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

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(1)

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## Applied Numerical Mathematics

www.elsevier.com/locate/apnum

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

a

b

1

a

a

### ,∗,

2

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:

Availableonline3September2018

Keywords:

Second-ordercone Absolutevalueequations SmoothingNewtonalgorithm

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

1. Introduction

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

Thestandardabsolutevalueequation(AVE)isintheformof

Ax

B

x

b

(1)

where A

n×n, B

n×n, B

0,andb

n.Here

x

### |

meansthecomponentwiseabsolutevalue ofvectorx

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

(2)

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

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 xi

,i.e.,

x

x1

x2

xn

T

n;however,

x

### |

intheSOCAVE(2) denotesthevector satisfying

x2

x

### ◦

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

n

n

1

### )

,alsocalledtheLorentz cone,is deﬁnedas

n

x1

x2

n1

x2

x1

where

###  · 

denotestheEuclideannorm.Ifn

1,then

### K

nisthesetofnonnegativereals

### R

+.Ingeneral,a generalsecond- ordercone

### K

couldbetheCartesianproductofSOCs,i.e.,

n1

nr

### .

Forsimplicity,wefocusonthesingleSOC

### K

nbecausealltheanalysiscanbecarriedovertothesettingofCartesianproduct.

TheSOCisaspecialcaseofsymmetricconesandcanbeanalyzedunderJordanproduct,see[9].Inparticular,foranytwo vectorsx

x1

x2

n1and y

y1

y2

### ) ∈ R × R

n1,theJordanproduct ofx and y associatedwith

nisdeﬁnedas

x

y

xTy y1x2

x1y2

###  .

TheJordanproduct,unlikescalarormatrixmultiplication,isnotassociative,whichisamainsourceofcomplicationinthe analysisofoptimizationproblemsinvolvedSOC,see[5,6,10] andreferencesthereinformoredetails. Theidentityelement underthis Jordanproduct is e

1

0

0

T

### ∈ R

n. Withthesedeﬁnitions, 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

n1,thespectraldecompositionofx withrespectto SOCisgivenby

x

1

x

u(x1)

2

x

u(x2)

(3)

(3)

where

i

x

x1

1

i

x2

fori

1

2 and u(xi)

1 2

1

1

i xxT2

2

T

if

x2

0

1 2

1

1

i

TT if

x2

0

(4)

with

### ω∈ R

n1 beinganyvector satisfying

1.Thetwo scalars

1

x

and

2

x

### )

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

nisuniqueifx2

### =

0.Itisknownthatthespectralvaluesandspectralvectorspossesthefollowingproperties:

(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

21

x

22

x

.

### K

n,andx be theprojectionof

### −

x ontothedualcone

n

of

### K

n,wherethedualcone

n

isdeﬁnedby

n

y

n

x

y

0

x

n

### }

.Infact,thedualconeof

n isitself,i.e.,

n

### = K

n.Duetothespecialstructureof

### K

n,theexplicitformulaof projectionofx

x1

x2

n1onto

### K

n isobtainedin[5,6,8–10] asbelow:

x+

x if x

n

0 if x

n

u otherwise

where u

x1+x2

2 x1+x2

2

x2

x2

### ⎦ .

Similarly,theexpressionofxcanbewrittenoutas

x

0 if x

n

x if x

n

w otherwise

where w

x1−2x2

x1−x2 2

x2

x2

### ⎦ .

Itiseasytoverifythatx

x+

x−and

x+

1

x

+u(x1)

2

x

+u(x2) x

1

x

+u(x1)

2

x

+u(x2)

where

+

max

0

for

### α ∈ R

.As fortheexpression of

x

### |

associatedwithSOC.There is analternative wayvia the so-calledSOC-functiontoobtaintheexpressionof

x

### |

,whichcanbefoundin[2,3].Inanycase,itcomesoutthat

x

1

x

+

1

x

+

u(x1)

2

x

+

2

x

+

u(x2)

1

x

u(x1)

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

++

### × R → R

iscalledasmoothingfunctionof

t

### |

ifitsatisﬁesthefollowing:

(i)

### φ

iscontinuouslydifferentiableat

t

++

; (ii) lim

μ0

t

t

foranyt

### ∈ R

.

Givenasmoothingfunction

### φ

,wefurtherdeﬁneavector-valuedfunction

++

n

nas

x

1

x

u(x1)

2

x

u(x2) (5)

where

++isaparameter,

1

x

,

2

x

### )

arethespectralvaluesofx,andu(x1),u(x2)arethespectral vectorsofx.Conse- quently,

isalsosmoothon

++

### × R

n.Moreover,itiseasytoverifythat

lim

μ0+

x

1

x

u(x1)

2

x

u(x2)

x

### |

(4)

whichmeans eachfunction

x

### )

servesasa smoothingfunctionof

x

### |

associatedwithSOC.Withthisobservation, for theSOCAVE(2),wefurtherdeﬁnethefunction H

x

++

n

n by

H

x

Ax

B

x

b

++and x

n

### .

(6)

Proposition2.1.Supposethatx

x1

x2

### ) ∈ R × R

n1hasthespectraldecompositionasin(3)–(4).LetH

++

n

### → R

nbe deﬁnedasin(6).Then,

(a) H

x

### ) =

0 ifandonlyifxsolvestheSOCAVE(2);

(b) H iscontinuouslydifferentiableat

x

++

### × R

nwiththeJacobianmatrixgivenby H

x

1 0

B∂(μμ,x) A

B∂(μx,x)

(7)

where

x

1

x

u

(1)

x

2

x

u

(2) x

x

x

∂φ(μ,x1)

x1 I if x2

0

b c

xT2

x2 cxx2

2 aI

b

a

xx2xT2

22

if x2

0

with

a

2

x

1

x

2

x

1

x

b

1

2

2

x

x1

1

x

x1

(8)

c

1 2

2

x

x1

1

x

x1

### .

Proof. (a)First,weobservethat

H

x

0

0 and Ax

B

x

b

0

Ax

B

x

b

0 and

0

### .

Thisindicatesthatx isasolutiontotheSOCAVE(2) ifandonlyif

x

isasolutiontoH

x

0.

(b)Since

x

### )

iscontinuously differentiable on

++

### × R

n,itisclearthat H

x

### )

iscontinuously differentiableon

++

### × R

n.Thus,itremainstocomputetheJacobianmatrixofH

x

.Notethat

x

1

x

u(x1)

2

x

u(x2)

1 2

1

x

2

x

1

x

xx2T

2

2

x

xxT2

2

if x2

0

1

2

1

x

2

x

1

x

T

2

x

T

if x2

0

1 2

1

x

2

x

1

x

2

x

xx¯2

2

1

x

2

x

x¯xn

2

if x2

0

1

x

2

x

0

0

if x2

0

(5)

wherex2

x2

xn

n1,

2

n

### ) ∈ R

n1.Fromchainrule,itistrivialthat

x

1

x

u

(1)

x

2

x

### ∂μ

u

(2) x Inordertocompute ∂(μ,x)

x ,forsimplicity,wedenote

x

1 2

1

x

2

x

n

x

### ⎥ ⎦ .

Toproceed,wediscusstwocases.

(i)Forx2

0,wecompute

1

x

x1

1

x

x1

2

x

x1

1

x

1

x

1

x

x1

2

x

2

x

2

x

x1

1

x

1

x

2

x

2

x

2b

and

1

x

x

i

1

x

x

i

2

x

x

i

1

x

1

x

1

x

x

i

2

x

2

x

2

x

x

i

1

x

1

x

xi

x2

2

x

2

x

xi

x2

2

x

2

x

1

x

1

x

x

i

x2

2

x

x1

1

x

x1

x

i

x2

2c x

i

x2

i

2

n

Moreover,

i

x

x1

2

x

x1

1

x

x1

x

i

x2

2c

xi

x2

i

2

n

Similarly,wehave

2

x

x

2

2

x

x

2

1

x

x2

x

2

x2

2

x

1

x

x¯2

x2

x

2

2b

x2

x2

x2

2

2

x

1

x

1

x2

x

2

x2

x2

3

2a

2

b

a

x2

x2

x2

2

wherea meansa

2

x

1

x

2

x

1

x

### )

.Ingeneral,mimickingthesamederivationyields

i

x

x

j

2a

2

b

a

x¯xi·¯xi

22 if i

j

2

b

a

x¯xi·¯xj

22 if i

j

Tosumup,weobtain

x

x

b c

xT2

x2 cxx2

2 aI

b

a

x2x2T

x22

### ⎦

whichisthedesiredresult.

(6)

(ii)Forx2

0,itiscleartosee

1

x

x1

2

x1

x1 and

1

x

x

i

0 for i

2

n

Since

i

x

0 fori

2

### ,· · · ,

n,itgives τi(xμ1,x)

0.Moreover,

2

x

x2

lim

¯ x20

2

x1

x

2

0

0

2

x1

0

0

x2

lim

¯ x20

x1

x2

x1

x2

x2

x2

x2

lim

¯ x20

x1

x2

x1

x2

x2

lim

¯ x20

x1

x2

x2

x1

x2

x2

### |)(

as L’Hopital’s rule

lim

¯ x20

x1

x2

x1

x2

x1

x2

x1

x2

2

x1

x1

Thus,weobtain

i

x

x

j

2∂φ(μx,x1)

1 if i

j

0 if i

j

### ,

whichisequivalenttosaying

x

x

x1

x1 I

### .

Fromalltheabove,weconcludethat

x

x

∂φ(μ,x1)

x1 I if x2

0

b c

xT2

x2 cxx2

2 aI

b

a

x2x

T

x222

if x2

0

### .

Thus,theproofiscomplete.

### 2

Lemma2.1.SupposethatM

N

n×n.Let

min

M

### )

denotetheminimumsingularvalueofM,and

max

N

### )

denotethemaximum singularvalueofN.Then,thefollowinghold.

(a)

min

M

max

N

ifandonlyif

min

MTM

max

NTN

. (b) If

min

MTM

max

NTN

,thenMTM

### −

NTN ispositivedeﬁnite.

Proof. Theproofisstraightforwardorcanbefoundinusualtextbookofmatrixanalysis,soweomitithere.

Lemma2.2.LetA

S

### ∈ R

n×nandA besymmetric.SupposethattheeigenvaluesofA andS STarearrangedinnon-increasingorder.

Then,foreachk

1

2

### ,· · · ,

n,thereexistsanonnegativerealnumber

ksuchthat

min

S ST

k

max

S ST

and

k

S A ST

k

k

A

### 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

x

### ))

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

x

### ))

isnonsingular.

(7)

Theorem2.1.ConsideraSOCAVE(2) with

min

A

max

B

### )

.LetH bedeﬁnedasin(6).Supposethat

++

### × R → R

isa smoothingfunctionof

t

.If

1

dtd

t

### ) ≤

1 issatisﬁed,thentheJacobianmatrixH

x

### )

isnonsingularforany

### μ >

0.

Proof. From the expression of H

x

### )

givenas in(7), we know that H

x

### )

is nonsingularif andonly if the matrix A

### +

B∂(μx,x) isnonsingular.Thus,itsuﬃcestoshowthatthematrix A

### +

B∂(μx,x) isnonsingularundertheconditions.

Supposenot,thatis,thereexistsavector0

v

n suchthat

A

B

x

x

v

0

whichimpliesthat

vTATA v

vT

x

x

T

BTB

x

x v

### .

(9)

Forconvenience,wedenoteC

### :=

∂(μx,x).Then,itfollowsthat vTATA v

### =

vTCTBTBC v.ApplyingLemma2.2,thereexistsa constant

suchthat

min

CTC

max

CTC

and

max

CTBTBC

max

BTB

### ).

Notethatifwecanprovethat

0

min

CTC

max

CTC

1

we willhave

max

CTBTBC

max

BTB

### )

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

min

A

max

B

### )

) and applyingLemma2.1,we obtain vTATA v

vTCTBTBC v.

x

isnonsingularfor

### μ >

0.

Thus, inlightoftheabovediscussion,itsuﬃces toclaim0

min

CTC

max

CTC

### ) ≤

1.Tothisend,we discusstwo cases.

Case1: Forx2

0,wecomputethatC

### =

∂φ (μx1,x1)I.Since

1

∂φ (μx1,x1)

1,itisclearthat0

CTC

1 for

### μ >

0.Then, theclaimisdone.

Case2: Forx2

### =

0,usingthefact thatthe matrixMTM isalways positivesemideﬁniteforanymatrix M

### ∈ R

m×n,we see thattheinequality

min

CTC

### ) ≥

0 alwaysholds.Inordertoprove

max

CTC

### ) ≤

1,weneedtofurtherarguethatthematrix I

### −

CTC ispositivesemideﬁnite.First,wewriteout

I

CTC

1

b2

c2

2bcxx2T2

2bcxx2

2

1

a2

I

a2

b2

c2

xx2x2T

22

If

1

∂φ (μx1i(x))

### <

1,thenweobtain b2

c2

1

2

1

x

x1

2

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

x2xT2

x2

2

1

a2

I

x2x2T

x2

2

1

b2

c2

4b2c2 1

b2

c2

x2x2T

x2

2

### .

(10)

Ontheotherhand,bytheMeanValueTheorem,wehave

2

x

1

x

2

x

1

x

where

1

x

2

x

### ))

.Toproceed,weneedtofurtherdiscusstwosubcases.

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