The New Butterfly Relaxation Method for Mathematical Programs with Complementarity Constraints

Transcription

1 The New Butterfly Relaxaton Method for Mathematcal Programs wth Complementarty Constrants Dussault, J.-P. Haddou, M. Mgot, T Abstract We propose a new famly of relaxaton schemes for mathematcal programs wth complementarty constrants that extend the relaxaton of Kadran, Dussault, Bechakroun from 2009 and the one of Kanzow and Schwartz from We dscuss the propertes of the sequence of relaxed non-lnear programs as well as statonary propertes of lmtng ponts. A sub-famly of our relaxaton schemes has the desred property of convergng to a M-statonary pont. We ntroduce new constrant qualfcatons, MPCC- CRSC and MPCC-GCRSC, to prove convergence of our method. In partcular, the latter s the weakest known constrant qualfcatons that ensure boundedness of the sequence generated by the method. A comprehensve numercal comparson between exstng relaxatons methods s performed on the lbrary of test problems MacMPEC and shows promsng results for our new method. Keywords: non-lnear programmng - MPCC - MPEC - relaxaton methods - statonary pont - constrant qualfcaton - CRSC AMS Subject Classfcaton: 90C30, 90C33, 49M37, 65K05 1 Introducton We consder the Mathematcal Program wth Complementarty Constrants mn f(x) s.t. g(x) 0, h(x) = 0, x Rn 0 G(x) H(x) 0, (MPCC) wth f : R n R, h : R n R m, g : R n R p and G, H : R n R q that are assumed contnuously dfferentable. The notaton 0 u v 0 for two vectors u and v n R q s a shortcut for u 0, v 0 and u v = 0 for all {1,..., q}. Ths problem has become an actve subject n the lterature n the last two decades. The wde varety of applcatons [1, 17, 6] to cte a few, that can be cast as a MPCC s one of the reasons for ths popularty. (MPCC) s clearly a non-lnear programmng problem and n general most of the functons nvolved n the formulaton are non-convex. In ths context solvng the problem means fndng a local mnmum. Even so, ths goal apparently modest s hard to acheve n general due to the degenerate nature of the MPCC. Therefore, numercal methods that consder only frst order nformaton may be expected to compute a statonary pont. The wde varety of approaches wth ths am computes the KKT condtons, whch requre that some constrant qualfcaton holds at the soluton to be an optmalty condton. However, t s well-known that Département d Informatque, faculté des Scences, Unversté de Sherbrooke, Canada. Ths research was partally supported by NSERC grant. IRMAR-Insa, Rennes, France IRMAR-Insa, Rennes, France, e-mal: tang.mgot@nsa-rennes.fr. Ths research was partally supported by a french grant from l Ecole des Docteurs de l UBL and le Consel Régonal de Bretagne. 1

2 these constrant qualfcatons never hold n general for (MPCC). For nstance, the classcal Mangasaran- Fromowtz constrant qualfcaton that s very often used to guarantee convergence of algorthms s volated at any feasble pont. Ths s partly due to the geometry of the complementarty constrant that always has an empty relatve nteror. These ssues have motvated the defnton of enhanced constrant qualfcatons and optmalty condtons for (MPCC) as n [21, 20, 35, 10] to cte some of the earlest research. In [11], Flegel and Kanzow provde an essental result that defnes the rght necessary optmalty condton to (MPCC). Ths optmalty condton s called M(Mordukhovch)-statonary condton. The name comes from the fact that those condtons are derved by usng Mordukhovch normal cone n the usual optmalty condtons of (MPCC). In vew of the constrant qualfcatons ssues that pledge the (MPCC), the relaxaton methods provde an ntutve answer. The complementarty constrant s relaxed usng a parameter so that the new feasble doman s not thn anymore. It s assumed here that the classcal constrants g(x) 0 and h(x) = 0 are not more dffcult to handle than the complementarty constrant. Fnally, as the relaxng parameter s reduced, convergence to the feasble set of (MPCC) s obtaned smlarly to a homotopy technque. These methods have been suggested n the lterature back n 2000 by Scheel and Scholtes n [35, 36] replacng the complementarty by G (x)h (x) t 0, {1,..., q}. (SS) Ths natural approach was later extended by Demguel, Fredlander, Nogales and Scholtes n [5] by also relaxng the postvty constrants G(x) t, H(x) t. Although, n [5], the motvaton of the authors was not to decrease the two parameters smultaneously. In [29], Ln and Fukushma mprove ths relaxaton by expressng the same set wth two constrants nstead of three. Ths mprovement leads to mproved constrant qualfcaton satsfed by the relaxed sub-problem. Even so, the feasble set s not modfed ths mproved regularty does not come as a surprse, snce constrant qualfcaton measures the way the feasble set s descrbed and not necessarly the geometry of the feasble set tself. In [39], the authors consder a relaxaton of the same type but only around the corner G(x) = H(x) = 0. In the correspondng papers t has been shown that under sutable condtons provdng convergence of the methods, converge to some spurous pont, called C-statonary pont, may stll happen. The convergence to M-statonary beng guaranteed only under some second-order condton. It s to be noted that dfferent methods used n the lterature such as nteror-pont methods, smoothng of an NCP functon and elastc net methods share a lot of common propertes wth the relaxaton from [36] and ts extenson. A new perspectve for those schemes has been motvated n [22] provdng an approxmaton scheme wth convergence to M-statonary pont by consderng (G (x) t)(h (x) t), {1,..., q}. (KDB) Ths s not a relaxaton snce the feasble doman of (MPCC) s not ncluded n the feasble set of the sub-problems. The method has been extended has a relaxaton method n [25] through an NCP functon φ: φ(g (x) t, H (x) t), {1,..., q}, (KS) The man am of ths paper s to contnue ths dscusson and extend the relaxaton of Kanzow and Schwartz by ntroducng the new butterfly relaxaton. The key assumpton necessary to guarantee convergence of the method reles very often on some MPCC constrant qualfcaton. In [26, 19, 24] the authors analyze the exstng methods and proves convergence under some mld constrant qualfcatons. The defnton of a new MPCC constrant qualfcaton allows to pursue ths dscusson and convergence of (KDB) and ts extenson has been shown under MPCC-CCP n [32]. Furthermore, the author proves that ths s the weakest MPCC constrant qualfcaton that assures convergence of these methods. In ths paper, we contnue the dscusson by provdng convergence result for the butterfly method. The MPCC-CCP condton s no longer suffcent for ths purpose and so we ntroduce a new MPCC constrant qualfcaton called MPCC-GCRSC. 2

3 In Secton 2, we ntroduce classcal defntons and results from non-lnear programmng and MPCC theory. Ths secton s completed by the defnton of new constrant qualfcatons for MPCC called MPCC- CRSC and MPCC-GCRSC n Defnton 2.7. In Secton 3, we defne the relaxaton scheme wth the new butterfly relaxaton. In Secton 4, we prove theoretcal results on convergence and exstence of the multpler of the relaxed sub-problems. We also provde an analyss on the convergence of approxmate statonary ponts. We prove that the butterfly method has smlar propertes as the best methods n the lterature. Fnally, n Secton 5, we provde an extensve numercal study by gvng detals on the mplementaton, comparson wth other methods as well as an example that llustrates the numercal dffcultes that mght occur. 2 Prelmnares (MPCC) s obvously a non-lnear programmng problem. Most of the numercal approaches used n nonlnear programmng compute necessary optmalty condtons that requre some constrant qualfcatons (CQs) defned n Sect. 2.1 to ensure exstence of Lagrange multplers at a local mnmum. Even so, (MPCC) belongs to ths class of problem t s requred to develop enhanced statonary condtons. Indeed, n a systematc way, feasble ponts of (MPCC) may fal to satsfy even the weakest constrant qualfcatons for non-lnear programmng. Talored optmalty condtons and constrant qualfcatons for (MPCC) are presented n Sect Non-Lnear Programmng Let a general non-lnear program be mn f(x) s.t. g(x) 0, h(x) = 0, x Rn (NLP) wth h : R n R m, g : R n R p and f : R n R. Denote F the feasble regon of (NLP), the set of actve ndces I g (x) := { {1,..., p} g (x) = 0}. Let the generalzed Lagrangan L r (x, λ) be L r (x, λ) := rf(x) + g(x) T λ g + h(x) T λ h, where λ = (λ g, λ h ) s the vector of Lagrange multpler. We call a KKT pont or a statonary pont a couple (x, λ) wth x F such that x L 1 (x, λ) = 0, λ g 0 and g(x) T λ g = 0. We remnd that the tangent cone of a set X at x X s a closed cone defned by T X (x ) := {d R n τ 0 and X x k x s.t. τ(x k x ) d}. Another useful tool for our study s the lnearzed cone of (NLP) at x F defned by L (x ) := {d R n g (x) T d 0 ( I g (x )), h (x) T d = 0 ( = 1,..., m)}. In the context of solvng non-lnear programs, that s fndng a local mnmum of (NLP), one wdely used technque s to compute necessary condtons. The prncpal tool s the Karush-Kuhn-Tucker (KKT) condtons. Let x be a local mnmum of (NLP) that satsfes a constrant qualfcaton, then there exsts y M 1 (x ) such that (x, y ) s a KKT pont of (NLP). Constrant qualfcatons are used to ensure the exstence of the ndex-1 multpler at x. We now defne some of the classcal constrant qualfcatons. Note that there exst a wde varety of such notons and we defne here those that are essental for our purpose. Defnton 2.1. Let x F. (a) Lnear Independence CQ (LICQ) holds at x f the famly of gradents { g (x ) ( I g (x )), h (x ) ( = 1,..., m)} s lnearly ndependent. (b) Constant Rank CQ (CRCQ) holds at x f there exsts δ > 0 such that for any subsets I 1 I g (x ) and I 2 {1,..., m} the famly of gradents { g (x) ( I 1 ), h (x) ( I 2 )} has a constant rank for all x B δ (x ). 3

4 (c) Mangasaran-Fromovtz CQ (MFCQ) holds at x f the famly of gradents { h (x )( = 1,..., m)} s lnearly ndependent and there exsts a d R n such that g (x ) T d < 0 ( I g (x )) and h (x ) T d = 0 ( = 1,..., m). (d) Constant Rank n the Subspace of Components (CRSC) holds at x f there exsts δ > 0 such that the famly of gradents { g (x) ( J ), h (x ) ( = 1,..., m)} has the same rank for every x B δ (x ), where J := { I g (x ) g (x ) L (x ) }. Remark 2.1. The defnton of MFCQ gven here s the most classcal. It can be shown usng some theorem of the alternatve that ths defnton s equvalent to the famly of actve gradents beng postvely lnearly ndependent. In the last defnton, C denotes the polar of a cone C, defned as C := {z R n z T d 0 d C}. Constant rank of the subspace component, CRSC, was ntroduced recently n [2]. Ths latter defnton consders an unusual set denoted J, that can be vewed as the set of ndces of the gradents of the actve constrants whose Lagrange multpler f they exst may be non-zero. A local mnmum s characterzed by the fact that there s no feasble descent drecton for the objectve functon of (NLP), that s f(x ) T F (x ), where T denotes the polar cone of T. On the other hand, the KKT condtons buld f usng a lnearzaton of the actve constrants. In a classcal way, we say that a pont x F satsfes Gugnard CQ f T F (x ) = L (x ) and Abade CQ f T F (x ) = L (x ). In practce, t s very dffcult to fnd a pont that conforms exactly to the KKT condton. Hence, an algorthm may stop when such condtons are satsfed approxmately. Ths has motvated the defnton of the CCP condton n [3]. Defnton 2.2. We say that a pont x F satsfes the Cone-Contnuty Property f the set-valued mappng R n x K(x) such that K(x) := { m λ g (x) + µ h (x) : λ R +, µ R} s outer semcontnuous. I g(x ) =1 It s to be noted here that K(x) depends on x, snce t consders only actve constrants at x. Clearly, K(x ) s a closed convex cone and concdes wth the polar lnearzed cone L (x ). Moreover, K(x) s always nner semcontnuous due to the contnuty of the gradents and the defnton of K(x). For ths reason, outer semcontnuty s suffcent for the contnuty of K(x) at x. Fnally, t has been shown n [3] that CCP s strctly stronger than ACQ and weaker than CRSC. In the context of numercal computatons, t s almost never possble to compute statonary ponts. Hence, t s of nterest to consder ɛ-statonary ponts. Defnton 2.3. Gven a general non-lnear program (NLP) and ɛ 0. We say that (x, y) R n R p+m s a ɛ-statonary pont (or a ɛ-kkt pont) f t satsfes L(x, λ) ɛ, h(x) ɛ, g (x) ɛ, λ 0, λ g (x) ɛ {1,..., p}. 2.2 Mathematcal Program wth Complementarty Constrants We now specalze the general notons above to our specfc case of (MPCC). Let Z be the set of feasble ponts of (MPCC). Gven x Z, we denote I +0 := { {1,..., q} G (x ) > 0 and H (x ) = 0}, I 0+ := { {1,..., q} G (x ) = 0 and H (x ) > 0}, I 00 := { {1,..., q} G (x ) = 0 and H (x ) = 0}. 4

5 In order to derve weaker optmalty condtons, we consder an enhanced Lagrangan functon. Let L r MP CC be the generalzed MPCC-Lagrangan functon of (MPCC) such that L r MP CC(x, λ) := rf(x) + g(x) T λ g + h(x) T λ h G(x) T λ G H(x) T λ H wth λ := (λ g, λ h, λ G, λ H ) R p R m R q R q. It s clear that we cannot expect to compute usual KKT ponts snce classcal constrant qualfcatons, n general, do not hold, so we ntroduce weaker statonary concepts as n [35, 20]. Defnton 2.4. A pont x Z s sad Weak (W)-statonary f there exsts λ = (λ g, λ h, λ G, λ H ) R p + R m R q R q such that x L 1 MP CC(x, λ) = 0, λ g = 0 / I g(x ), λ G I = 0, +0 λh I0+ = 0. Clarke (C)-statonary f x s weak-statonary and I 00, λ G λh 0. Alternatvely or Abade (A)-statonary f x s weak-statonary and I 00, λ G 0 or λ H 0. Mordukhovch (M)-statonary f x s weak-statonary and I 00, ether λ G > 0, λ H > 0 or λ G λh = 0. Strong (S)-statonary f x s weak-statonary and I 00, λ G 0, λ H 0. Relatons between these defntons are straghtforward from the defntons. Local optmal soluton s often denoted Boulgand (B)-statonary pont n the lterature, but ths wll not be used here. In a classcal way from the lterature, we extend the varous constrant qualfcatons for (NLP) to (MPCC). MPCC CQ denotes ths extenson of usual CQ. Abade CQ and Gugnard CQ are the weakest constrant qualfcatons n non-lnear programmng. Unfortunately, Abade condton s very unlkely to be satsfed wth (MPCC). Indeed, the tangent cone, T Z, s closed but n general not convex and the classcal lnearzed cone of (MPCC) s polyhedral for (MPCC) and therefore convex. That s why we defne a specfc cone for (MPCC) denoted L MP CC as n [35, 9, 31] L MP CC (x ) := {d R n g (x ) T d 0 I g (x ), h (x ) T d = 0 = 1,..., m, G (x ) T d = 0 I 0+, H (x ) T d = 0 I +0, G (x ) T d 0 I 00, H (x ) T d 0 I 00, ( G (x ) T d)( H (x ) T d) = 0 I 00 }. Ths cone s no longer polyhedral and s not necessarly convex. However due to [9], one always has the followng nclusons: T Z (x ) L MP CC (x ) L (x ). Defnton 2.5. Let x Z. We say that MPCC-ACQ holds at x f T Z (x ) = L MP CC (x ) and MPCC- GCQ holds at x f T Z (x ) = L MP CC (x ). The followng theorem s a keystone to defne necessary optmalty condtons for (MPCC). Theorem 2.1 ([11]). A local mnmum of (MPCC) that satsfes MPCC-GCQ or any stronger MPCC CQ s a M-statonary pont. 5

6 The polar of the MPCC-lnearzed cone s a key tool n the defnton of constrant qualfcatons. It s, however, not trval to compute. Therefore, we ntroduce the followng: P M (x ) := {d R n (λ g, λ h, λ G, λ H ) R p + R m R q R q wth λ G λ H = 0 or λ G > 0, λ H > 0 I 00, d = m λ g g (x ) + λ h h (x ) I g(x ) I 0+ I 00 λ G G (x ) =1 I +0 I 00 λ H H (x )}. Remark 2.2. When MPCC-GCQ holds at x, due to [7], one gets the followng ncluson: L MP CC (x ) P M (x ). We now ntroduce some constrant qualfcatons that wll be used n the sequel. One of the man constrant qualfcatons used n the lterature of (MPCC) s the MPCC-LICQ, see [37] for a dscusson on ths CQ. In a smlar way we may extend CRCQ as n [15]. A condton that s smlar was used n [25, 19] to prove convergence of relaxaton methods for (MPCC). As ponted out n Theorem 2.1, the correct sgn of the multpler λ G, λh for I 00 n the necessary optmalty condtons for (MPCC) are the sgn of M-statonary ponts. Ths motvates the defnton of MPCC-GMFCQ that specalzes the MPCC-MFCQ and the MPCC-LICQ by takng nto account those sgns of multplers for I 00. Defnton 2.6. Let x Z. 1. MPCC-LICQ holds at x f the gradents { g (x ) ( I g (x )), h (x ) ( = 1,..., m), G I 00 I 0+(x ), H I 00 I +0(x )} are lnearly ndependent. 2. MPCC-MFCQ holds at x f the only soluton of I g(x ) λ g g (x ) + m λ h h (x ) =1 wth λ g 0 ( I g(x )) s the trval soluton. 3. MPCC-GMFCQ holds at x f the only soluton of I g(x ) λ g g (x ) + m λ h h (x ) =1 I 0+ I 00 λ G G (x ) I 0+ I 00 λ G G (x ) I +0 I 00 λ H H (x ) = 0 I +0 I 00 λ H H (x ) = 0 wth λ g 0 ( I g (x )) and ether λ G λh = 0 ether λ G > 0, λ H > 0 for all I 00 s the trval soluton. Note here that MPCC-MFCQ and MPCC-GMFCQ have been defned usng the alternatve form of MFCQ mentoned n the Remark The New MPCC-GCRSC and MPCC-CRSC Constrant Qualfcatons In a smlar way as for MPCC-MFCQ and MPCC-GMFCQ, we extend the defnton of CRSC constrant qualfcaton to ntroduce the MPCC-CRSC and the MPCC-GCRSC, whch are new n the MPCC lterature. Defnton 2.7. Let x Z. 6

7 (a) MPCC-CRSC holds at x f there exsts δ > 0 such that the famly of gradents { g (x) ( I 1 ), h (x) ( = 1,..., m), G (x) ( I 0+ I 00 ), H (x) ( I 00 I +0 )} has the same rank for every x B δ (x ), where I 1 := { I g (x ) g (x ) L MP CC (x ) }. (b) MPCC-GCRSC holds at x f for any partton I I 00 0 I 00 0 = I 00 such that I g(x ) + I 00 0 λ g g (x ) + m λ h h (x ) =1 λ G G (x ) + I 00 0 I 0+ I λ H H (x ) = 0, λ G G (x ) I +0 I λ H H (x ) wth λ g 0 ( I g(x )),λ G and λ H 0 ( I++), 00 λ G > 0 ( I 0), 00 λ H ( I0 ) 00 > 0, there exsts δ > 0 such that the famly of gradents { g (x) ( I 1 ), h (x) ( = 1,..., m), G (x) ( I 3 ), H (x) ( I 4 )} has the same rank for every x B δ (x ), where I 1 := { I g (x ) g (x ) P M (x )}, I 3 := I 0+ { I++ G 00 (x ) P M (x )} I 0, 00 I 4 := I +0 { I++ H 00 (x ) P M (x )} I0. 00 In the specal case where there s no partton of I 00 that satsfes the condton of the defnton above, then obvously the gradents are lnearly ndependent and so MPCC-GMFCQ holds at x. Furthermore, MPCC-GCRSC s weaker than MPCC-CRCQ. Indeed, MPCC-CRCQ requres that every famly of lnearly dependent gradents remans lnearly dependent n some neghborhood, whle the MPCC- GCRSC condton consders only the famly of gradents that are lnearly dependent wth coeffcents that have M-statonary sgns. We now state that ths new noton of MPCC-GCRSC s actually a MPCC CQ by provng that t mples MPCC-CCP. Defnton 2.8. We say that a feasble pont x satsfes the MPCC-CCP f the set-valued mappng R n x K MP CC (x) such that K MP CC (x) := { I g(x ) s outer semcontnuous at x, that s λ g g (x) + m λ h h (x) λ G G (x) λ H H (x) : =1 I 0+ I 00 I +0 I 00 λ g R + and, ether λ G λ H = 0 ether λ G > 0, λ H > 0 for I 00 } lm sup x x K MP CC (x) K MP CC (x ). In ths context, the outer lmt s taken n the sense of Kuratowsk-Panlevé correspondng to the Defnton 5.4 gven n [34]. Ths defnton s motvated by sequental optmalty condtons from [3] for non-lnear programmng and extended for (MPCC) n [32], where t has been proved to be a MPCC constrant qualfcaton. The followng results gve a characterzaton of some sequences that satsfy MPCC-CRSC and MPCC- GCRSC at ther lmt pont. Note that ths result s essental for the convergence proof of relaxaton methods 7

8 for (MPCC) that wll be studed n the next secton snce t proves boundedness of approxmate statonary sequences. Durng the process of an teratve algorthm, we are nterested n the study of accumulaton ponts of sequences computed by the relaxaton method. It s common to compute sequences that satsfy the followng assumptons. Assumpton 2.1. Let {x k } and 0 {λ k } R p + R m R q R q be such that x k x and () f(x k ) + p =1 λ g,k g (x k ) + m =1 λ h,k h (x k ) q =1 λ G,k G (x k ) q =1 λ H,k H (x k ) 0, () / I g (x ) lm k λ g,k λ k = 0, I +0 lm k λ G,k λ k = 0 and I 0+ lm k λ H,k λ k = 0, () the famly of gradents of non-vanshng multplers n () are lnearly ndependent. Ths condton may correspond to some knd of sequental optmalty condtons. Note that assumpton () s not restrctve. Accordng to Lemma D.1, we can buld a sequence of multplers that satsfes () and (), such that the gradents correspondng to non-vanshng multplers n equaton () are lnearly ndependent for all k N. Ths may change the multplers, but prevously postve ones wll stay at least non-negatve and vanshng multplers wll reman zero. The frst step n our analyss s to prove that the sequences of multplers satsfyng Assumpton 2.1 are bounded. Theorem 2.2. Gven two sequences {x k },{λ k } that satsfy Assumpton 2.1. Suppose that x k x Z, and MPCC-CRSC holds at x. Then, the sequence {λ k } s bounded. Proof. Let {w k } be a sequence defned such that w k := j I g(x ) λ g,k j g j (x k ) + m =1 λ h,k h (x k ) λ G,k j j I 0+ I 00 G j (x k ) j I +0 I 00 λ H,k j H j (x k ). (1) We prove by contradcton that the sequence {λ k } s bounded. If {λ k } were not bounded, there would exst a subsequence such that λ k λ k λ 0. Here we consder a subsequence K, where the famly of lnearly ndependent gradents of non-vanshng multplers s the same for all k K. Note that ths can be done wth no loss of generalty, snce there s a fnte number of such subsequences and altogether they form a partton of the sequence. Note that condtons () and () gve that lm k w k = lm k f(x k )/ λ k = 0. Thus, dvdng by λ k and passng to the lmt n (1) yelds I g(x ) λ g g (x ) + m λ h h (x ) =1 I 0+ I 00 λg G (x ) I +0 I 00 λh H (x ) = 0, wth λ g j = 0 for j / I g(x ), λ G j = 0 for j I+0 and λ H j = 0 for j I 0+ by (). It follows that the gradents wth non-zero multplers nvolved n the prevous equaton are lnearly dependent. MPCC-CRSC guarantees that these gradents reman lnearly dependent n a whole neghborhood. Ths, however, s a contradcton to the lnear ndependence of these gradents gven by Assumpton 2.1. Here, we used that for all k suffcently large supp( λ) supp(λ k ). Consequently, the sequence {λ k } s bounded. 8

9 The followng result s smlar to Theorem 2.2 and focus on the case where the lmt pont s a M-statonary pont. Theorem 2.3. Gven two sequences {x k },{λ k } that satsfy Assumpton 2.1. Suppose that x k x Z, and MPCC-GCRSC holds at x. Furthermore, assume that I 00 ether lm k λ G,k λ k Then, the sequence {λ k } s bounded. lm k λ H,k λ k = 0 or lm k λ G,k λ k > 0, lm k λ H,k λ k > 0. (2) Proof. The proof s completely smlar to Theorem 2.2. Assumng that {λ k } s not bounded, we can extract a subsequence such that λ k λ k λ 0. Dvdng by λ k and passng to the lmt n the equaton (1) yelds I g(x ) λ g g (x ) + m λ h h (x ) =1 I 0+ I 00 λg G (x ) I +0 I 00 λh H (x ) = 0, wth λ g j = 0 for j / I g(x ), λ G j = 0 for j I+0, λ H j = 0 for j I 0+ and ether λ G j λ H j = 0 or λ G j > 0, λ H j > 0 for j I 00 by () and (2). It s clear that the famly of gradents consdered n the defnton of MPCC-GCRSC corresponds to the gradents wth non-zero multplers n the prevous equaton. Indeed, by lnear dependence of the gradents at x any gradent whose multpler s non-zero may be formulated as a lnear combnaton of the other gradents. Therefore, those gradents wth non-vanshng multplers belong to the polar of the M-lnearzed cone. MPCC-GCRSC guarantees that these gradents reman lnearly dependent n a whole neghborhood, whch contradcts () n Assumpton 2.1. Thus, the sequence {λ k } s bounded. We conclude ths secton by a consequence of Theorem 2.3 that states an essental result for ths secton, namely MPCC-GCRSC s a MPCC constrant qualfcaton. Corollary 2.1 (Corollary 2.2,[7]). MPCC-GCRSC mples MPCC-CCP. We sum up ths secton n Fgure 1 by gvng the relatonshp between the varous MPCC CQ defned here. Note that MPCC-CRSC does not necessarly mples MPCC-GCRSC due to Remark 2.2 (page 6). MPCC-LICQ = MPCC-MFCQ = = MPCC-CRSC MPCC-GMFCQ = MPCC-GCRSC = MPCC-CCP Fgure 1: Relatons between the MPCC constrant qualfcatons. 3 The Butterfly Relaxaton Methods The focus of ths paper s on relaxaton methods to solve (MPCC). The sketch of such a method behaves as follows: we consder a non-lnear parametrc program R tk, where the complementarty constrants have been relaxed usng a parameter t k > 0. A sequence {x k+1 } of statonary ponts of R tk s then computed for each value of t k > 0. Such statonary ponts are computed usng teratve methods that requre an ntal pont. 9

10 We use the prevous statonary pont as an ntal pont. For {t k } convergng to zero the sequence {x k+1 } should converge to a statonary pont of (MPCC). Accordng to Secton 2.2, our am s to compute a M-statonary pont of (MPCC). A motvaton to consder such methods s that the sequence of relaxed non-lnear program may satsfy some constrant qualfcaton and then are more tractable for classcal non-lnear methods. We consder a famly of contnuously dfferentable non-decreasng concave functons θ : R ], 1] such that ( ) x θ(0) = 0, θ t1 (x) := θ t 1 > 0 and lm θ t 1 (x) = 1 x R ++, t1 0 t 1 completed n a smooth way for negatve values by consderng θ t1 (z < 0) = zθ (0)/t 1. Examples of such functons are θt 1 1 (x) = x x+t 1 and θt 2 1 (x) = 1 exp x t 1. Those functons have already been used n the context of complementarty constrants n [16]. Usng ths famly of functons, we denote F 1 (x; t 1, t 2 ) := H (x) t 2 θ t1 (G (x)) and F 2 (x; t 1, t 2 ) := G (x) t 2 θ t1 (H (x)). We propose a new famly of relaxatons wth two postve parameters (t 1, t 2 ) defned such that for all {1,..., q} Φ B (G(x), H(x); t 1, t 2 ) = 0 = Φ B (G(x), H(x); t 1, t 2 ) = F 1 (x; t 1, t 2 )F 2 (x; t 1, t 2 ), and Φ B (G(x), H(x); t 1, t 2 ) s extended n a contnuously dfferentable as a functon wth negatve values for mn(f 1 (x; t 1, t 2 ), F 2 (x; t 1, t 2 )) < 0 and as a functon wth postve values otherwse. Ths new relaxaton uses two parameters t 1 and t 2 such that t 2 θ (0) t 1. (3) Ths condton ensures that the ntersecton pont between the sets {x R n F 1 (x; t 1, t 2 ) = 0} and {x R n F 2 (x; t 1, t 2 ) = 0} s reduced to the orgn. In other words, the two branches of the relaxaton does not cross each other. A typcal choce wll be to take t 2 = o(t 1 ) motvated by strong convergence propertes as dscussed n Secton 4.1. One way to wrte (Bu.) for t 2 < θ (0)t 1 uses the NCP functon from [25] by consderng { F1 (x; t Φ B 1, t 2 )F 2 (x; t 1, t 2 ), f F 1 (x; t 1, t 2 ) + F 2 (x; t 1, t 2 ) 0, (G(x), H(x); t 1, t 2 ) := ( F 1 (x; t 1, t 2 ) 2 + F 2 (x; t 1, t 2 ) 2) otherwse. 1 2 (Bu.) Ths formulaton wll be used n the numercal part n Secton 5 and n the study of convergence of approxmate ponts n Secton 4.3. Most of the results presented n ths artcle are only senstve to the descrpton of the constrant at ts boundary. Our motvaton s to consder regularzaton of the complementarty constrant, so we can also add a regularzaton of the postvty constrants parametrzed by t. Fgure 2 llustrates the feasble set of the relaxed complementarty constrant for t 2 = 2t 1 as well as the nfluence of the parameters on the relaxaton. Ths method s smlar to the methods (KDB) from [22] and (KS) from [25] n the sense that they can also be wrtten as a product of two functons. The man dfference s that handlng two parameters allows brngng the two wngs of the relaxaton closer. Ths observaton motvated to consder algorthmc propertes of a class of relaxaton methods n a recent workng paper [8]. A comparson of the feasble set of these methods can be seen n Fgure 3. We now ntroduce some notatons that wll be extensvely used n the sequel. Snce the butterfly relaxaton uses two parameters we denote t := (t 1, t 2 ) to smplfy the notaton and by extenson t k := (t 1,k, t 2,k ). 10

11 Fgure 2: Feasble set of the butterfly relaxaton for θ t1 (z) = parameters. z z+t 1 wth t 2 = 2t 1 and nfluence of the Fgure 3: The feasble set of the butterfly relaxaton, the approxmaton from [22] and the relaxaton from [25]. 11

12 Let Xt, t B be the feasble set of RB t, t, whch corresponds to the non-lnear program related to the butterfly relaxaton of the complementarty constrants defned n (Bu.), that s mn x R nf(x) s.t g(x) 0, h(x) = 0, G(x) te, H(x) te, Φ B (G(x), H(x); t) 0, (R B t, t ) where e denotes the vector of all ones, and X B t, t := {x Rn g(x) 0, h(x) = 0, G(x) te, H(x) te, Φ B (G(x), H(x); t) 0}. The sets of ndces used n the sequel are defned n the followng way I G (x; t) := { = 1,..., q G (x) + t = 0}, I H (x; t) := { = 1,..., q H (x) + t = 0}, I GH (x; t) := { = 1,..., q Φ B (G(x), H(x); t) = 0}, I 0+ GH (x; t) := { I GH(x; t) F 1 (x; t) = 0, F 2 (x; t) > 0}, I +0 GH (x; t) := { I GH(x; t) F 1 (x; t) > 0, F 2 (x; t) = 0}, I ++ GH (x; t) := { I GH(x; t) F 1 (x; t) > 0, F 2 (x; t) > 0}, I 00 GH(x; t) := { I GH (x; t) F 1 (x; t) = F 2 (x; t) = 0}. Several relatons between these sets follow drectly from the defnton of the relaxaton. For nstance, t holds that I G I GH = I H I GH =. The followng two lemmas gve more nsghts on the relaxaton. Lemma 3.1. Let x Xt, t B, then t s true for the relaxaton (Bu.) that: (a) { I GH (x; t) F 1 (x; t) = 0, F 2 (x; t) < 0} = { I GH (x; t) F 1 (x; t) < 0, F 2 (x; t) = 0} = ; (b) I GH (x; t) = G (x) 0, H (x) 0. Proof. Case (a) s drect consderng that Φ B (G(x), H(x); t 1, t 2 ) 0 for F 1 (x; t) + F 2 (x; t) < 0. By symmetry of the relaxaton t s suffcent to assume that F 1 (x; t) = H (x) t 2 θ t1 (G (x)) = 0 for some = 1,..., q. Then, by defnton of F 2 (x; t) t holds that F 2 (x; t) = G (x) t 2 θ t1 (H (x)) = G (x) t 2 θ t1 (t 2 θ t1 (G (x))), so G (x) 0 snce n the other case by defnton of the functon θ t would follow that F 2 (x; t) = G (x)(1 (θ (0)t 2 /t 1 ) 2 ), whch would be negatve f G (x) < 0. Fnally, G (x) 0 mples that H (x) 0 snce F 1 (x; t) = 0. The followng lemma sum up some of the key features of the relaxaton. Lemma 3.2. X B t, t satsfy the followng propertes: 1. X B 0,0 = Z; 2. X B t a, t a X B t b, t b for all 0 < ta,2 t a,1 < t b,2 t b,1 and 0 < t a < t b ; 3. t, t 0X B t, t = Z. 12

13 If the feasble set of the (MPCC) s non-empty, then the feasble set of the relaxed sub-problems are also non-empty for all t 0. If for some parameter t 0 the set Xt, t B s empty, then t mples that Z s empty. Fnally, f a local mnmum of Rt, t B already belongs to Z, then t s a local mnmum of the (MPCC). We focus n the sequel on the propertes of these new relaxaton schemes. We prove that the method converges to an A-statonary pont n Theorem 4.1 and to a M-statonary pont, Theorem 4.2, wth some relaton between the sequences {t 2,k } and {t 1,k }. The man motvaton to consder relaxaton methods for (MPCC) s to solve a sequence of regular problems. Under classcal assumptons, the butterfly relaxed non-lnear programs satsfy the Gugnard CQ, Theorem 4.3. Fnally, numercal results wll be presented n Sect. 5 and show that these new methods are very compettve compared to exstng methods. Before movng to our man results regardng convergence and regularty propertes of the butterfly relaxaton, we provde some useful results on the asymptotc behavor of functons θ t1 and Φ B (G(x), H(x); t). Drect computaton gves the gradent of Φ B (G(x), H(x); t) n the followng lemma. Lemma 3.3. For all {1,..., q}, the gradent of Φ B (G(x), H(x); t) w.r.t. x for the relaxaton (Bu.) s gven by ( F1 (x; t) t 2 θ t 1 (G (x))f 2 (x; t) ) G (x) x Φ B + ( F 2 (x; t) t 2 θ t (G(x), H(x); t) = 1 (H (x))f 1 (x; t) ) H (x) f F 1 (x; t) F 2 (x; t), ( t2 θ t 1 (G (x))f 1 (x; t) F 2 (x; t) ) G (x) + ( t 2 θ t 1 (H (x))f 2 (x; t) F 1 (x; t) ) H (x) f F 1 (x; t) < F 2 (x; t). The followng result llustrates the behavor of functons θ t1 and ther dervatves. The proof of ths result s gven n Appendx C. Lemma 3.4. Gven two sequences {t 1,k } and {t 2,k }, whch converge to 0 as k goes to nfnty and k N, (t 1,k, t 2,k ) R We have for any z R + lm k t 2,kθ t1,k (z) = 0. Furthermore, let {z k } be such that lm k z k = 0. Then, ether z k = O(t 1,k ) and so there exsts a constant C θ [0, θ (0)] such that lm t 2,kθ t 2,k t k 1,k (z k ) = C θ lm, k t 1,k otherwse,.e z k = ω(t 1,k ), then lm t 2,kθ t k 1,k (z k ) θ t 2,k (1) lm. k t 1,k We conclude ths secton by an example that shows that the butterfly relaxaton may mprove relaxatons from [22] and [25]. Indeed, t llustrates an example where there are no sequence of statonary ponts that converge to some undesrable pont. Example 3.1. mn x 1 s.t x 1 1, 0 x 1 x 2 0. x R 2 In ths example, there are two statonary ponts: an S-statonary pont (1, 0) that s the global mnmum and a M-statonary pont (0, 0), whch s not a local mnmum. Unlke the relaxatons (KDB) and (KS) where for t k = 1 k a sequence xk = (t k 2t k ) T, wth λ Φ,k = k, may converge to (0, 0) as k goes to nfnty, there s no sequences of statonary pont that converges to ths undesrable pont wth the butterfly relaxaton. 13

14 4 Theoretcal Propertes The study of theoretcal propertes of the butterfly relaxaton method s splt nto three parts: convergence of the sequence of statonary ponts, exstence of Lagrange multplers for the relaxed non-lnear program and convergence of the sequence of approxmate statonary ponts. 4.1 Convergence In ths secton, we focus on the convergence propertes of the butterfly relaxaton method and the constrant qualfcatons guaranteeng convergence of the sequence of statonary ponts generated by the method. Our am s to compute a M-statonary pont or at least to provde a certfcate f we converge to an undesrable pont. Relaxaton methods that converge to M-statonary ponts are ntroduced n [22] and [25]. C-statonary ponts are also frequently encountered n these relaxatons methods as n [36] and [29]. We prove n Theorem 4.1 that the butterfly relaxaton converges to an A-statonary pont and provde a certfcate ndependent of the multplers n the case t converges to undesrable ponts. Ths result s mproved to a convergence to M-statonary ponts for some choces on the parameters t 2 and t 1 n Theorem 4.2. Another man concern n the lterature s to fnd the weakest constrant qualfcaton, whch ensures convergence of the method. Ths has been extensvely studed n the thess [38] and related papers mentoned heren, where they prove convergence of most of the exstng relaxaton methods n the lterature under a hypothess close to MPCC-CRCQ. More recently n [32] the author proves convergence of the relaxaton from [22] and [25] under MPCC-CCP. Convergence of the butterfly relaxaton under MPCC-CRSC s proved n Proposton 4.1. An mproved result for some choces of the parameter t 2 and t 1 s gven n Proposton 4.2 that uses our new constrant qualfcaton denoted MPCC-GCRSC. Example 4.2 shows that our methods may not converge under MPCC- CCP snce t requres boundedness of some multplers. Theorem 4.1. Gven two sequences {t k } and { t k } of postve parameters satsfyng (3) and decreasng to zero as k goes to nfnty. Let {x k, λ g,k, λ h,k, λ G,k, λ H,k, λ Φ,k } be a sequence of ponts from R n R p R m R 3q that are statonary ponts of Rt B k, t k for all k N wth x k x. Assume that the sequence s bounded, where for all {1,..., q} η G,k η H,k := λ G,k := λ H,k Then, x s an A-statonary pont. + λ Φ,k + λ Φ,k {λ g,k, λ h,k, η G,k, η H,k } (4) ( ) t 2,k θ t 1,k (G (x k ))F 2 (x k ; t k ) F 1 (x k ; t k ), ( ) t 2,k θ t 1,k (H (x k ))F 1 (x k ; t k ) F 2 (x k ; t k ). The boundedness assumpton on the sequence (4) s a classcal assumpton and s guaranteed under some constrant qualfcaton as shown n the next Proposton 4.1. Proof. Frst, we dentfy the expressons of the multplers of the complementarty constrant n Defnton 2.4 through the statonary ponts of R B t k, t k. Let {x k, λ g,k, λ h,k, λ G,k, λ H,k, λ Φ,k } be a sequence of statonary ponts of R B t k, t k for all k N. The representaton of Φ B mmedately gves Φ B (G(xk ), H(x k ); t k ) = 0, I 00 GH (xk ; t k ) for all k N. Thus, we can wrte f(x k ) = p =1 λ g,k g (x k ) + m =1 λ h,k h (x k ) q =1 η G,k G (x k ) q =1 η H,k H (x k ), (5) 14

15 where η G,k = η H,k = λ G,k, f I G (x k ; t k ), λ Φ,k t 2,k θ t 1,k (G (x k ))F 2 (x k ; t k ), f I 0+ GH (xk ; t k ), λ Φ,k F 1 (x k ; t k ), f I +0 GH (xk ; t k ), 0, otherwse, λ H,k, f I H (x k ; t k ), λ Φ,k t 2,k θ t 1,k (H (x k ))F 1 (x k ; t k ), f I +0 GH (xk ; t k ), λ Φ,k F 2 (x k ; t k ), f I 0+ GH (xk ; t k ), 0, otherwse. Notce that { = 1,..., q F 1 (x k ; t k ) = 0} mples that I 0+ GH (xk ; t k ) IGH 00 (xk ; t k ) or symmetrcally { F 2 (x k ; t k ) = 0} mples that I +0 GH (xk ; t k ) IGH 00 (xk ; t k ) by concavty and t 2,k θ (0) t 1,k for all k N. We assume that the sequence {λ g,k, λ h,k, η G,k, η H,k } s bounded, then t converges, up to a subsequence, to some lmt denoted by {λ g,, λ h,, η G,, η H, }. These multplers are well-defned snce and for k suffcently large I G (x k ; t k ) I GH (x k ; t k ) ( {1,..., q} \ I 00 GH(x k ; t k ) ) =, I H (x k ; t k ) I GH (x k ; t k ) ( {1,..., q} \ I 00 GH(x k ; t k ) ) =, supp(λ G,k ) I G (x k ; t k ), supp(λ H,k ) I H (x k ; t k ), supp(λ G,k ) I GH (x k ; t k ), supp(η G,k ) I GH (x k ; t k ) ({1,..., q} \ I 00 GH(x k ; t k )), supp(η H,k ) I GH (x k ; t k ) ({1,..., q} \ I 00 GH(x k ; t k )). Moreover, for k suffcently large t holds supp(λ G, ) supp(λ G,k ), supp(λ H, ) supp(λ H,k ), supp(η G, ) supp(η G,k ), supp(η H, ) supp(η H,k ). The proof that shows convergence of the sequence and W-statonary of x wll be gven n Secton 4.3 by Lemma 4.1 on page 22 for ɛ k = 0. Let us now verfy that x s an A-statonary pont by computng the multplers for ndces I 00. We denote I 0,k G := { = 1,..., q ηg,k = 0} and I 0,k H := { = 1,..., q ηh,k = 0}, and I 0, G, I0, H the sets for ηg, and η H,. Consder the varous possbles cases: 1. If supp(λ G, ) supp(λ H, ), then for k suffcently large supp(λ G,k ) supp(λ H,k ). One has λ G,k 0, λ H,k 0 and G (x k ) = H (x k ) = t k. 2. If supp(λ G, ) supp(η H, ), then for k suffcently large supp(λ G,k ) supp(η H,k ). One has 0, G (x k ) = t k and necessarly I GH (x k ; t k ), whch s not possble. λ G,k 3. The case supp(η G, ) supp(λ H, ) s completely smlar. 4. If supp(λ G, ) I 0, H, then ηg, 0 and η H, = If I 0, G supp(λh, ), then η G, = 0 and η H, 0. 15

16 6. If I 0, G I0, H, then ηg, = η H, = If I 0, G supp(ηh, ), then I 0,k G and then I GH (x k ; t k ). η G,k supp(ηh,k ). Snce η G,k = 0 and η H,k free, one has λ Φ,k 0 = 0 F 1 (x k ; t k ) = t 2,k θ t 1,k (G (x k ))F 2 (x k ; t k ) or λ Φ,k = 0. Moreover t 2,k θ t 1,k (G (x k )) > 0, so ether λ Φ,k η G, = η H, = 0. = 0 or F 1 (x k ; t k ) = F 2 (x k ; t k ) = 0. It follows that 8. The case supp(η G, ) λ H 0 s completely smlar to the prevous case and leads to η G, = η H, = If supp(η G, ) supp(η H, ), then supp(η G,k ) supp(η H,k ) for k suffcently large and I GH (x k ; t k ). (a). IGH 00 (xk ; t k ) mples that F 1 (x k ; t k ) = F 2 (x k ; t k ), therefore G(x k ) = H(x k ) = 0 and η G, η H, = 0. (b). If I 0+ GH (xk ; t k ), then F 1 (x k ; t k ) = 0 0 < H (x k ) = t 2,k θ t1,k (G (x k )) < t 2,kθ (0) t 1,k G (x k ), = therefore F 2 (x k ; t k ) > 0. Assume λ Φ,k non-negatve constant C such that s not bounded, then gong through the lmt there s a and so η H, has and so η G, and so η G, = C. If λ Φ,k 0 and η H, = 0 and η H, < 0. lm k λφ,k F 2 (x k ; t k ) = C 0, s bounded, t corresponds to the case C = 0. Furthermore ether one lm k t 2,kθ t 1,k (G (x k )) 0 0. Ether one has lm k t 2,kθ t 1,k (G (x k )) = 0 (c). The case I +0 GH (xk ; t k ) s completely smlar to the prevous case. Indces that correspond to the frst eght cases and 9.a) are ndces that satsfy S-statonary condton. Furthermore, the ndces n cases 9.b) and 9.c), when the constant C = 0, also have the sgn of S-statonary ndces. M- and A-statonary ndces may appear only n the case 9.b) when C 0 and ether t 2,k θ t 1,k (G (x k )) = 0 or t 2,k θ t 1,k (G (x k )) > 0 for I +0 GH (xk ; t k ) and symmetrcally n case 9.c). The followng proposton proves the boundedness of the sequence of multplers under MPCC-CRSC by a drect applcaton of Theorem 2.2. Here, we focus on the sequence of multplers {λ g,k, λ h,k, η G,k, η H,k } defned n (4), where we assume that the gradents assocated wth the non-vanshng multplers n ths sequence are lnearly ndependent. Followng the dscusson before Theorem 2.2, ths can be done wthout loss of generalty. 16

17 Proposton 4.1. Gven two sequences {t k } and { t k } of postve parameters satsfyng (3) and decreasng to zero. Let {x k, λ g,k, λ h,k, λ G,k, λ H,k, λ Φ,k } be a sequence of ponts that are statonary ponts of Rt B k, t k for all k N wth x k x such that MPCC-CRSC holds at x. Furthermore, assume that the famly of gradents of non-vanshng multplers n (5) are lnearly ndependent for all k N. Then, the sequence {λ g,k, λ h,k, η G,k, η H,k }, defned n (4), s bounded. Proof. In order to apply Theorem 2.2, we prove that Assumpton 2.1 for {x k, λ g,k, λ h,k, η G,k, η H,k } s verfed here. Denote η k := λ g,k, λ h,k, η G,k, η H,k. Snce {x k, λ g,k, λ h,k, λ G,k, λ H,k, λ Φ,k } are statonary ponts of Rt B k, t k, the proof of Theorem 4.1 showed that the equaton (5) holds true. It follows from ths equaton that () s satsfed. Let us now verfy condton (). By defnton of {λ g,k } t holds that I g (x k ) I g (x ) and so / I g (x ) lm k λg,k follow symmetrcally. Notce that λ G,k G (x ) > 0. = 0. Let I +0, we verfy that lm k η G,k η k = 0. The case I 0+ lm k η G,k η k = 0 wll = 0, snce statonary condton mples that λ G,k (G (x k ) + t k ) = 0 and G (x k ) η G,k η k Assume by contradcton that lm k 0. By defnton of η G,k and snce λ G,k = 0, ths mples that λ Φ,k > 0. As a consequence I GH and n partcular I 0+ GH. Indeed, f I+0 GH, t would follow that F 2 (x k ; t k ) = G (x k ) t 2,k θ t1,k (H (x k )) = 0, whch would contradct G (x k ) G (x ) > 0. Besdes, t also holds that λ H,k = 0, snce supp(λ Φ,k ) supp(λ H,k ) =. Here, we used that by defnton of the relaxaton t always holds that I H I GH =. These smplfcatons yelds η G,k = λ Φ,k t 2,k θ t 1,k (G (x k ))F 2 (x k ; t k ) and η H,k = λ Φ,k F 2 (x k ; t k ). However, by hypothess on the sequences {t 1,k } and {t 2,k }, ths gves that 0 lm k η G,k η k lm k η G,k η H,k lm k t 2,kθ t 1,k (G (x k )) = 0, leadng to a contradcton, so I +0 η lm G,k k η k = 0. Fnally, the lnear ndependence assumpton and (5) gve condton (). In concluson, {x k, λ g,k, λ h,k, η G,k, η H,k } satsfes Assumpton 2.1, and the result follows by a straghtforward applcaton of Theorem 2.2. In [32], the author proves smlar convergence results for the relaxatons [22] and [25] usng the very weak constrant qualfcaton MPCC-CCP, obtaned by dervng the sequental optmalty condtons from [3] n non-lnear programmng to (MPCC). However, ths constrant qualfcaton does not ensure boundedness of the sequence of multplers (4), whch s necessary for the proof of our prevous theorem. The followng example shows that the result of Proposton 4.1 s sharp snce convergence cannot be ensured assumng only that MPCC-GCRSC or even MPCC-GMFCQ holds at the lmt pont. Example 4.1. Consder the followng two-dmensonal example mn x 2 s.t 0 x 1 + x 2 x R 2 2 x 1 0. MPCC-GMFCQ holds at (0, 0) T. However, MPCC-CRSC obvously fals to hold at ths pont. The pont (0, 0) T s even not a W-statonary pont. In ths case, there exsts a sequence of statonary ponts of the relaxaton such that {x k } converges to the orgn. Gven a sequence {x k }, wth {1} I GH (x k ; t k ), such that x k (0, 0) T then λ G,k = λ H,k = 0 and we can choose λ Φ,k that satsfes The sequence {x k } converges to an undesrable pont. η G,k = η H,k = 1 2x k. 2 17

18 The man reason for ths behavour s that MPCC-GMFCQ does not gve strong enough condtons n the neghbourhood of a pont that s not a M-statonary pont. The result of the Theorem 4.1 can be tghtened f we consder a partcular choce of parameter. It s an essental result, snce t shows that a subfamly of the butterfly relaxaton has the desred property to converge to a M-statonary pont. Theorem 4.2. Consder the same assumptons as n Theorem 4.1. If, n addton, we assume t 2,k = o(t 1,k ), then, x s a M-statonary pont. Proof. Theorem 4.1 proves that x s an A-statonary pont. Thus, t remans to verfy that there s no ndex I 00 such that η G, η H, < 0. In the proof of the Theorem 4.1 we consdered all the possble cases, and t follows that the case η G, η H, < 0 may only occur n the case (9).(b) (and by symmetry (9).(c)). In partcular, n (9).(b) the sequences {t 2,k }, {t 1,k } and {x k } satsfy lm k t 2,kθ t 1,k (G (x k )) > 0. However, by Lemma 3.4 ths s mpossble when t 2,k = o(t 1,k ). In Theorem 4.2, we assumed that the sequence (4) s bounded. The followng result gves an equvalent result to Proposton 4.1 n the case t 2,k = o(t 1,k ) wth a weaker constrant qualfcaton. The proof of ths result, smlar to the one of Proposton 4.1, follows by a straghtforward applcaton of Theorem 2.3. Proposton 4.2. Gven two sequences {t k } and { t k } of postve parameters satsfyng (3), t 2,k = o(t 1,k ), and decreasng to zero as k goes to nfnty. Let {x k, λ g,k, λ h,k, λ G,k, λ H,k, λ Φ,k } be a sequence of ponts that are statonary ponts of Rt B k, t k for all k N wth x k x such that MPCC-GCRSC holds at x. Furthermore, assume that the famly of gradents of non-vanshng multplers n (5) are lnearly ndependent for all k N. Then, the sequence (4) s bounded. To conclude, any sequence {x k } wth t 2,k = o(t 1,k ) that satsfes MPCC-GCRSC at ts lmt pont converges to a M-statonary pont. The followng example shows that ths result s sharp, snce t llustrates a stuaton where MPCC- GCRSC does not hold and the method converges to an undesrable W-statonary pont. Ths phenomenon only happens f the sequence of multplers (4) s unbounded. Example 4.2. Consder the problem mn x R x2 2 2 s.t 0 x 2 1 x 1 + x The feasble set s Z = {(x 1, x 2 ) T R 2 x 1 = 0} {(x 1, x 2 ) T R 2 x 1 = x 2 2}. (0, 0) T s the unque M-statonary, wth (λ G, λ H = 0). It s easy to verfy that MPCC-CCP holds at ths pont. However, MPCC-GCRSC fals to hold at any pont (0, a R) T snce the gradent of x 2 1 s non-zero for x 0. Consder a sequence such that for (t 1,k, t 2,k ) suffcently small F 2 (x k ; t k ) = 0 and x k 1 = t 2,k θ t 1,k (x k 1 + a 2 ), x k 2 = a, λ Φ,k F 1 (x k ; t k ) = 1 t 2,k θ t 1,k (x k 1 + a2 ). Obvously, the sequence x k goes to x = (0, a 0) T, whch s not a W-statonary pont. Indeed, we have η G,k = 1 t 2,k θ t 1,k (x k 1 + a2 ) and ηh,k =

19 4.2 Exstence of Lagrange Multplers for the Relaxed Sub-Problems In ths part, we study some regularty propertes of the relaxed non-lnear programs. Indeed, n order to guarantee the exstence of a sequence of statonary ponts, the relaxed non-lnear programs must satsfy some constrant qualfcatons n the neghborhood of the lmt pont. The butterfly relaxaton satsfes Gugnard CQ as stated n Theorem 4.3, whch s equvalent n terms of regularty to the relaxaton (KS). Theorem 4.3. Let x Z, satsfyng MPCC-LICQ. Then, there exsts t > 0 and a neghborhood U(x ) of x such that: t (0, t ] : x U(x ) X B t, t = standard GCQ holds at x for RB t, t. Proof. Let x U(x ) Xt, t B. We know that L Xt, t(x) B T X B (x). So, t s suffcent to show the converse t, t ncluson. The lnearzed cone of Rt, t B s gven by L X B (x) = {d R n g (x) T d 0, I g (x), h (x) T d = 0, = 1,..., m, t, t G (x) T d 0, I G (x; t), H (x) T d 0, I H (x; t), Φ B (G(x), H(x); t) T d 0, I 0+ GH (x; t) I+0 GH (x; t)}, usng that Φ B (G(x), H(x); t) = 0 for all I00 GH (x, t). Let us compute the polar of the tangent cone. Consder the followng set of non-lnear constrants parametrzed by z Xt, t B and I I00 GH (z; t), defned by S I (z) := {x R n g(x) 0, h(x) = 0, G(x) te, H(x) te, Φ B (G(x), H(x); t) 0, / I 00 GH(z; t), F 1 (x; t) 0, F 2 (x; t) 0, I, F 1 (x; t) 0, F 2 (x; t) 0, I c }, where I c I = I 00 GH (z; t) and I Ic =. Snce z X B t, t, t s obvous that z S I(z). By constructon of U(x ) and t, the gradents { g (x ) ( I g (x )), h (x ) ( = 1,..., m), G (x ) ( I 00 I 0+ ), H (x ) ( I +0 I 00 )} reman lnearly ndependent for all x U(x ) by contnuty of the gradents and we have I g (x) I g (x ), I G (x; t) I 00 I 0+, I H (x; t) I +0 I 00, IGH(x; 00 t) I +0 GH (x; t) I00 I 0+, IGH(x; 00 t) I 0+ GH (x; t) I+0 I 00. Therefore, by Lemma A.1, MFCQ holds for (6) at x. Furthermore, by Lemma D.2 and snce MFCQ n partcular mples Abade CQ t follows that By [4, Theorem 3.1.9], passng to the polar, we get By [4, Theorem 3.2.2], we know that L SI (x) (x) = {v R n v = T X B (x) = t, t I I 00 GH (x;t) T SI (x) (x) = I IGH 00 (x;t) L SI (x) (x). + I g(x) T X B (x) = t, t I I 00 GH (x;t) L SI (x) (x). λ g g (x) + I +0 GH (x;t) I0+ I m λ h h (x) =1 I G (x; t) λ G G (x) I H (x; t) λ H H (x) Φ λ Φ B (G(x), H(x); t) λ G G (x) + λ G G (x) GH (x;t) I I c λ H H (x) + I c λ H H (x) : λ g, λ G, λ H, λ Φ 0}. (6) 19

20 For v T X B (x), we have v L t, t SI (x) (x) for all I IGH 00 (x; t). If we fx such I, then there exsts some multplers λ h and λ g, λ G, λ H, λ Φ 0 so that v = + I g(x) λ g g (x) + I +0 GH (x;t) I0+ GH (x;t) I m λ h h (x) =1 λ Φ Φ B (G(x), H(x); t) λ G G (x) + I c λ G G (x) I I G (x; t) λ G G (x) I H (x; t) λ H H (x) + I c λ H H (x). λ H H (x) Now, t also holds that v L SI c (x) (x) and so there exsts some multplers λ h and λ g, λ G, λ H, λ Φ 0 such that v = + I g(x) λ g g (x) + I +0 GH (x;t) I0+ GH (x;t) + I m λ h h (x) =1 λ Φ Φ B (G(x), H(x); t) λ G G (x) I c λ G G (x) + I I G (x; t) λ G G (x) I H (x; t) λ H H (x) I c λ H H (x). λ H H (x) By the constructon of t and U(x ), the gradents nvolved here are lnearly ndependent and so the multplers n both prevous equatons must be equal. Thus, the multplers λ G and λ H wth ndces n I I c vansh. Therefore, v L X B (x) and as v has been chosen arbtrarly then T t, t X B (x) L t, t X B (x), whch concludes t, t the proof. Ths result s sharp as shown by the followng example, snce Abade CQ does not hold except for the specal case where t 2 θ (0) = t 1. In ths case, a stronger result than Theorem 4.3 can be found n [7]. Example 4.3. Consder the problem mn f(x) s.t. 0 x 1 x 2 0. x R 2 At x = (0, 0) T t holds that Φ B (G(x), H(x); t) = (0, 0) T and so L X B (x ) = R 2, whch s obvously t, t dfferent from the tangent cone at x for t 2 θ (0) < t 1 and t > 0. However, when t 2 θ (0) = t 1 the tangent cone s the whole space and thus Abade CQ holds at x n ths case. The followng example shows that we cannot have a smlar result usng MPCC-GMFCQ. Example 4.4. Consder the set 0 x 1 + x 2 2 x 1 0. MPCC-GMFCQ holds at x = (0, 0) T, snce the gradents are lnearly dependent but only wth coeffcents λ G = λ H that does not satsfy the condton gven n Defnton 2.6. Now, we can choose a sequence of ponts such that x k x and F 2 (x k ; t k ) = 0, t 2,k θ t 1,k (H(x k )) 1. Snce G(x ) = H(x ) t holds that F 2 (x ; 0) = (0 0) T and so MFCQ does not hold for (6). 20