

x

)

^{.Ingeneral,mimickingthesamederivationyields}

∂ τ

_i

( μ ,

x

)

∂

x

¯

_j

=



2a

+

²

(

b

−

^a

)

_^x¯_x^i·¯xi

2² if i

=

^j

,

2

(

b

−

^a

)

_^x¯_x^i·¯xj

2² if i

=

^j

.

Tosumup,weobtain

∂ ( μ ,

x

)

∂

x

=

⎡

⎣

^{b c}

x^T₂

x2 c_^x_x²

2 aI

+ (

^b

−

^a

)

_^x2x2T

x2²

⎤

⎦

whichisthedesiredresult.

(ii)Forx2

=

^{0,itiscleartosee}

∂ τ

1

( μ ,

x

)

∂

x₁

=

²

∂φ ( μ ,

x₁

)

∂

x₁ and

∂ τ

1

( μ ,

x

)

∂

x

¯

_i

=

^{0 for i}

=

²

,· · · ,

ⁿ

.

Since

τ

i

( μ ,

x

) =

^{0 fori}

=

²

, · · · ,

^{n,itgives ∂τi}_∂⁽_x^μ₁^,x)

=

^0.Moreover,

∂ τ

₂

( μ ,

x

)

∂ ¯

x2

=

^lim

¯ x₂→⁰

τ

₂

( μ ,

x₁

,

x

¯

₂

,

0

, · · · ,

⁰

) − τ

₂

( μ ,

x₁

,

0

, · · · ,

⁰

)

¯

x2

=

^lim

¯ x2→0

φ ( μ ,

x₁

+ |¯

^x2

|) − φ( μ ,

x₁

− |¯

^x2

|)

¯

x₂

¯

x₂

|¯

^x2

|

=

lim

¯ x2→⁰

φ ( μ ,

x1

+ |¯

x2

|) − φ( μ ,

x1

− |¯

x2

|)

|¯

x2

|

=

^lim

¯ x₂→⁰

∂φ ( μ ,

x₁

+ |¯

^x2

|)

∂( |¯

^x2

|) − ∂φ ( μ ,

x₁

− |¯

^x2

|)

∂( |¯

^x2

|) (

as L’Hopital’s rule

)

=

^lim

¯ x2→0

∂φ ( μ ,

x₁

+ |¯

^x2

|)

∂(

x₁

+ |¯

^x2

|) + ∂φ ( μ ,

x₁

− |¯

^x2

|)

∂(

x₁

− |¯

^x2

|)

=

²

∂φ ( μ ,

x₁

)

∂

x₁

.

Thus,weobtain

∂ τ

i

( μ ,

x

)

∂

x

¯

j

=



2^∂φ(_∂^μ_x^,x1)

1 if i

=

^j

,

0 if i

=

^j

,

whichisequivalenttosaying

∂( μ ,

x

)

∂

x

= ∂φ ( μ ,

x₁

)

∂

x₁ I

.

Fromalltheabove,weconcludethat

∂( μ ,

x

)

∂

x

=

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

∂φ(μ,x1)

∂x1 I if x₂

=

⁰

,

⎡

⎣

^{b c}

x^T₂

^x2 c_^x_x²

2 aI

+ (

b

−

a

)

^x2x

T

^x22²

⎤

⎦

if x2

=

⁰

.

Thus,theproofiscomplete.

2

Now,we areready toanswerthe question aboutwhatkindofsmoothingfunctionscan beadopted inthesmoothing typealgorithm.Twotechnicallemmasareneededtowardstheanswer.

Lemma2.1.SupposethatM

,

N

∈ R

^n×n.Let

σ

min

(

M

)

denotetheminimumsingularvalueofM,and

σ

max

(

N

)

denotethemaximum singularvalueofN.Then,thefollowinghold.

(a)

σ

min

(

M

) > σ

max

(

N

)

ifandonlyif

σ

min

(

M^TM

) > σ

max

(

N^TN

)

. (b) If

σ

min

(

M^TM

) > σ

max

(

N^TN

)

,thenM^TM

−

^{NTN ispositivedefinite.}

Proof. Theproofisstraightforwardorcanbefoundinusualtextbookofmatrixanalysis,soweomitithere.

2

Lemma2.2.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[11,Corollary4.5.11] foraproof.

2

We point out that the crucial key, which guarantees a smoothing function can be employed in the smoothing type algorithm,isthenonsingularity oftheJacobian matrix H^

( μ ,

x

))

givenin(7).Asbelow,we provideunderwhatcondition theJacobianmatrix H^

( μ ,

x

))

isnonsingular.

Theorem2.1.ConsideraSOCAVE(2) with

σ

min

(

A

) > σ

max

(

B

)

.LetH bedefinedasin(6).Supposethat

φ : R

++

× R → R

^isa smoothingfunctionof

|

^t

|

^.If

−

¹

≤

_dt^d

φ ( μ ,

t

) ≤

^{1 issatisfied,thentheJacobianmatrixH}

( μ ,

x

)

isnonsingularforany

μ >

0.

Proof. From the expression of H^

( μ ,

x

)

givenas in(7), we know that H^

( μ ,

x

)

is nonsingularif andonly if the matrix A

+

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

+

^B∂(_∂^μ_x^{,x) is}nonsingularundertheconditions.

Supposenot,thatis,thereexistsavector0

=

^v

∈ R

^{n suchthat}



A

+

^B

∂ ( μ ,

x

)

∂

x



v

=

⁰

whichimpliesthat

v^TA^TA v

=

^vT

 ∂( μ ,

x

)

∂

x



T

B^TB

∂ ( μ ,

x

)

∂

x v

.

(9)

Forconvenience,wedenoteC

:=

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

=

^{vTCTBTBC v.ApplyingLemma2.2,thereexistsa} constant

ˆθ

suchthat

λ

_min

(

C^TC

) ≤ ˆθ ≤ λ

max

(

C^TC

)

and

λ

max

(

C^TB^TBC

) = ˆθλ

max

(

B^TB

).

Notethatifwecanprovethat

0

≤ λ

min

(

C^TC

) ≤ λ

max

(

C^TC

) ≤

¹

,

we willhave

λ

max

(

C^TB^TBC

) ≤ λ

max

(

B^TB

)

.Then,bytheassumptionthattheminimumsingular valueof A strictlyexceeds the maximumsingular value of B (i.e.,

σ

min

(

A

) > σ

max

(

B

)

) and applyingLemma2.1,we obtain v^TA^TA v

>

v^TC^TB^TBC v.

Thiscontradictstheidentity(9),whichshowstheJacobianmatrix H^

( μ ,

x

)

isnonsingularfor

μ >

0.

Thus, inlightoftheabovediscussion,itsuffices toclaim0

≤ λ

min

(

C^TC

) ≤ λ

max

(

C^TC

) ≤

^{1.Tothisend,we discusstwo} cases.

Case1: Forx2

=

^{0,wecomputethatC}

=

^{∂φ (}_∂^μ_x1^,x1)I.Since

−

¹

≤

^{∂φ (}_∂^μ_x1^,x1)

≤

^{1,itisclearthat0}

≤ λ(

^CTC

) ≤

^{1 for}

μ >

0.Then, theclaimisdone.

Case2: Forx2

=

^{0,usingthefact thatthe matrixMTM isalways positive}semidefiniteforanymatrix M

∈ R

^{m×n,we see} thattheinequality

λ

min

(

C^TC

) ≥

^{0 alwaysholds.Inordertoprove}

λ

max

(

C^TC

) ≤

^{1,weneedtofurtherarguethatthematrix} I

−

^{CTC ispositive}semidefinite.First,wewriteout

I

−

^CTC

=

⎡

⎣

¹

−

^b2

−

^c2

−

^2bc_^x_x^2T_2

−

2bc_^x_x²

2

(

1

−

a²

)

I

+ (

a²

−

b²

−

c²

)

_^x_x^2x2T

2²

⎤

⎦ .

If

−

¹

<

^{∂φ (μ}_∂^,λ_x1^i(x))

<

1,thenweobtain b²

+

c²

=

¹

2

 ∂φ ( μ , λ

1

(

x

))

∂

x1



2

+

∂φ ( μ , λ

2

(

x

))

∂

x1



2



<

1

.

Thisindicatesthat1

−

^b2

−

^c2

>

0.Byconsidering

[

¹

−

^b2

−

^c2

]

^asan1

×

^{1 matrix,thissays}

[

¹

−

^b2

−

^c2

]

^{ispositivedefinite.}

Hence,itsSchurcomplementcanbecomputedasbelow:

(

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

.

(10)

Ontheotherhand,bytheMeanValueTheorem,wehave

φ ( μ , λ

2

(

x

)) − φ( μ , λ

1

(

x

)) = ∂φ ( μ , ξ )

∂ξ (λ

2

(

x

) − λ

1

(

x

)),

where

ξ ∈ (λ

1

(

x

), λ

2

(

x

))

.Toproceed,weneedtofurtherdiscusstwosubcases.