Predictor-corrector interior-point algorithm for P∗(κ)-linear complementarity problems based on a new type of algebraic equivalent transformation technique

European Journal of Operational Research xxx (xxxx) xxx

ContentslistsavailableatScienceDirect

European Journal of Operational Research

journalhomepage:www.elsevier.com/locate/ejor

Continuous Optimization

Predictor-corrector interior-point algorithm for P _∗ ( κ ) ^-linear

complementarity problems based on a new type of algebraic equivalent transformation technique

Zsolt Darvay

^a

, Tibor Illés

^b

, Petra Renáta Rigó

^b,∗

aFaculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania

bCorvinus Center for Operations Research at Corvinus Institute for Advanced Studies, Corvinus University of Budapest, Hungary; on leave from Department of Differential Equations, Faculty of Natural Sciences, Budapest University of Technology and Economics

a rt i c l e i nf o

Article history:

Received 10 September 2020 Accepted 24 August 2021 Available online xxx Keywords:

Interior-point methods

P ∗(κ)-linear complementarity problem Predictor-corrector algorithm Polynomial iteration complexity

a b s t r a c t

Weproposeanewpredictor-corrector(PC)interior-pointalgorithm(IPA)forsolvinglinearcomplementar- ityproblem(LCP)withP_∗(κ)^{-matrices.TheintroducedIPAusesanewtypeofalgebraicequivalenttrans-} formation(AET)onthecenteringequationsofthesystemdefiningthecentralpath.Thenewtechnique wasintroduced byDarvayand Takács(2018)forlinearoptimization.Thesearchdirectiondiscussedin thispapercanbederivedfrompositive-asymptotickernelfunctionusingthefunctionϕ(^t)=t² inthe newtypeofAET.WeprovethattheIPAhasO

(¹+4κ)^√nlog3n4^μ0

iterationcomplexity,whereκ ^isan upperboundofthehandicapoftheinputmatrix.Tothebestofourknowledge,thisisthefirstPCIPA forP_∗(κ)^{-LCPswhichisbasedonthissearchdirection.}

© 2021TheAuthor(s).PublishedbyElsevierB.V.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

The linear complementarity problem (LCP) is a well-known problemwhichincludeslinearprogramming(LP)andlinearlycon- strained(convex) quadraticprogrammingproblem(QP),asspecial cases. The mostimportantbasic resultsrelatedto LCPs are sum- marized inthebooksofCottle, Pang, &Stone(1992)andKojima, Megiddo, Noma,& Yoshise (1991). Many classical applications of LCPs can be found in different fields, such as optimization the- ory,gametheory,economics,engineering,etc.Cottleetal.(1992); Ferris & Pang (1997). Forexample,bimatrix games canbe trans- formed into LCPs under specific assumptions Lemke & Howson (1964). Kojima & Saigal (1979) used the degree theory in order tostudyLCPs.Furthermore,theArrow–Debreucompetitivemarket equilibriumproblemwithlinearandLeontiefutility functionscan be also givenasLCP (Ye,2008). More recentwork ofBrás, Eich- felder, & Júdice (2016) connected the copositivity testing of ma- trices and solvability of special LCPs. Darvay, Illés, Povh, & Rigó (2020b) publisheda PC IPAforsufficientLCPs using thefunction

∗Corresponding author at: Corvinus Center for Operations Research at Corvinus Institute for Advanced Studies, Corvinus University of Budapest, Hungary.

E-mail addresses: darvay@cs.ubbcluj.ro (Zs. Darvay), tibor.illes@uni-corvinus.hu (T. Illés), petra.rigo@uni-corvinus.hu (P.R. Rigó).

ϕ

¯(^t)=t−√

t forAET, but testednumerically their algorithm be- yond the class of sufficient matrices, too. Numerical results pro- ducedbythedevelopedPCIPAfortestingcopositivityofmatrices usingLCPswereverypromising.Sloan&Sloan(2020)showedthat solvabilityofLCPsrelatedtoquittinggamesensures theexistence ofdifferent

ε

-equilibriumsolutions.Thereisnoreportedcomputa- tionalstudyonthistypeofapplicationofLCPs,yet.

In the LCP we want to find vectors x,s∈Rⁿ, that satisfy the constraints

−Mx+s=q, xs=0, x,s≥0, (LCP) whereM∈R^n×n,q∈Rⁿ andxsdenotestheHadamardproductof vectors x ands. The following notations are used to denote the feasibleregion,theinteriorandthesolutionssetofLCP:

F:=

{ (

^x,s

)

∈Rⁿ×Rⁿ:−Mx+s=q

}

, F⁺:=

{ (

^x,s

)

∈Rⁿ₊×Rⁿ₊:−Mx+s=q

}

^{, and}

F^∗:=

{ (

^x,s

)

∈F:xs=0

}

^.

WedenotedbyRⁿthen-dimentionalnonnegativeorthantandby Rⁿ₊thepositiveorthant,respectively.WecallaproblemP_∗(

κ

)^-LCP iftheproblem’smatrixof(LCP)isP_∗(

κ

)^-matrix,i.e.

(

¹+4

κ )

i∈I+(x)

x_i

(

^Mx

)

i+

i∈I−(x)

x_i

(

^Mx

)

i≥0,

∀

^x∈Rⁿ, (1)

https://doi.org/10.1016/j.ejor.2021.08.039

0377-2217/© 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Please citethisarticleas:Z.Darvay,T.IllésandP.R.Rigó,Predictor-correctorinterior-pointalgorithmforP_∗(

κ

)^-linearcomplementarity problemsbasedonanewtypeofalgebraicequivalenttransformationtechnique,European JournalofOperationalResearch,https://doi.

where

I₊

(

^x

)

⁼

{

^1≤i≤n:xi

(

^Mx

)

i>0

}

^and

I₋

(

^x

)

=

{

¹≤i≤n:xi

(

^Mx

)

i<0

}

and

κ

≥0isanonnegativerealnumber.Wewillassumethrough- out thepaper that F⁺=∅, there isan initial point (^x0,s⁰)∈F⁺ and M is a P_∗(

κ

)^{-matrix. The class of P}∗ matrices is the set of all P_∗(

κ

)^{-matrices, where}

κ

≥0. Väliaho (1996) showed that the classofP_∗-matricesisequivalenttotheclassofsufficientmatrices givenbyCottle,Pang,&Venkateswaran(1989).ThehandicapofM (Väliaho, 1996) is the smallestvalue of

κ

ˆ(^M)≥0 such that M is P_∗(

κ

^ˆ(^M))^{-matrix.Väliaho(1996)alsoprovedthatamatrixMisP}∗

ifandonlyifthehandicap

κ

ˆ(^M)^{ofMisfinite.}

There are several methods for solving LCPs with different matrices, such as simplex (Csizmadia, Csizmadia, & Illés, 2018;

van de Panne & Whinston, 1964; 1969; Wolfe, 1959), criss-cross (Csizmadia & Illés, 2006; Csizmadia, Illés, & Nagy, 2013; den Hertog, Roos, & Terlaky, 1993; Fukuda, Namiki, & Tamura, 1998;

Fukuda&Terlaky,1997)orotherpivot(Lemke,1968;vandePanne, 1974)algorithms.However,theIPAsforsolvingLCPsreceivedmore attention inlast decades (Kojimaetal., 1991). Itshould be men- tioned that LCPs belong to the class of NP-complete problems (Chung, 1989). In spite of thisfact, due to the results of Kojima et al.(1991),ifwe suppose that the problem’smatrixhas P_∗(

κ

)^- property,theIPAssolvingthesekindofLCPsusuallyhavepolyno- mial complexityinthehandicapoftheproblem’smatrix,thesize oftheproblemandthebitsizeofthedata.However,notethatthe worst-caseiterationcomplexityoftheIPAsforLCPdependsonthe upper bound of the handicap ofthe matrix M. de Klerk & Nagy (2011)showedthat thehandicapofa P_∗(

κ

)^{-matrixmaybe expo-} nentialinitsbitsize.Thismeansthatifthehandicapofthematrix isexponentiallylargeinthesizeandbitsizeoftheproblem,then the known complexityboundsof IPAsmay not be polynomial in theinputsizeoftheLCP.

Potra & Liu (2005) proposed an IPAfor sufficient LCPs which uses awideneighbourhood ofthecentral pathandthealgorithm doesnot depend onthe handicap oftheproblem. There are sev- eral knownIPAs not dependingon thehandicap of thesufficient matrix,suchastheIPAsgivenbyPotra&Sheng(1997),Potra&Liu (2005),Illés&Nagy(2007),Liu&Potra(2006)andLešaja&Potra (2019).TheIPAsforsolvingsufficientLCPshavebeenalsoextended togeneralLCPs(Illés,Nagy,&Terlaky,2010a;2010b).Illés,Nagy,&

Terlaky (2009, 2010a)generalizedlarge-update, affinescaling and PCIPAsforsolvingLCPswithgeneralmatrices.

The PC IPAs perform a predictor and one or more corrector stepsinamainiteration.Theaimofthepredictorstepistoreach optimality, hence afteran affine-scaling stepa certain amount of deviationfromthecentralpathisallowed.Thegoalofthecorrec- torstepistoreturnintheneighbourhoodofthecentralpath.The PC IPAsturnedout tobe efficientinpractice.The firstPCIPAfor LO was givenby Mehrotra (1992) and Sonnevend,Stoer, & Zhao (1991).Potra & Sheng(1996, 1997) definedPC IPAs forsufficient LCPs.Mizuno,Todd,&Ye(1993)gavethefirstPCIPAforLOwhich uses only one corrector step in a main iteration andthese IPAs werenamedMizuno–Todd–Ye(MTY)typePCIPAs.Miao(1995)ex- tended the MTYIPA giveninMizuno etal.(1993) toP_∗(

κ

)^-LCPs.

Followingthisresult,severalMTYtypePCIPAshavebeenproposed amongothersbyIllés&Nagy(2007),Kheirfam(2014)andDarvay etal.(2020b).InDarvay etal.(2020b) theauthorsgavea unified framework to determinethe Newton systemsandscaled systems incaseofPCIPAsusingtheAETtechnique.

Barrierfunctionsareoftenusedforthedeterminationofsearch directionsincaseofIPAs.Byconsidering self-regularkernelfunc- tions, Peng, Roos, & Terlaky (2002)reduced the theoretical com- plexity oflarge-update IPAs. Later on, Lešaja & Roos (2010) pro- vided a unifiedanalysis ofIPAs forP_∗(

κ

)^{-LCPs that are basedon}

eligiblekernelfunctions. Tunçel& Todd(1997) consideredforthe first time areparametrization ofthecentral path system. Karimi, Luo, & Tunçel (2017) used entropy-based search directions for LPworkingin a wide neighbourhood ofthe central path.Darvay (2003)proposed the AETtechniquefordefining search directions incaseofIPAsforLO.Hedividedbothsidesofthenonlinearequa- tionofthe centralpathsystemby thebarrierparameter

μ

^.After

thatheappliedacontinuouslydifferentiable,invertible,monotone increasingfunction

ϕ

¯^:(

ξ

²,∞)→R,where0≤

ξ

<1,onthemod- ifiednonlinearequationofthecentralpathproblem.Themajority ofthepublishedIPAsforsufficientLCPsdoesnotuseanytransfor- mationofthecentralpathequations,whichmeansthattheseIPAs usetheidentity map intheAET techniqueinordertodefine the searchdirections.Darvay (2003,2005) usedthesquarerootfunc- tionintheAETtechniqueforLO.Lateron,Darvay,Papp,&Takács (2016)introduced an IPAforLObasedon thedirectionusingthe function

ϕ

¯(^t)=t−√

t.In her Ph.D. thesis,Rigó (2020) presented severalIPAs that usethe function

ϕ

¯(^t)=t−√

t inthe AET tech- nique.Recently,Kheirfam&Haghighi(2016)haveproposedanIPA forP_∗(

κ

)^{-LCPswhich usesthefunction}

ϕ

^¯(^t)=₂₍₁^√₊^t√_t) inthe AET technique.Haddou,Migot,&Omer(2019)haverecentlyintroduced a family of smooth concave functions which leads to IPAs with thebestknowniterationbound.TheAETtechniquehasbeenalso extended toLCPs (Achache,2010;Asadi & Mansouri, 2012;2013;

Asadi, Mansouri, & Darvay, 2017; Asadi, Zangiabadi, & Mansouri, 2016;Kheirfam,2014;Mansouri&Pirhaji,2013).

Zhang & Xu (2011) used the equivalent form v²=v of the centering equation, where v=

xsμ,

μ

>0. They considered the xs=

μ

^vtransformation.Darvay&Takács(2018)introducedanew methodfordeterminingclassofsearchdirectionsusinganewtype of AET of the centering equations. They modified the nonlinear equation v²=vby applyingcomponentwiselyacontinuously dif- ferentiablefunction

ϕ

^:(

ξ

²,∞)→R, 0≤

ξ

<1 to the both sides of this equation. The properties of this function

ϕ

^{will be pre-}

sented in Section 2.2. The relationship between the functions

ϕ

and

ϕ

¯ ^{willbediscussedlaterasanoveltyofthispaper.InDarvay}

& Takács (2018)the authors considered thefunction

ϕ

(^t)=t² in order to determine the new search directions. Zhang, Huang, Li,

& Lv (2020) extended the feasible IPA given in Darvay & Takács (2018) to P_∗(

κ

)^-LCPs. Furthermore, Takács & Darvay (2018) gen- eralizedthe approachfordetermining search directionsproposed inDarvay& Takács(2018)toSOandtheyshowedthatthe corre- spondingkernelfunction isapositive-asymptotic kernelfunction.

Thepositive-asymptotickernelfunctionwasintroducedbyDarvay

&Takács (2018)anddiffers fromtheclassofkernelfunctionsin- troducedbyBai,ElGhami,&Roos(2004).

In this paper we introduce a new PC IPA for solving P_∗(

κ

)^- LCPs which uses the new type of AET givenin Darvay & Takács (2018) for LO. The proposed IPA applies the function

ϕ

(^t)=t² onthe modifiednonlinearequation v²=vin ordertoobtain the searchdirections.Inthissense,thecorresponding kernelfunction is a positive-asymptotic kernel function. Similar to Darvay et al.

(2020b) we present the method for determining the Newton systems and scaled systems in case of PC IPAs using this new type of AET. We also present the complexity analysis of the proposed PC IPA. Due to the used search direction we have to ensureduring thewhole process oftheIPA thatthe components of the vector v are greater than ^√₂², which makes the analysis more difficult.In spiteof thisfact, we show that the introduced IPA has O((¹+4

κ

)^√nlog3n₄^μ0) ^iteration complexity, where

κ

^is

theupperbound onthehandicapofmatrixM,

μ

^{0 isthestarting,}

averagecomplementaritygapand

ε

^isthefinaldisplacementfrom thecomplementarity gap,respectively.This isthefirst PCIPAfor solving P_∗(

κ

)^{-LCPs which usesthe function}

ϕ

(^t)=t² in thenew typeofAET.

Zs. Darvay, T. Illés and P.R. Rigó European Journal of Operational Research xxx (xxxx) xxx

The paper isorganized asfollows.In Section 2 we give some basicconcepts andusefulresultsabouttheP_∗(

κ

)^-LCPsandP∗(

κ

)^- matrices. Furthermore,inSection2.2,dependingontherepresen- tation of the nonlinear equation of the central path, a new way ofapplyingtheAETisdiscussedandcomparedtotheearlierused AET technique.The usual,butimportant,scaling techniqueisdis- cussedtogetherwiththeuniquesolvabilityoftheNewton-system, aswell.InSection 3wepresenta methodfordeterminingsearch directions in case of PC IPAs for P_∗(

κ

)^{-LCPs by using the new} type of AETapproach. InSection 4,the newPCIPAis presented.

While, Section 5 contains the complexity analysis of the intro- duced PC IPA with the new search directions. In Section 6 nu- merical computationsarepresentedandcomparedto thecompu- tational performance of an earlierintroduced PCIPA appeared in Darvay etal.(2020b)thatuseddifferentfunction

ϕ

^intheAET.In

Section7someconcludingremarksareenumerated.

2. Algebraicequivalenttransformationtechniqueofthecentral pathequations

Inthissectionwe summarizeimportantdefinitionsandresults related to P_∗(

κ

)^-LCPs. Furthermore, we introduce the AET ofthe central path equations. Following the steps of Darvay & Takács (2018),firstwederiveaknown,equivalentdescriptionofthecen- tralpathandthenweapplytheAETapproach,seeSection2.2.An importantnoveltyofthepaperisthatinthissectionwe compare thetwodifferentAETtechniquesintroducedinDarvay(2003)and Darvay &Takács(2018),respectively.Aninteresting observationis relatedtothefactthatthesamesearchdirectionscanbeobtained indifferentways.

2.1. CentralpathofsufficientLCPs

Thecentralpathproblemfor(LCP)is:

−Mx+s=q, x,s>0, xs=

μ

^{e, (2)}

where e denotes the n-dimensional vector of ones and

μ

>0. Kojima et al. (1991) showed that the sequence

{

(^x(

μ

),s(

μ

))

| μ

>0

}

^{of solutions lying on the central path}

parameterisedby

μ

>0approachesasolution(^x,s)^ofthe(LCP). Illés, Roos, andTerlaky gave an elementaryconstructive proof fortheexistenceanduniqueness ofthecentral pathforsufficient LCPsinanunpublishedmanuscriptin1997.Theconstructiveproof of Illés etal.appears in Theorem3.6 inthe Ph.D.thesis of Nagy (2009).

SimilarlytoDarvay&Takács(2018),weusex,s>0and

μ

>0, henceweobtain:

xs=

μ

^e⇔xs

μ

^=e⇔

xs

μ

^=e⇔x

μ

^s=

xs

μ

^.

Nowthecentralpathproblemfor(LCP)canbeequivalentlystated as

−Mx+s=q, x,s>0, xs

μ

⁼

xs

μ

^{. (3)}

Different forms of the central path problem (2) and (3) will be usedlaterintheAETcontext.

An importantresultwasprovedinLemma4.1ofKojimaetal.

(1991),whichplaysimportantroleinthesolvabilityoftheNewton system. AnimportantcorollaryofLemma4.1presentedinKojima etal.(1991)isthefollowing.

Corollary 2.1. LetM∈R^n×n be a P_∗(

κ

)^-matrix,x,s∈Rⁿ₊.Then, for alla_ϕ∈Rⁿthesystem

−M

^x+

^s=0

S

^x+X

^s=a_ϕ (4)

hasauniquesolution(^x,^s)^{,whereX andS arethediagonalma-} tricesobtainedfromthevectorsxands.

2.2. RelationshipbetweenthetwodifferenttypesofAETapproaches

Thegoal oftheAET techniqueintroducedby Darvay (2003)is to represent the central path in a different way and to derive Newton-systemfromtheserepresentationsdependingonthecon- tinuously differentiable, invertible, monotone increasing function

ϕ

¯^:(

ξ

²,∞)→R,where0≤

ξ

<1.

Now,we canapplytheAETtothecentralpathprobleminthe form(2)or(3).IncaseofapplyingtheAETmethodto(2),weob- tainthefollowingformofthecentralpath

−Mx+s=q, x,s>0,

ϕ

¯

_xs

μ

=

ϕ

¯

(

^e

)

. (5)

However, iftheAET isapplied to (3), usingthecontinuously dif- ferentiable, invertible function

ϕ

^:(

ξ

²,∞)→R, where 0≤

ξ

<1, thenusingtheideapresentedinDarvay&Takács(2018),weget

−Mx+s=q, x,s>0,

ϕ

_xs

μ

=

ϕ

xs

μ

. (6)

The following interesting question arises: if we use different transformedformsofthe centralpath(say (5)and(6)), isitnec- essarytousesomeextracriteriononfunctions

ϕ

^{?Ananswerwill}

begivenattheendofthissubsection.

An interesting observationis the connectionbetween systems (5)and(6).Forthis,let

ϕ

¯^:(

ξ

²,∞)→R

ϕ

¯

(

^t

)

=

ϕ (

^t

)

−

ϕ (

^√t

)

. (7) Thisleadsto

ϕ

¯

_xs

μ

=

ϕ

_xs

μ

−

ϕ

xs

μ

. (8)

Hence,wehave

ϕ

¯

_xs

μ

=

ϕ

¯

(

^e

)

⇔

ϕ

_xs

μ

−

ϕ

xs

μ

=

ϕ (

^e

)

−

ϕ (

^√e

)

⇔

ϕ

_xs

μ

=

ϕ

xs

μ

.

Majority of the published IPAs using the AET, derives the Newton-system from(5),while only few,like the onesproposed byDarvay&Takács(2018),andZhangetal.(2020)appliestheAET to(6).Wefollowthesecondapproachtoderivethecorresponding Newton-system.

Foran (^x,s)∈F⁺ ouraimistofindsearch directions^{xand ssuchthat}

−M

(

^x+

^x

)

+

(

^s+

^s

)

=q,

ϕ

xs

μ

⁺

x

^s+s

^x+

^x

^s

μ

=

ϕ

xs

μ

⁺

x

^s+s

^x+

^x

^s

μ

,

Weneglect thequadraticterms andapplyTaylor’s theoremto the function

ϕ

¯(^t)=

ϕ

(^t)−

ϕ

(^√t)^{. Hence, after some} calculations weobtain(4)with

a_ϕ=

μ

⁻

ϕ

_xs

μ

+

ϕ

_xs

μ

ϕ

_xs

μ

−₂√¹xs μ

ϕ

_xs

μ

. (9)

Now, from the denominator of the obtained fractional expres- sion,itisclearthatweneed extraassumptiononthefunction

ϕ

^,

namely

2t

ϕ

(

^t2

)

⁻

ϕ

(

^t

)

^{>0, (10)}

forallt>

ξ

^,with0≤

ξ

<1.

Lemma 2.2. Consider

ϕ

¯^:(

ξ

²,∞)→R as given in (7). Then,

ϕ

¯^: (

ξ

²,∞)→Ris monotone increasingifand onlyifcondition(10) is satisfiedforthefunction

ϕ

^.

Proof. Using(7)wehave

ϕ

¯(^t)=

ϕ

(^t)−₂^1√_t

ϕ

(^√t)^.Hence,

ϕ

¯

(

^t

)

>0,

∀

^t>

ξ

^{2 ifandonlyif}

ϕ

(

^t

)

− 1

2√

t

ϕ

(

^√t

)

>0,

∀

^t>

ξ

². (11) Considering change of variable u:=√

t in the second part of (11)weobtaincondition(10).

Dependingontheusedfunctions

ϕ

^{wecanhavedifferentvec-}

tors a_ϕ.InDarvayetal.(2020b) andRigó (2020)theauthorspre- sentedthefunctions

ϕ

¯^{alreadyusedintheliteratureincaseofIPAs} inordertoderivecomplexityresultsfordifferentclassofoptimiza- tionproblems,includingLOandsufficientLCPs,aswell.

Now,ifafunction

ϕ

^{satisfyingcondition(10)isapplied to(6),}

thenusing(7)andLemma2.2weimmediatelyobtainanIPAwith

ϕ

¯^{appliedto(5).However,ifafunction}

ϕ

^{¯ satifying}

ϕ

^¯(^t)>0isap- pliedto(5)andwederiveanIPA,wedonothaveguaranteethata correpondingfunction

ϕ

^{exists,duetothefactthattheconnection}

between

ϕ

¯^and

ϕ

^{isgivenasafunctionalequationgiveninEq.(7).}

Thus, we do not havein this case immediatelyanother descrip- tion oftheIPA.Inother words,we shouldconsiderthefollowing question: can we find a corresponding function

ϕ

^:(

ξ

²,∞)→R for a given

ϕ

¯^:(

ξ

²,∞)→R, 0≤

ξ

<1? To answer this, we give counterexamples. Using thedefinition of thefunction

ϕ

¯ ^givenin (7),wehavelim_t→0

ϕ

¯(^t)=

ϕ

¯(¹)=0.However,thefunctions

ϕ

¯^are monotone increasing. Hence, all the functions

ϕ

¯ ^{that are defined} on the whole interval (⁰,∞)^{, i.e.}

ξ

=0, are counterexamples. It wouldbeinterestingtodefineaclassofmonotoneincreasingfunc- tions

ϕ

¯ ^{for which we can assign} corresponding functions

ϕ

^{. For}

this, weshouldsolvethefunctionalequation

ϕ

(^t)−

ϕ

(^√t)=

ϕ

¯(^t) foragivenfunction

ϕ

¯^:(

ξ

²,∞)→R.Thisleadstofurtherresearch topics.

3. SearchdirectionsincaseofthenewtypeofAETtechnique

Inthissectionwepresentamethodtodeterminesearchdirec- tions incaseofIPAsforP_∗(

κ

)^{-LCPs,byusingthenewtypeofAET} approachpresentedinSection2.2.

3.1. Scaling Letusconsider v=

xs

μ

^{, d=}

x

s, dx=d⁻¹_√

^x

μ

⁼

v

^x x , ds= d

^s

√

μ

⁼

v

^s

s . (12)

From(12)weobtain

^x=xdx

v and

^s=sds

v . (13)

Hence, if we substitute these in the second equation of system (4)weget

xsdx

v +xsds

v =

μ

^2v

ϕ (

^v

)

−

ϕ (

^v2

)

2v

ϕ

(

^v2

)

−

ϕ

(

^v

)

^{. (14)}

The transformedNewton system (4)witha_ϕ given in(9),ob- tainedfrom(6)by applyingthe AETandthen scaling it,leadsto thefollowingformofthescaledNewton-system:

−M¯dx+ds=0,

dx+ds=p_ϕ, (15)

whereM¯=DMD,D=diag(^d)^and p_ϕ= 2

ϕ (

^v

)

⁻

ϕ (

^v2

)

2v

ϕ

(

^v2

)

−

ϕ

(

^v

)

^{. (16)}

From Theorem 3.5 proposed in Kojima et al. (1991) and Corollary 2.1 it can be proved that system (15)has unique solu- tion.

It should be mentioned that if we use the function

ϕ

^:

₁

2,∞

→R,

ϕ

(^t)=t, which satisfies condition (10),then we have

p_ϕ=2v−2v²

2v−e . (17)

Interestinglyenoughthatexactlythesamevectorp_ϕcanbede- rivediftheAET isapplied to(5)withfunction

ϕ

¯(^t)=t−√

t.For details seepapersDarvay,Illés, Kheirfam,& Rigó (2020a);Darvay et al. (2016) for LO and Darvay, Illés, & Majoros (2021); Darvay etal.(2020b) forsufficientLCPs.Thiscan beprovedby using(7), because in this case we have

ϕ

¯(^t)=

ϕ

(^t)−

ϕ

(^√t)=t−√

t. Fur- thermore, if we apply the AET to system (6) using the function

ϕ

(^t)=t²,then weobtain thesamesystemasifwe apply

ϕ

¯(^t)=

ϕ

(^t)−

ϕ

(^√t)=t²−t to system(5).It shouldbe mentioned, that thisfunctionhasnotbeenusedinthe literatureintheAET tech- nique.Hence,thefunction

ϕ

(^t)=t²usedintheAETapproachand appliedto(6)leadstonovelsearchdirectionsdiscussedinthispa- per.

Inthefollowingsubsectionwegive ageneralmethodofdeter- miningthescaledpredictorandscaledcorrectorsystemsincaseof PCIPAsusingthisnewtypeofAET.

3.2. SearchdirectionsincaseofPCIPAs

Darvay et al.(2020b) gave a generalframework to determine thescaledsystemsincaseofPCIPAsforsufficientLCPs.Following thestepsoftheir method,wegivefirstlythescaledcorrectorsys- tem,whichcoincideswithsystem(15).Thissystemhastheunique solution: d^c_x=(^I+M¯)^−1pϕ, d^c_s=M¯(^I+M¯)^−1pϕ. Analogous to the formulagivenin(13)wecandefine^cx= ^{x d}_v^cx and^cs= ^{s d}_v^cs.The differencebetweenthismethodandtheone presentedinDarvay etal.(2020b)isthatwe havedifferentvalue ofthevectorp_ϕ due totheusedfunction

ϕ

(^t)=t² intheAETtechnique.Inthetrans- formed Newton system(4)we decompose a_ϕ givenin(9) inthe followingwayusingtheideapresentedinDarvayetal.(2020b): a_ϕ=f

(

^x,s,

μ )

+g

(

^x,s

)

, (18) where f:Rⁿ₊×Rⁿ₊×R→Rⁿ with f(^x,s,0)=0 and g:Rⁿ₊×Rⁿ₊→Rⁿ. We set

μ

=0 in (18), because we would like to make as greedy predictor step as possible. From Darvay etal.(2020b)weobtain

−M¯dx+ds=0, dx+ds= vg

(

^x,s

)

xs , (19)

where M¯ =DMD. The unique solution of system (19) is d_x^p=(^I+M¯)^−1vg(xs^x,s) and d_s^p=M¯(^I+M¯)^−1vg(xs^x,s). The differ- ence between this approach and the one given in Darvay et al.

(2020b) liesinthedifferentvalueofthe vectora_ϕ andofg(^x,s)^. Using (13) we can obtain the predictor search directions from ^px= ^{x d}v^px and^ps= ^{s d}v^sp. It should be mentioned that the de- composition(18)isnottrivialandwehavenoguaranteethatsuch decompositionexistsforallfunctions

ϕ

^{suitableforAET.}

Zs. Darvay, T. Illés and P.R. Rigó European Journal of Operational Research xxx (xxxx) xxx 4. NewPCIPAforP_∗(

κ

)^{-LCPsbasedonanewsearchdirection}

Inthissection weintroduce aPCIPAusingtheAET technique presentedinSection 2.2.We dealwiththe function

ϕ

^:

₁

2,∞

→ R,

ϕ

(^t)=t²,soweobtain

p_ϕ= v−v³

2v²−e. (20)

It should be mentioned that the condition 2t

ϕ

(^t2)−

ϕ

(^t)>

0,

∀

^t>

ξ

^{issatisfiedinthiscase,where}

ξ

= ^√2².Notethatwecan associate a corresponding kernel function to the search direction determined by thefunction

ϕ

^{inthe new type ofAET approach.}

In this way,we obtain a positive-asymptotic kernel function,see Darvay&Takács(2018);Rigó &Darvay(2018):

ψ

^:

√

2 2 ,∞

→R,

ψ (

^t

)

^=t2₄^−1−log

(

^2t2−1

)

8 .

Let us define the centrality measure

δ

^:Rⁿ₊×Rⁿ₊×R+→R∪

{

∞

}

^as

δ (

^x,s,

μ )

^:=

δ (

^v

)

^:=

^pϕ

2 =1 2

2^vv⁻²−^v3e

^{. (21)}

Beside this, we give the

τ

-neighbourhood of afixed point ofthe centralpathas

N2

( τ

^,

μ )

:=

{ (

^x,s

)

∈F⁺:

δ (

^x,s,

μ )

≤

τ}

^{, (22)}

where

δ

(^x,s,

μ

) ^{isgiven in(21),}

τ

^{is a thresholdparameter and}

μ

>0isfixed.

First,weneedtofindthedecompositionofa_ϕ asitisgivenin (18):

a_ϕ=

μ

^xs

2

(

^2xs−

μ

^e

)

⁻ xs

2 ,

hence f(^x,s,

μ

)=₂₍₂^μ_xs^x₋^s_μe), which satisfies the condition f(^x,s,0)=0 and g(^x,s)=−^{x s}₂. In this case, the transformed Newtonsystem(4)with(9)isthefollowing:

−M

^x+

^s=0, S

^x+X

^s=

μ

^xs

2

(

^2xs−

μ

^e

)

⁻ xs

2 . (23)

Note that some IPAs use firstly corrector steps andafter that predictorstep,seePotra(2008).Ouralgorithmalsoperformsfirstly a corrector step if the initial interior point is not well centered and after that a predictor one. The PC IPA starts with (^x0,s⁰)∈ N2(

τ

,

μ

) ^forwhich

δ

(^x0,s⁰,

μ

)≤

τ

^{.Inacorrectorstepwe obtain}

d^c_x andd^c_sby solving

−M¯d^c_x+d^c_s=0, d^c_x+d^c_s= v−v³

2v²−e, (24)

where we used the scaling notations considered in Section 3.1, M¯ =DMD and D=diag(^d)^{. From Theorem 3.5 given in Kojima} etal.(1991)andCorollary2.1itcanbeprovedthatsystem(24)has uniquesolution:

d^c_x=

(

^I+M¯

)

⁻¹₂^v_v⁻₂₋^v3_e^{, dc}s=M¯

(

^I+M¯

)

⁻¹₂^v_v⁻₂₋^v3_e^.

From

^cx=xd^c_x

v and

^cs= sd^c_s

v (25)

the^{cxandcssearchdirectionscanbeeasilyobtained.Let} x^c=x+

^cx, s^c=s+

^cs.

Considerthefollowingnotations:

v^c=

x^cs^c

μ

^{, dc=}

x^c

s^c, D⁺=diag

(

^dc

)

, M¯⁺=D⁺MD⁺.

Then,thescaledpredictorsystemis

−M¯⁺d_x^p+d_s^p=0, d_x^p+d_s^p=−v^c

2, (26)

whichhasthesolution

d^p_x=−

(

^I+M¯⁺

)

^−1v₂^c, d^p_s=−M¯⁺

(

^I+M¯⁺

)

^−1v₂^c. (27) Then,using

^px= x^c

v^cd_x^p and

^ps= s^c

v^cd_s^p, (28)

the search directions ^{px andps can be easily} calculated. The iterateafterapredictorstepis

x^p=x^c+

θ

^px, s^p=s^c+

θ

^ps,

μ

^p=

1−

θ

2

μ

^,

where

θ

∈(⁰,1)^{istheupdateparameter.}

5. AnalysisofthePCIPA

Inthefirstpartoftheanalysiswedealwiththecorrectorstep.

5.1. Thecorrectorstep

Inthecorrectorpartoftheproposed PCIPAweusetheclassi- calsmall-updatestepofIPAs.Therefore,theresultsofZhangetal.

(2020) can be applied to analyse the corrector steps ofthe pro- posedPC IPA.It should be mentioned thatthe defaultvalue

τ

=

1

16(¹+4κ) givenin Algorithm4.1issmallerthan theupperbounds

Algorithm4.1 PCIPAforsufficient LCPsbasedona newtypeof AET.

Let

>0betheaccuracyparameter,0<

θ

<1theupdateparam- eter (default value

θ

=₄₍₁₊₄¹_κ)^√_n) and

τ

^{the proximity parameter}

(default value

τ

=₁₆₍₁₊₄¹ _κ)). Furthermore, a known upper bound

κ

^{of the handicap}

κ

^ˆ(^M) ^{is given. Assume that for} (^x0,s⁰) ^the

x⁰

T

s⁰=n

μ

^0,

μ

⁰>0holdssuchthat

δ

(^x0,s⁰,

μ

⁰)≤

τ

^andx_μ^0s0⁰ >

1 2e. begin

k:=0; while

x^k

T

s^k>

^do

begin

(correctorstep)

compute(^cxk,^csk)^{fromsystem(24)using(25);}

let(^xc)^k:=x^k+^cxkand(^sc)^k:=s^k+^csk;

(predictorstep)

compute(^pxk,^psk)^{fromsystem(26)using(28);}

let(^xp)^k:=(^xc)^k+

θ

^pxkand(^sp)^k:=(^sc)^k+

θ

^psk;

(updateoftheparametersandtheiterates) x^k+1:=(^xp)^k, s^k+1:=(^sp)^k,

μ

^k+1:=

1−^θ₂

μ

^k; k:=k+1;

end end

ofcentralitymeasuresgiveninthefollowingtheoremandlemma, hencewecanusetheseresultsintheanalysisofthecorrectorstep.

Furthermore, a detaileddescription ofhow the defaultvalues of the parameters have been chosen is given in Section 5.4. In the nexttheoremthestrictfeasibilityofthefull-NewtonIPAisproved, wherev^c=

x^cs^c μ .