Duality for nonsmooth mathematical programming problems with equilibrium constraints

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1 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 DOI /s R E S E A R C H Open Access Dualty for nonsmooth mathematcal programmng problems wth equlbrum constrants Sy-Mng uu 1*, Shash Kant Mshra 2 and Yogendra Pandey 2 * Correspondence: esmguu@mal.cgu.edu.tw 1 College of Management, Chang ung Unversty and Research Dvson, Chang ung Memoral Hosptal, Taoyuan, Tawan Full lst of author nformaton s avalable at the end of the artcle Abstract In ths paper, we consder the mathematcal programs wth equlbrum constrants (MPECs) n Banach space. The objectve functon and functons n the constrant part are assumed to be lower semcontnuous. We study the Wolfe-type dual problem for the MPEC under the convexty assumpton. A Mond-Wer-type dual problem s also formulated and studed for the MPEC under convexty and generalzed convexty assumptons. Condtons for weak dualty theorems are gven to relate the MPEC and two dual programs n Banach space, respectvely. Also condtons for strong dualty theorems are establshed n an Asplund space. MSC: 90C30; 90C46 Keywords: mathematcal programmng problems wth equlbrum constrants; Wolfe-type dual; Mond-Wer dual; convexty; nonsmooth analyss 1 Introducton Luo et al. [1] presented a comprehensve study of MPEC. Flegel and Kanzow [2] obtaned short and elementary proof of the optmalty condtons for MPEC usng the standard Frtz-John condtons. Further, Flegel and Kanzow [3] ntroduced a new Abade-type constrant qualfcaton and a new Slater-type constrant qualfcaton for the MPEC and proved that new Slater-type CQ mples new Abade-type CQ. Ye [4] consdered MPEC and ntroduced varous statonary condtons and establshed that t s suffcent for beng globally or locally optmal under some generalzed convexty assumpton and obtaned new constrant qualfcatons. Outrataet al. [5] derved necessary optmalty condtons for those MPECs whch can be treated by the mplct programmng approach and proposed a soluton method based on the bundle technque of nonsmooth optmzaton. Flegel et al. [6] consdered optmzaton problems wth a dsjunctve structure of the feasble set and obtaned optmalty condtons for dsjunctve programs wth applcaton to MPEC usng ugnard-type constrant qualfcatons. Movahedan and Nobakhtan [7] ntroduced nonsmooth strong statonarty, M-statonarty and generalzed the Abade and ugnard-type constrant qualfcatons for nonsmooth MPEC. Movahedan and Nobakhtan [8] ntroduced a nonsmooth type of the M-statonary condton based on the Mchel-Penot subdfferental and establshed the Frtz-John-type, Kuhn-Tucker-type M-statonary necessary condtons for the 2016 uu et al. Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a lnk to the Creatve Commons lcense, and ndcate f changes were made.

2 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 2 of 15 nonsmoothmpec. Further,Movahedan andnobakhtan [9] establshed necessary optmalty condtons for Lpschtz MPEC on Asplund space and suffcent optmalty condtons for nonsmooth MPEC n Banach space. We refer to the recent results of Ardal et al. [10], Cheu and Lee [11], uo and Ln [12], uo et al. [13, 14] and Ye and Zhang [15], and the references theren for more detals related to the MPEC. FollowngLuo et al. [1]andMovahedan and Nobakhtan [9], we consder the followng mathematcal programmng problem wth equlbrum constrants (MPEC): (MPEC) mnf (z) subject to: g(z) 0, h(z)=0, (z) 0, H(z) 0, (z), H(z) =0, where X s a Banach space, f : X R s a lower sem-contnuous (lsc) functon, g : X R k, h : X R p, : X R l and H : X R l are functons wth lsc components. The use of equlbrum constrants n modelng process engneerng problems s a relatvely new and exctng feld of research; see Raghunathan and Begler [16]. Hydroeconomc rver basn models (HERBM) based on mathematcal programmng are conventonally formulated as explct aggregate optmzaton problems wth a sngle, aggregate objectve functon. Brtz et al. [17] proposed a new soluton format for hydroeconomc rver basn models, based on a multobjectve optmzaton problem wth equlbrum constrants, whch allowed, nter ala, to express spatal externaltes resultng from asymmetrc access to water use. Wolfe [18] formulateda dualprogramfora nonlnear programmngproblem. Motvated by a specfc problem, namely the mathematcal descrpton of the rotatng heavy chan, Toland [19, 20] ntroduced the noton of dualty and establshed the dualty theory for nonconvex optmzaton problems. Rockafellar [21, 22] studed fundamental dualty theory for convex programs usng a conjugate functon and establshed a generalzed verson of the Fenchelís dualty theorem. In the last four decades there has been an extensve nterest n the dualty theory of nonlnear programmng problems; see Mangasaran [23] andmshra and org [24]. To the best of our knowledge, the dual problem to a nonsmooth MPEC has not been gven n the lterature as yet. In ths paper, we ntroduce Wolfe-type and Mond-Wer-type dual programs to the nonsmooth MPEC. We have establshed weak and strong dualty theorems relatng the nonsmooth MPEC and the two dual programs. The paper s organzed as follows: n Secton 2, we gve some prelmnares, defntons, and results. In Secton 3, wederveweakand strong dualty theorems relatng to the nonsmooth MPEC and the two dual models under convexty and generalzed convexty assumptons. 2 Prelmnares In ths secton, we gve some notatons, basc defntons, and prelmnary results, whch wll be used later n the paper. The Clarke-Rockafellar subdfferental of f s defned by c f (x)= { x X : x, v f (x; v), v X },

3 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 3 of 15 where f (x; v)=sup nf ɛ>0 γ >0 δ>0 >0 sup y B(x;γ ) f (y) f (x)+δ t (0,) f (y + tw) f (y) nf w B(v;ɛ) t s the Clarke-Rockafellar drectonal dervatve. Defnton 2.1 (Rockafellar [25]) The lsc functon f : X R {+ } s drectonally Lpschtzan at x f for some y X, lm sup sup x f x t 0 y y f (x + ty ) f (x ) t <. The functon f : X R {+ } s sad to be radally nonconstant (rnc) f x, y X, z (x, y), wth f (z) f (x),.e., one cannot fnd any lne segment on whch f s constant. Defnton 2.2 (Avrel et al. [26]) The lsc functon f : X R {+ } s sad to be a quasconvex functon, f for any x, y X,onehas f (z) max { f (x), f (y) }, x, y X, z [x, y], where [x, y]={x + t(y x):t (0, 1)}. Defnton 2.3 (Clarke [27]) The lsc functon f : X R {+ } s sad to be a convex functon at x X,f,forallx X, f (x) f ( x)+ ξ, x x, ξ c f ( x). Defnton 2.4 (Aussel [28]) The lsc functon f : X R {+ } s sad to be pseudoconvex functon at x X,f,forallx X, ξ, x x 0, for some ξ c f ( x) f (x) f ( x), f (x)<f ( x) ξ, x x <0, ξ c f ( x). Theorem 2.5 (Aussel [28]) Let f : X R {+ } be lsc, quasconvex and rnc on a convex open set U X. Moreover, assume that f s fnte at x Uandf ( x;0)>. Then, for each x U, f (x) f ( x) ξ c f ( x): ξ, x x 0. ven a feasble vector z for the MPEC,we defne the followngndex sets: I g := I g ( z):= { =1,2,...,k : g ( z)=0 }, α := α( z)= { =1,2,...,l : ( z)=0,h ( z)>0 },

4 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 4 of 15 β := β( z)= { =1,2,...,l : ( z)=0,h ( z)=0 }, γ := γ ( z)= { =1,2,...,l : ( z)>0,h ( z)=0 }. The set β s known as a degenerate set. If β s empty, the vector z s sad to satsfy the strct complementarty condton. Movahedan and Nobakhtan [8] ntroduced a nonsmooth type of M-statonary va the Mchel-Penot subdfferental for fnte-dmensonal spaces. Further, Movahedan and Nobakhtan [9] extend the M-statonary noton to nonsmooth MPEC n terms of the Clarke-Rockafellar subdfferental n Banach spaces. The followng defnton of the M-statonary pont for the nonsmooth MPEC s taken from Defnton 3.1 n Movahedan and Nobakhtan [9]. Defnton 2.6 Afeasblepont z of MPEC s called the Mordukhovch statonary pont f there exsts =( g, h,, H ) R k+p+2l, such that the followng condtons hold: 0 c f ( z)+ I g g cg ( z)+ h ch ( z) [ c ( z)+ H c H ( z) ], g I g 0, γ =0, H α =0, ether >0, H >0or H =0, β. The followng defnton of the no nonzero abnormal multpler constrant qualfcaton for MPEC s taken from Defnton 3.3 n Movahedan and Nobakhtan [9]. Defnton 2.7 Let z be a feasble pont of MPEC. We say that the No Nonzero Abnormal Multpler Constrant Qualfcaton (NNAMCQ) s satsfed at z f there s no nonzero vector =( g, h,, H ) R k+p+2l,suchthat 0 I g g cg ( z)+ h ch ( z) [ c ( z)+ H c H ( z) ], g I g 0, γ =0, H α =0, ether >0, H >0or H =0, β. Defnton 2.8 (Mordukhovch [29]) A Banach space X s Asplund, or t has the Asplund property, f every convex contnuous functon φ : U R defned on an open convex subset U of X s a Frechet dfferental on a dense subset of U. In the followng theorem,movahedan and Nobakhtan [9] proved a necessary optmalty condton for Lpschtz MPEC on Asplund spaces. Theorem 2.9 Let z be a local optmal pont for the MPEC where X s an Asplund space and all of the functons are locally Lpschtz around z. Then z s an M-statonary pont provded that the NNAMCQ holds at z. Now, dvde the ndex sets as follows. Let J + := { : h >0 }, J := { : h <0 }, β + := { β : >0, H >0 },

5 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 5 of 15 β + := { β : =0, H >0 }, β := { β : =0, H <0 }, β H + := { β : H =0, >0 }, β H := { β : H =0, <0 }, α + := { α : >0 }, α := { α : <0 }, γ + := { γ : H >0 }, γ := { γ : H <0 }. 3 Dualty In ths secton, we formulate and study a Wolfe-type dual problem for the MPEC under the convexty assumpton. A Mond-Wer-type dual problem s also formulated and studed for the MPEC under convexty and generalzed convexty assumptons. The Wolfe-type dual problem s formulated as follows: WDMPEC( z) max f (u)+ g g (u)+ u, I g h h (u) [ (u)+ H H (u) ] subject to 0 c f (u)+ I g g cg (u)+ h ch (u) [ c (u)+ H c H (u) ], (3.1) g I g 0, γ =0, H α =0, ether >0, H >0or H =0, β, where =( g, h,, H ) R k+p+2l. Theorem 3.1 (Weak dualty) Let zbefeasbleformpec where X s a Banach space,(u, ) feasble for WDMPEC( z), and ndex sets I g, α, β, γ defned accordngly. Suppose that f, g ( I g ), h ( J + ), ( α β H ), H ( γ β ) are convex at u and radally nonconstant. Also, assume that h ( J ), ( α + β H + β+ ), H ( γ + β + β+ ) are drectonally Lpschtzan, convex at u, and radally nonconstant. If α γ β β H = φ, then, for any z feasble for the MPEC,we have f (z) f (u)+ I g g g (u)+ h h (u) [ (u)+ H H (u) ]. Proof Let z be any feasble pont for MPEC. Then we have g (z) 0, I g and h (z)=0, =1,2,...,p. Snce f s convex at u, f (z) f (u) ξ, z u, ξ c f (u). (3.2) Smlarly, we have g (z) g (u) ξ g, z u, h (z) h (u) ξ h, z u, ξ g c g (u), I g, (3.3) ξ h c h (u), J +, (3.4)

6 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 6 of 15 h (z)+h (u) ξ h, z u, (z)+ (u) ξ, z u, ξ h c h (u), J, (3.5) ξ c (u), α + β H + β+, (3.6) H (z)+h (u) ξ H, z u, ξ H c H (u), γ + β + β+. (3.7) If α γ β β H = φ, multplyng (3.3)-(3.7)byg 0( I g), h >0( J + ), h >0 ( J ), >0( α + β H + β+ ), H >0( γ + β + β+ ), respectvely, and addng (3.2)-(3.7), we get f (z) f (u)+ I g g g (z) I g g g (u)+ h h (z) h h (u) (z) + (u) H H (z)+ ξ + g ξ g + h ξ h I g H H (u) [ ξ + H ξ H ], z u. From (3.1), there exst ξ c f (u), ξ g c g (u), ξ h c h (u), ξ c (u), and ξ H c H (u), such that ξ + Ig g ξ g + h ξ h [ ξ + H ξ H ] =0. So, f (z) f (u)+ I g g g (z) I g g g (u)+ h h (z) h h (u) (z)+ (u) H H (z)+ H H (u) 0. Now, usng the feasblty of z for MPEC, that s, g (z) 0, h (z)=0, (z) 0, H (z) 0, we get f (z) f (u) I g g g (u) h h (u)+ (u)+ H H (u) 0. Hence, f (z) f (u)+ I g g g (u)+ h h (u) [ (u)+ ] H H (u). Ths completes the proof. The followng corollary s a drect consequence of Theorem 3.1.

7 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 7 of 15 Corollary 3.2 Let z befeasbleformpec where all constrant functons g, h,, H are affne and ndex sets I g, α, β, γ defned accordngly. Then, for any z feasble for the MPEC and (u, ) feasble for WDMPEC( z), we have f (z) f (u)+ I g g g (u)+ h h (u) [ (u)+ H H (u) ]. Analogously, we have the followng result for Asplund spaces. Theorem 3.3 (Weak dualty) Let z befeasbleformpec where X s an Asplund space, (u, ) feasble for WDMPEC( z) and ndex sets I g, α, β, γ defned accordngly. Suppose that f, g ( I g ), h ( J + ), ( α β H ), H ( γ β ) are convex at u and radally nonconstant. Also, assume that h ( J ), ( α + β H + β+ ), H ( γ + β + β+ ) are drectonally Lpschtzan, convex at u, and radally nonconstant. If α γ β β H = φ, then, for any z feasble for the MPEC, we have f (z) f (u)+ I g g g (u)+ h h (u) [ (u)+ H H (u) ]. Proof The proof follows the lnes of the proof of Theorem 3.1. Theorem 3.4 (Strong dualty) Assume z s a locally optmal soluton of MPEC where X s an Asplund space, such that NNAMCQ s satsfed at z and the ndex sets I g, α, β, γ are defned accordngly. Let f, g ( I g ), h ( J + ), h ( J ), ( α β H ), ( α + β H + β+ ), H ( γ β ), H ( γ + β + β+ ) satsfy the assumpton of the Theorem 3.3. Then there exsts, such that ( z, ) s an optmal soluton of WDMPEC( z) and the respectve objectve values are equal. Proof Snce z s a locally optmal soluton of MPEC and the NNAMCQ s satsfed at z, hence, by Theorem 2.9, =( g, h,, H ) R k+p+2l, such that the nonsmooth M- statonarty condtons for MPEC are satsfed, that s, there exst ξ c f ( z), ξ g c g ( z), ξ h c h ( z), ξ c ( z), and ξ H c H ( z), such that 0= ξ + I g g ξ g + h ξ h [ ξ + H ξ H ], g I g 0, γ =0, H α =0, ether >0, H >0or H =0, β. Therefore, ( z, )sfeasbleforwdmpec( z). By Theorem 3.3,wehave f ( z) f (u)+ I g g g (u)+ h h (u) [ (u)+ H H (u) ], (3.8) for any feasble soluton (u, ) forwdmpec( z). Also, from the feasblty condton of MPEC and WDMPEC( z), that s, for I g ( z), g ( z)=0,andh ( z)=0, ( z)=0, α β,

8 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 8 of 15 H ( z)=0, β γ,wehave f ( z)=f ( z)+ I g g g ( z)+ h h ( z) [ ( z)+ H H ( z) ]. (3.9) Usng (3.8)and(3.9), we have f ( z)+ I g g g ( z)+ h h ( z) [ ( z)+ H H ( z) ] f (u)+ I g g g (u)+ h h (u) [ (u)+ H H (u) ]. Hence, ( z, ) s an optmal soluton for WDMPEC( z) andtherespectveobjectvevalues are equal. Example 3.1 Consder the followng MPEC n R 2 : MPEC(1) mn z 1 + z2 2 subject to : z 1 + z 2 0, z 2 0, ( ) z 2 z1 + z 2 =0. Now, we formulate Wolfe-type dual problem WDMPEC( z) for MPEC(1): max u, u 1 + u 2 2 [ ( u 1 + u 2 ) + H ( u 2 ) ] subject to ( ) ( ) ( ) ( ) 0 ξ = η H 0, 0 2u where ξ, η [ 1, 1]. If β s non-empty, then ether >0, H >0, or H =0. If we take the pont z = (0, 0) from the feasble regon, then the ndex sets α(0, 0) and γ (0, 0) are empty sets, but β := β(0, 0) s non-empty. Also, from solvng a constrant equaton n the feasble regon of WDMPEC(0, 0), we get = ξ η and H = ξ η 2u 2,where η 0.Snceβ s non-empty, we consder a β +, β +, β+ H to decde the feasble regon of WDMPEC(0, 0). It s clear that the assumptons of Theorem 3.1 are satsfed, so Theorem 3.1holdsbetween MPEC(1) and WDMPEC(0, 0). It s clear that z = (0, 0) s the optmal soluton of MPEC(1) and NNAMCQ s satsfed at z. Hence, the assumptons of the Theorem 3.4 are satsfed. Then, by Theorem 3.4, there

9 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 9 of 15 exsts such that ( z, )s anoptmalsolutonof WDMPEC(0,0)andtherespectve values are equal. We now prove the dualty relaton between the mathematcal programmng problem wth equlbrum constrants (MPEC) and the followng Mond-Wer-type dual problem MWDMPEC( z) max f (u) u, subject to 0 c f (u)+ I g g cg (u)+ h ch (u) [ c (u)+ H c H (u) ], (3.10) g (u) 0 ( I g ), h (u)=0 ( =1,...,p), (u) 0 ( α β), H (u) 0 ( β γ ), g I g 0, γ =0, H α =0, ether >0, H >0or H =0, β, where =( g, h,, H ) R k+p+2l. Theorem 3.5 (Weak dualty) Let zbefeasbleformpec where X s a Banach space,(u, ) be feasble for MWDMPEC( z), and the ndex sets I g, α, β, γ are defned accordngly. Suppose that f, g ( I g ), h ( J + ), ( α β H ), H ( γ β ) are convex at u and radally nonconstant. Also, assume that h ( J ), ( α + β H + β+ ), H ( γ + β + β+ ) are drectonally Lpschtzan, convex at u, and radally nonconstant. If α γ β β H = φ, then, for any z feasble for the MPEC, we have f (z) f (u). Proof Snce f s convex at u, f (z) f (u) ξ, z u, ξ c f (u). (3.11) Smlarly, we have g (z) g (u) ξ g, z u, h (z) h (u) ξ h, z u, ξ g c g (u), I g, (3.12) ξ h c h (u), J +, (3.13) h (z)+h (u) ξ h, z u, (z)+ (u) ξ, z u, ξ h c h (u), J, (3.14) ξ c (u), α + β + H β+, (3.15) H (z)+h (u) ξ H, z u, ξ H c H (u), γ + β + β+. (3.16) If α γ β β H = φ, multplyng (3.12)-(3.16)byg 0( I g), h >0( J + ), h >0 ( J ), >0( α + β H + β+ ), H >0( γ + β + β+ ), respectvely, and addng

10 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 10 of 15 (3.11)-(3.16), we get f (z) f (u)+ I g g g (z) I g g g (u)+ h h (z) h h (u) (z) + (u) H H (z)+ ξ + g ξ g + h ξ h I g H H (u) [ ξ + H ξ H ], z u. From (3.10), there exst ξ c f (u), ξ g c H (u), such that c g (u), ξ h c h (u), ξ c (u), and ξ H ξ + Ig g ξ g + h ξ h [ ξ + H ξ H ] =0. So, f (z) f (u)+ I g g g (z) I g g g (u)+ h h (z) h h (u) (z)+ (u) H H (z)+ H H (u) 0. Now, usng the feasblty of z and u for MPEC and MWDMPEC( z),respectvely, we get f (z) f (u). Ths completes the proof. The followng corollary s a drect consequence of Theorem 3.5. Corollary 3.6 Let z befeasbleformpec where all constrant functons g, h,, H are affne and the ndex sets I g, α, β, γ defned accordngly. Then, for any z feasble for the MPEC and (u, ) feasble for MWDMPEC( z), we have f (z) f (u). Analogously, we have the followng result for Asplund spaces. Theorem 3.7 (Weak dualty) Let z befeasbleformpec where X s an Asplund space, (u, ) be feasble for MWDMPEC( z) and the ndex sets I g, α, β, γ are defned accordngly. Suppose that f, g ( I g ), h ( J + ), ( α β H ), H ( γ β ) are convex at u and radally nonconstant. Also, assume that h ( J ), ( α + β H + β+ ), H ( γ + β + β+ ) are drectonally Lpschtzan, convex at u, and radally nonconstant. If α

11 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 11 of 15 γ β β H = φ, then, for any z feasble for the MPEC, we have f (z) f (u). Proof The proof follows the lnes of the proof of Theorem 3.5. Theorem 3.8 (Strong dualty) Assume z s a locally optmal soluton of MPEC where X s an Asplund space, such that NNAMCQ s satsfed at z and the ndex sets I g, α, β, γ defned accordngly. Let f, g ( I g ), h ( J + ), h ( J ), ( α β H ), ( α + β + H β+ ), H ( γ β ), H ( γ + β + β+ ) satsfy the assumpton of the Theorem 3.7. Then there exsts, such that ( z, ) s an optmal soluton of MWDMPEC( z), and the respectve objectve values are equal. Proof z s a locally optmal soluton of MPEC and the NNAMCQ s satsfed at z,bytheorem 2.9, =( g, h,, H ) R k+p+2l, such that the nonsmooth M-statonarty condtons for MPEC are satsfed, that s, there exst ξ c f ( z), ξ g c g ( z), ξ h c h ( z), ξ c ( z) and ξ H c H ( z), such that 0= ξ + I g g ξ g + h ξ h [ ξ + H ξ H ], g I g 0, γ =0, H α =0, ether >0, H >0or H =0, β. Snce z s an optmal soluton for MPEC, we have g g ( z)=0, I g h h ( z)=0, ( z)=0, H H ( z)=0. Therefore, ( z, ) s feasblefor MWDMPEC( z). Also, by Theorem 3.7, for any feasble (u, ), we have f ( z) f (u). Thus, ( z, ) s an optmal soluton for MWDMPEC( z) andtherespectveobjectvevalues are equal. Ths completes the proof. Example 3.2 Consder the followng MPEC problem n R 2 : MPEC mn z 1 + z 2 subject to z 1 + z 2 0, z 2 z 1 0, ( z1 + z 2 )( z2 z 1 ) =0. The Mond-Wer-type dual problem MWDMPEC( z)forthempecs max u, u 1 + u 2

12 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 12 of 15 subject to ( ) ( ) ( ) ( ) 0 ξ 1 = ξ 2 H η, (3.17) where ξ 1, ξ 2, η [ 1, 1], ( u 1 + u 2 ) 0, (3.18) H( u 2 u 1 ) 0, (3.19) f β s non-empty, then ether >0, H >0, or H =0. From (3.17), ξ 2 + H η = ξ 1,and + H =1,weget H = ξ 2 ξ 1 ξ 2 η and = ξ 1 η ξ 2 η,where ξ 2 η. If z = (0, 0), then the ndex sets α(0, 0) and γ (0, 0) are empty sets, but β(0, 0) s non-empty. It s clear that the assumptons of Corollary 3.6 are satsfed. So,Corollary 3.6 holds between MPEC and MWDMPEC(0, 0). Also, we can see that the NNAMCQ s satsfed at z. Then by Theorem 3.8 there exsts =(, H )suchthat( z, ) s an optmal soluton of MWDMPEC(0, 0) and the optmal values are equal. Now, we establsh weak and strong dualty theorems for the MPEC and ts Mond-Wertype dual problem under generalzed convexty assumptons. Theorem 3.9 (Weak dualty) Let zbefeasbleformpec where X s a Banach space,(u, ) be feasble for MWDMPEC( z), and the ndex sets I g, α, β, γ are defned accordngly. Suppose that f s pseudoconvex at z, g ( I g ), h ( J + ), ( α β H ), H ( γ β ) are quasconvex at u and radally nonconstant. Also, assume that h ( J ), ( α + β H + β+ ), H ( γ + β + β+ ) are drectonally Lpschtzan, quasconvex at u, and radally nonconstant. If α γ β β H = φ, then, for any z feasble for the MPEC, we have f (z) f (u). Proof Supposethat,forsomefeasblepontz, suchthatf (z)<f (u), then, by pseudoconvexty of f at u,wehave ξ, z u <0, ξ c f (u). (3.20) From (3.10), there exst ξ g c g (u) ( I g ), ξ h α β)and ξ H c H (u)( β γ ), such that I g g ξ g h ξ h + α β ξ + β γ H c h (u) ( = 1,...,p), ξ c (u) ( ξ H c f (u). (3.21)

13 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 13 of 15 By (3.20), we get ( g ξ g h ξ h + I g α β ξ + β γ H ξ H ), z u <0. (3.22) For each I g, g (z) 0 g (u). Hence, by Theorem 4.4 n [9], we have ξ, z u 0, ξ c g (u), I g. (3.23) Smlarly, we have ξ, z u 0, ξ c h (u), J +. (3.24) Now, for any feasble pont u of MWDMPEC( z), and for each J,0= h (u)=h (z). On the other hand, (z) (u), α + β H +,and H (z) H (u), γ + β +.Snce all of these functons are drectonally Lpschtzan, by Theorem 2.5,weget ξ, z u 0, ξ c h (u), J, (3.25) ξ, z u 0, ξ c (u), α + β + H, (3.26) ξ, z u 0, ξ c H (u), γ + β +. (3.27) From equaton (3.23)-(3.27), t s clear that ξ g, z u 0 ( I g ), ξ h, z u 0 ( J +), ξ h, z u 0 ( J ), ξ, z u 0, α + β H +, ξ H, z u 0, γ + β +. Snce α γ β β H = φ,wehave α β ξ, z u 0, g ξ g, z u 0, I g β γ H ξ H, z u 0, h ξ h, z u 0. Therefore, ( g ξ g h ξ h + I g α β ξ + β γ H ξ H ), z u 0, whch contradcts (3.22). Hence, f (z) f (u). Ths completes the proof. Analogously, we have the followng result for Asplund spaces. Theorem 3.10 (Weak dualty) Let z befeasbleformpec where X s an Asplund space, (u, ) be feasble for MWDMPEC( z), and the ndex sets I g, α, β, γ are defned accordngly.

14 uu et al. Journal of Inequaltes and Applcatons (2016) 2016:28 Page 14 of 15 Suppose that f s pseudoconvex at z, g ( I g ), h ( J + ), ( α β H ), H ( γ β ) are quasconvex at u and radally nonconstant. Also, assume that h ( J ), ( α + β H + β+ ), H ( γ + β + β+ ) are drectonally Lpschtzan, quasconvex at u, and radally nonconstant. If α γ β β H = φ, then, for any z feasble for the MPEC, we have f (z) f (u). Proof The proof follows the lnes of the proof of Theorem 3.9. Theorem 3.11 (Strong dualty) Assume z s a locally optmal soluton of MPEC where X s an Asplund space, such that NNAMCQ s satsfed at z, and the ndex sets I g, α, β, γ are defned accordngly. Let f, g ( I g ), h ( J + ), h ( J ), ( α β H ), ( α + β H + β+ ), H ( γ β ), H ( γ + β + β+ ) satsfy the assumpton of Theorem Then there exsts, such that ( z, ) s an optmal soluton of MWDMPEC( z), and the respectve objectve values are equal. Proof The proof follows the lnes of the proof of Theorem 3.8, nvokng Theorem Results and dscusson We have studed mathematcal programs wth equlbrum constrants (MPECs). The objectve functon and functons n the constrant part are assumed to be lower semcontnuous. We studed the Wolfe-type dual problem for the MPEC under the convexty assumpton. A Mond-Wer-type dual problem was also formulated and studed for the MPEC under convexty and generalzed convexty assumptons. Condtons for weak dualty theorems were gven to relate the MPEC and two dual programs n Banach space, respectvely. Also condtons for strong dualty theorems were establshed n an Asplund space. We also dscussed the cases when all the constrant functons are affne. Two numercal examples were gven to llustrate the Wolfe-type dualty and the Mond-Wer-type dualty wth our MPECs, respectvely. Competng nterests The authors declare that they have no competng nterests. Authors contrbutons YP conceved of the study and drafted the manuscrpt ntally. S-M partcpated n ts desgn, coordnaton and fnalzed the manuscrpt. SKM outlned the scope and desgn of the study. All authors read and approved the fnal manuscrpt. Author detals 1 College of Management, Chang ung Unversty and Research Dvson, Chang ung Memoral Hosptal, Taoyuan, Tawan. 2 Department of Mathematcs, Banaras Hndu Unversty, Varanas, , Inda. Acknowledgements The authors would lke to express ther deep apprecaton to the helpful comments of revewers. The research of S-M uu was partally supported by NSC E MY3 of the Natonal Scence Councl, Tawan and BMRPD017 of Chang ung Memoral Hosptal, Tawan. The research of Yogendra Pandey was supported by the Councl of Scentfc and Industral Research, New Delh, Mnstry of Human Resource Development, overnment of Inda. rant 09/013(0388)2011-EMR Receved: 14 July 2015 Accepted: 14 January 2016

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