Duality for nonsmooth mathematical programming problems with equilibrium constraints

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R E S E A R C H

Open Access

Duality for nonsmooth mathematical

programming problems with equilibrium

constraints

Sy-Ming Guu

1*

_{, Shashi Kant Mishra}

2

_{and Yogendra Pandey}

2

*_{Correspondence:}

[email protected]

1_{College of Management, Chang}

Gung University and Research Division, Chang Gung Memorial Hospital, Taoyuan, Taiwan Full list of author information is available at the end of the article

Abstract

In this paper, we consider the mathematical programs with equilibrium constraints (MPECs) in Banach space. The objective function and functions in the constraint part are assumed to be lower semicontinuous. We study the Wolfe-type dual problem for the MPEC under the convexity assumption. A Mond-Weir-type dual problem is also formulated and studied for the MPEC under convexity and generalized convexity assumptions. Conditions for weak duality theorems are given to relate the MPEC and two dual programs in Banach space, respectively. Also conditions for strong duality theorems are established in an Asplund space.

MSC: 90C30; 90C46

Keywords: mathematical programming problems with equilibrium constraints; Wolfe-type dual; Mond-Weir dual; convexity; nonsmooth analysis

1 Introduction

Luoet al. [] presented a comprehensive study of MPEC. Flegel and Kanzow [] ob-tained short and elementary proof of the optimality conditions for MPEC using the stan-dard Fritz-John conditions. Further, Flegel and Kanzow [] introduced a new Abadie-type constraint qualification and a new Slater-type constraint qualification for the MPEC and proved that new Slater-type CQ implies new Abadie-type CQ. Ye [] considered MPEC and introduced various stationary conditions and established that it is sufficient for being globally or locally optimal under some generalized convexity assumption and obtained new constraint qualifications.

Outrataet al.[] derived necessary optimality conditions for those MPECs which can be treated by the implicit programming approach and proposed a solution method based on the bundle technique of nonsmooth optimization. Flegelet al.[] considered optimization problems with a disjunctive structure of the feasible set and obtained optimality condi-tions for disjunctive programs with application to MPEC using Guignard-type constraint qualifications. Movahedian and Nobakhtian [] introduced nonsmooth strong stationar-ity, M-stationarity and generalized the Abadie and Guignard-type constraint qualifica-tions for nonsmooth MPEC. Movahedian and Nobakhtian [] introduced a nonsmooth type of the M-stationary condition based on the Michel-Penot subdifferential and estab-lished the Fritz-John-type, Kuhn-Tucker-type M-stationary necessary conditions for the

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nonsmooth MPEC. Further, Movahedian and Nobakhtian [] established necessary opti-mality conditions for Lipschitz MPEC on Asplund space and suﬃcient optiopti-mality condi-tions for nonsmooth MPEC in Banach space. We refer to the recent results of Ardaliet al. [], Chieu and Lee [], Guo and Lin [], Guoet al.[, ] and Ye and Zhang [], and the references therein for more details related to the MPEC.

Following Luoet al.[] and Movahedian and Nobakhtian [], we consider the following mathematical programming problem with equilibrium constraints (MPEC):

(MPEC) minf(z)

subject to:g(z)≤, h(z) = ,

G(z)≥, H(z)≥, G(z),H(z)= ,

whereXis a Banach space,f :X→Ris a lower semi-continuous (lsc) function,g:X→Rk, h:X→Rp_,_G_:_X_→_Rl_and_H_:_X_→_Rl_{are functions with lsc components.}

The use of equilibrium constraints in modeling process engineering problems is a rel-atively new and exciting ﬁeld of research; see Raghunathan and Biegler []. Hydroeco-nomic river basin models (HERBM) based on mathematical programming are conven-tionally formulated as explicit aggregate optimization problems with a single, aggregate objective function. Britzet al.[] proposed a new solution format for hydroeconomic river basin models, based on a multiobjective optimization problem with equilibrium con-straints, which allowed,inter alia, to express spatial externalities resulting from asymmet-ric access to water use.

Wolfe [] formulated a dual program for a nonlinear programming problem. Motivated by a speciﬁc problem, namely the mathematical description of the rotating heavy chain, Toland [, ] introduced the notion of duality and established the duality theory for non-convex optimization problems. Rockafellar [, ] studied fundamental duality theory for convex programs using a conjugate function and established a generalized version of the Fenchelís duality theorem. In the last four decades there has been an extensive interest in the duality theory of nonlinear programming problems; see Mangasarian [] and Mishra and Giorgi [].

To the best of our knowledge, the dual problem to a nonsmooth MPEC has not been given in the literature as yet.

In this paper, we introduce Wolfe-type and Mond-Weir-type dual programs to the non-smooth MPEC. We have established weak and strong duality theorems relating the nons-mooth MPEC and the two dual programs. The paper is organized as follows: in Section , we give some preliminaries, deﬁnitions, and results. In Section , we derive weak and strong duality theorems relating to the nonsmooth MPEC and the two dual models under convexity and generalized convexity assumptions.

2 Preliminaries

In this section, we give some notations, basic deﬁnitions, and preliminary results, which will be used later in the paper.

The Clarke-Rockafellar subdiﬀerential off is deﬁned by

∂cf(x) =

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where

f↑(x;v) =sup

>

inf

γ> δ> λ>

sup y∈B(x;γ)

f(y)≤f(x)+δ

t∈(,λ)

inf w∈B(v;)

f(y+tw) –f(y) t

is the Clarke-Rockafellar directional derivative.

Deﬁnition .(Rockafellar []) The lsc functionf :X→R∪ {+∞}is directionally Lip-schitzian atx¯if for somey∈X,

lim sup

x→¯fx t↓

sup y→y

f(x+ty) –f(x) t <∞.

The function f :X→R∪ {+∞}is said to be radially nonconstant (rnc) if∀x,y∈X, ∃z∈(x,y), withf(z)=f(x),i.e., one cannot ﬁnd any line segment on whichf is constant.

Deﬁnition .(Avrielet al.[]) The lsc functionf :X→R∪ {+∞}is said to be a qua-siconvex function, if for anyx,y∈X, one has

f(z)≤maxf(x),f(y), ∀x,y∈X,z∈[x,y],

where [x,y] ={x+t(y–x) :t∈(, )}.

Deﬁnition .(Clarke []) The lsc functionf :X→R∪ {+∞}is said to be a convex function atx¯∈X, if, for allx∈X,

f(x)≥f(x) +¯ ξ,x–x¯, ∀ξ∈∂cf(x).¯

Deﬁnition .(Aussel []) The lsc functionf:X→R∪ {+∞}is said to be pseudocon-vex function atx¯∈X, if, for allx∈X,

ξ,x–x¯ ≥, for someξ∈∂cf(x)¯ ⇒ f(x)≥f(x),¯

f(x) <f(x)¯ ⇒ ξ,x–x¯< , ∀ξ∈∂cf(x).¯

Theorem .(Aussel []) Let f :X→R∪ {+∞}be lsc,quasiconvex and rnc on a convex open set U⊂X.Moreover,assume that f is ﬁnite atx¯∈U and f↑(x; ) > –¯ ∞.Then,for each x∈U,

f(x)≤f(x)¯ ⇒ ∀ξ∈∂cf(x) :¯ ξ,x–x¯ ≤.

Given a feasible vectorz¯for the MPEC, we deﬁne the following index sets:

Ig:=Ig(z) :=¯

i= , , . . . ,k:gi(¯z) = 

,

α:=α(z) =¯ i= , , . . . ,l:Gi(z) = ,¯ Hi(¯z) > 

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β:=β(¯z) =i= , , . . . ,l:Gi(z) = ,¯ Hi(z) = ¯

,

γ :=γ(z) =¯ i= , , . . . ,l:Gi(z) > ,¯ Hi(z) = ¯

.

The setβis known as a degenerate set. Ifβis empty, the vectorz¯is said to satisfy the strict complementarity condition. Movahedian and Nobakhtian [] introduced a nonsmooth type of M-stationary via the Michel-Penot subdifferential for finite-dimensional spaces. Further, Movahedian and Nobakhtian [] extend the M-stationary notion to nonsmooth MPEC in terms of the Clarke-Rockafellar subdifferential in Banach spaces. The following definition of the M-stationary point for the nonsmooth MPEC is taken from Definition . in Movahedian and Nobakhtian [].

Deﬁnition . A feasible pointz¯of MPEC is called the Mordukhovich stationary point if there existsλ= (λg,λh,λG,λH)∈Rk+p+l_{, such that the following conditions hold:}

∈∂cf(¯z) +

i∈Ig

λg_i∂cgi(z) +¯ p

i=

λh_i∂chi(z) –¯ l

i=

λG_i∂cGi(¯z) +λHi ∂cHi(z)¯

,

λg_I_g≥, λG_γ = , λH_α = , eitherλ_iG> ,λH_i >  orλG_i λH_i = ,∀i∈β.

The following definition of the no nonzero abnormal multiplier constraint qualification for MPEC is taken from Definition . in Movahedian and Nobakhtian [].

Definition . Letz¯be a feasible point of MPEC. We say that the No Nonzero Abnormal Multiplier Constraint Qualification (NNAMCQ) is satisfied atz¯ if there is no nonzero vectorλ= (λg_,_λh_,_λG_,_λH₎_∈_Rk+p+l_{, such that}

∈

i∈Ig

λg_i∂cgi(z) +¯ p

i=

λh_i∂chi(¯z) – l

i=

λG_i∂cGi(z) +¯ λHi ∂cHi(¯z)

,

λg_I_g≥, λG_γ = , λH_α = , eitherλ_iG> ,λH_i >  orλG_i λH_i = ,∀i∈β.

Definition .(Mordukhovich []) A Banach spaceXis Asplund, or it has the Asplund property, if every convex continuous functionφ:U→Rdefined on an open convex sub-setUofXis a Frechet differential on a dense subset ofU.

In the following theorem, Movahedian and Nobakhtian [] proved a necessary optimal-ity condition for Lipschitz MPEC on Asplund spaces.

Theorem . Letz be a local optimal point for the MPEC where X is an Asplund space¯ and all of the functions are locally Lipschitz aroundz.¯ Thenz is an M-stationary point¯ provided that theNNAMCQholds atz.¯

Now, divide the index sets as follows. Let

J+:=i:λh_i > , J–:=i:λh_i < ,

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β_G+:=i∈β:λG_i = ,λH_i > , β_G–:=i∈β:λG_i = ,λH_i < ,

β_H+:=i∈β:λH_i = ,λG_i > , β_H–:=i∈β:λH_i = ,λG_i < ,

α+:=i∈α:λG_i > , α–:=i∈α:λG_i < ,

γ+:=i∈γ :λH_i > , γ–:=i∈γ:λH_i < .

3 Duality

In this section, we formulate and study a Wolfe-type dual problem for the MPEC under the convexity assumption. A Mond-Weir-type dual problem is also formulated and studied for the MPEC under convexity and generalized convexity assumptions. The Wolfe-type dual problem is formulated as follows:

WDMPEC(z)¯ max u,λ f(u) +

i∈Ig

λg_igi(u) + p

i=

λh_ihi(u) – l

i=

λG_iGi(u) +λHi Hi(u)

subject to

∈∂cf(u) +

i∈Ig

λg_i∂cgi(u) + p

i=

λh_i∂chi(u) – l

i=

λG_i∂cGi(u) +λHi ∂cHi(u)

, (.)

λg_I

g≥, λ

G

γ = , λHα = , eitherλGi > ,λiH>  orλGi λHi = ,∀i∈β,

whereλ= (λg,λh,λG,λH)∈Rk+p+l.

Theorem .(Weak duality) Letz be feasible for¯ MPECwhere X is a Banach space, (u,λ) feasible forWDMPEC(¯z),and index sets Ig,α,β,γ deﬁned accordingly.Suppose that f,gi

(i∈Ig),hi(i∈J+),Gi(i∈α–∪βH–),Hi(i∈γ–∪βG–)are convex at u and radially

noncon-stant.Also,assume that–hi(i∈J–), –Gi(i∈α+∪βH+∪β+), –Hi(i∈γ+∪βG+∪β+)are

directionally Lipschitzian,convex at u,and radially nonconstant.Ifα–∪γ–∪β_G–∪β_H–=φ, then,for any z feasible for theMPEC,we have

f(z)≥f(u) +

i∈Ig

λg_igi(u) + p

i=

λh_ihi(u) – l

i=

λG_i Gi(u) +λHi Hi(u)

.

Proof Letzbe any feasible point for MPEC. Then we have

gi(z)≤, ∀i∈Igandhi(z) = ,i= , , . . . ,p.

Sincef is convex atu,

f(z) –f(u)≥ ξ,z–u, ∀ξ∈∂cf(u). (.)

Similarly, we have

gi(z) –gi(u)≥

ξ_ig,z–u, ∀ξ_ig∈∂cgi(u),∀i∈Ig, (.)

hi(z) –hi(u)≥

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–hi(z) +hi(u)≥–

ξ_ih,z–u, ∀ξ_ih∈∂chi(u),∀i∈J–, (.)

–Gi(z) +Gi(u)≥–

ξ_iG,z–u, ∀ξ_iG∈∂cGi(u),∀i∈α+∪βH+∪β+, (.)

–Hi(z) +Hi(u)≥–

ξ_iH,z–u, ∀ξ_iH∈∂cHi(u),∀i∈γ+∪βG+∪β+. (.)

Ifα–∪γ–∪β_G–∪β_H–=φ, multiplying (.)-(.) byλ_ig≥ (i∈Ig),λih>  (i∈J+), –λhi > 

(i∈J–),λG_i >  (i∈α+∪β_H+ ∪β+),λH_i >  (i∈γ+∪β_G+∪β+), respectively, and adding (.)-(.), we get

f(z) –f(u) +

i∈Ig

λg_igi(z) –

i∈Ig

λg_igi(u) + p

i=

λh_ihi(z) – p

i=

λh_ihi(u) – l

i=

λG_iGi(z)

+

l

i=

λG_iGi(u) – l

i=

λH_i Hi(z) + l

i=

λH_i Hi(u)

≥ ξ+

i∈Ig λg_iξ_ig+

p

i=

λh_iξ_ih–

l

i=

λG_iξ_iG+λ_iHξ_iH,z–u

.

From (.), there existξ¯∈∂cf(u),ξ¯ig∈∂cgi(u),ξ¯ih∈∂chi(u),ξ¯iG∈∂cGi(u), andξ¯iH∈∂cHi(u),

such that

¯

ξ+

i∈Ig λg_iξ¯_ig+

p

i=

λh_iξ¯_ih–

l

i=

λG_iξ¯_iG+λH_i ξ¯_iH= .

So,

f(z) –f(u) +

i∈Ig

λg_igi(z) –

i∈Ig

λg_igi(u) + p

i=

λh_ihi(z) – p

i=

λh_ihi(u)

–

l

i=

λG_iGi(z) + l

i=

λG_iGi(u) – l

i=

λH_i Hi(z) + l

i=

λH_i Hi(u)≥.

Now, using the feasibility ofzfor MPEC, that is,gi(z)≤,hi(z) = ,Gi(z)≥,Hi(z)≥,

we get

f(z) –f(u) –

i∈Ig

λg_igi(u) – p

i=

λh_ihi(u) + l

i=

λG_i Gi(u) + l

i=

Hence,

f(z)≥f(u) +

i∈Ig

λg_igi(u) + p

i=

λh_ihi(u) – l

i=

λG_iGi(u) + l

i=

λH_i Hi(u)

.

This completes the proof.

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Corollary . Letz be feasible for¯ MPECwhere all constraint functions gi,hi,Gi,Hiare

aﬃne and index sets Ig,α,β,γ deﬁned accordingly.Then,for any z feasible for theMPEC

and(u,λ)feasible forWDMPEC(z),¯ we have

f(z)≥f(u) +

i∈Ig

λg_igi(u) + p

i=

λh_ihi(u) – l

i=

.

Analogously, we have the following result for Asplund spaces.

Theorem .(Weak duality) Letz be feasible for¯ MPECwhere X is an Asplund space, (u,λ)feasible forWDMPEC(¯z)and index sets Ig,α,β,γ deﬁned accordingly.Suppose that

f,gi(i∈Ig),hi(i∈J+),Gi(i∈α–∪βH–),Hi (i∈γ–∪βG–)are convex at u and radially

nonconstant.Also,assume that–hi(i∈J–), –Gi(i∈α+∪βH+∪β+), –Hi(i∈γ+∪βG+∪β+)

are directionally Lipschitzian,convex at u,and radially nonconstant.Ifα–_∪_γ–_∪_β–

G∪βH–= φ,then,for any z feasible for theMPEC,we have

f(z)≥f(u) +

i∈Ig

λg_igi(u) + p

i=

λh_ihi(u) – l

i=

.

Proof The proof follows the lines of the proof of Theorem ..

Theorem .(Strong duality) Assumez is a locally optimal solution of¯ MPECwhere X is an Asplund space,such thatNNAMCQis satisﬁed atz and the index sets I¯ g,α,β,γ

are deﬁned accordingly.Let f, gi (i∈Ig), hi (i∈J+), –hi (i∈J–),Gi (i∈α–∪βH–), –Gi

(i∈α+_∪_β+

H∪β+),Hi(i∈γ–∪βG–), –Hi(i∈γ+∪βG+∪β+)satisfy the assumption of the

Theorem..Then there existsλ¯,such that(z,¯ λ¯)is an optimal solution ofWDMPEC(¯z) and the respective objective values are equal.

Proof Sincez¯is a locally optimal solution of MPEC and the NNAMCQ is satisﬁed atz,¯ hence, by Theorem ., ∃¯λ= (λ¯g_,_λ¯h_,_λ¯G_,_λ¯H₎_∈_Rk+p+l_{, such that the nonsmooth}

M-stationarity conditions for MPEC are satisﬁed, that is, there existξ¯∈∂cf(z),¯ ξ¯ig∈∂cgi(z),¯

¯

ξ_ih∈∂chi(¯z),ξ¯iG∈∂cGi(z), and¯ ξ¯iH∈∂cHi(z), such that¯

 =ξ¯+

i∈Ig

¯

λg_iξ¯_ig+

p

i= ¯

λh_iξ¯_ih–

l

i=

_¯

λG_iξ¯_iG+λ¯H_i ξ¯_iH,

¯

λg_I

g≥, λ¯

G

γ = ,λ¯Hα = , eitherλ¯Gi > ,λ¯iH>  orλ¯Giλ¯Hi = ,∀i∈β.

Therefore, (¯z,λ¯) is feasible forWDMPEC(z). By Theorem ., we have¯

f(¯z)≥f(u) +

i∈Ig

λg_igi(u) + p

i=

λh_ihi(u) – l

i=

, (.)

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Hi(z) = ,¯ ∀i∈β∪γ, we have

f(¯z) =f(z) +¯

i∈Ig

¯

λg_igi(¯z) + p

i= ¯

λh_ihi(z) –¯ l

i=

_¯

λ_iGGi(z) +¯ λ¯Hi Hi(z)¯

. (.)

Using (.) and (.), we have

f(¯z) +

i∈Ig

¯

λg_igi(z) +¯ p

i= ¯

λh_ihi(z) –¯ l

i=

_¯

λ_iGGi(¯z) +λ¯Hi Hi(¯z)

≥f(u) +

i∈Ig

λg_igi(u) + p

i=

λh_ihi(u) – l

i=

.

Hence, (z,¯ λ¯) is an optimal solution forWDMPEC(z) and the respective objective values¯

are equal.

Example . Consider the following MPEC inR:

MPEC() min|z|+z

subject to :|z|+z≥,

–z≥,

z|z|+z= .

Now, we formulate Wolfe-type dual problemWDMPEC(z) for MPEC():¯

max u,λ |u|+u

 –

λG|u|+u+λH(–u)

subject to

 

=

ξ

u

–λG

η



–λH

 –

,

whereξ,η∈[–, ].

Ifβis non-empty, then either

λG> , λH> , or λGλH= .

If we take the pointz¯= (, ) from the feasible region, then the index setsα(, ) and

γ(, ) are empty sets, butβ:=β(, ) is non-empty. Also, from solving a constraint equa-tion in the feasible region of WDMPEC(, ), we getλG ₌ ξ

η and λ

H₌ ξ

η – u, where

η= . Sinceβ is non-empty, we consider aβ+, β_G+,β_H+ to decide the feasible region of WDMPEC(, ). It is clear that the assumptions of Theorem . are satisﬁed, so Theo-rem . holds between MPEC() andWDMPEC(, ).

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existsλ¯such that (¯z,λ¯) is an optimal solution ofWDMPEC(, ) and the respective values are equal.

We now prove the duality relation between the mathematical programming problem with equilibrium constraints (MPEC) and the following Mond-Weir-type dual problem

MWDMPEC(z)¯ max u,λ f(u)

subject to

∈∂cf(u) +

i∈Ig

λg_i∂cgi(u) + p

i=

λh_i∂chi(u) – l

i=

λG_i∂cGi(u) +λHi ∂cHi(u)

, (.)

gi(u)≥ (i∈Ig), hi(u) =  (i= , . . . ,p),

Gi(u)≤ (i∈α∪β), Hi(u)≤ (i∈β∪γ),

λg_I_g≥, λG_γ = , λH_α = , eitherλ_iG> ,λH_i >  orλG_i λH_i = ,∀i∈β,

whereλ= (λg_,_λh_,_λG_,_λH₎_∈_Rk+p+l_.

Theorem .(Weak duality) Letz be feasible for¯ MPECwhere X is a Banach space, (u,λ)

be feasible forMWDMPEC(z),¯ and the index sets Ig,α,β,γare deﬁned accordingly.Suppose

that f,gi(i∈Ig),hi(i∈J+),Gi(i∈α–∪βH–),Hi(i∈γ–∪βG–)are convex at u and radially

nonconstant.Also,assume that–hi(i∈J–), –Gi(i∈α+∪βH+∪β+), –Hi(i∈γ+∪βG+∪β+)

are directionally Lipschitzian,convex at u,and radially nonconstant.Ifα–∪γ–∪β_G–∪β_H–=

φ,then,for any z feasible for theMPEC,we have

f(z)≥f(u).

Proof Sincef is convex atu,

f(z) –f(u)≥ ξ,z–u, ∀ξ∈∂cf(u). (.)

Similarly, we have

gi(z) –gi(u)≥

ξ_ig,z–u, ∀ξ_ig∈∂cgi(u),∀i∈Ig, (.)

hi(z) –hi(u)≥

ξ_ih,z–u, ∀ξ_ih∈∂chi(u),∀i∈J+, (.)

–hi(z) +hi(u)≥–

ξ_ih,z–u, ∀ξ_ih∈∂chi(u),∀i∈J–, (.)

–Gi(z) +Gi(u)≥–

ξ_iG,z–u, ∀ξ_iG∈∂cGi(u),∀i∈α+∪βH+∪β+, (.)

–Hi(z) +Hi(u)≥–

ξ_iH,z–u, ∀ξ_iH∈∂cHi(u),∀i∈γ+∪βG+∪β+. (.)

Ifα–∪γ–∪β_G–∪β_H–=φ, multiplying (.)-(.) byλ_ig≥ (i∈Ig),λih>  (i∈J+), –λhi > 

(i∈J–_),_λG

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(.)-(.), we get

f(z) –f(u) +

i∈Ig

λg_igi(z) –

i∈Ig

λg_igi(u) + p

i=

λh_ihi(z) – p

i=

λh_ihi(u) – l

i=

λG_iGi(z)

+

l

i=

λG_iGi(u) – l

i=

λH_i Hi(z) + l

i=

λH_i Hi(u)

≥ ξ+

i∈Ig λg_iξ_ig+

p

i=

λh_iξ_ih–

l

i=

λG_iξ_iG+λ_iHξ_iH,z–u

.

From (.), there exist ξ¯ ∈∂cf(u), ξ¯ig ∈∂cgi(u), ξ¯ih ∈∂chi(u), ξ¯iG ∈∂cGi(u), and ξ¯iH ∈ ∂cHi(u), such that

¯

ξ+

i∈Ig λg_iξ¯_ig+

p

i=

λh_iξ¯_ih–

l

i=

λG_iξ¯_iG+λH_i ξ¯_iH= .

So,

f(z) –f(u) +

i∈Ig

λg_igi(z) –

i∈Ig

λg_igi(u) + p

i=

λh_ihi(z) – p

i=

λh_ihi(u)

–

l

i=

λG_iGi(z) + l

i=

λG_iGi(u) – l

i=

λH_i Hi(z) + l

i=

Now, using the feasibility ofzandufor MPEC andMWDMPEC(z), respectively, we get¯

f(z)≥f(u).

This completes the proof.

The following corollary is a direct consequence of Theorem ..

Corollary . Letz be feasible for¯ MPECwhere all constraint functions gi,hi,Gi,Hiare

aﬃne and the index sets Ig,α,β,γ deﬁned accordingly.Then,for any z feasible for the

MPECand(u,λ)feasible forMWDMPEC(z),¯ we have

f(z)≥f(u).

Theorem .(Weak duality) Letz be feasible for¯ MPECwhere X is an Asplund space,

(u,λ)be feasible forMWDMPEC(z)¯ and the index sets Ig,α,β,γ are deﬁned accordingly.

Suppose that f,gi(i∈Ig),hi(i∈J+),Gi(i∈α–∪βH–),Hi (i∈γ–∪βG–)are convex at u

and radially nonconstant.Also,assume that–hi(i∈J–), –Gi(i∈α+∪βH+∪β+), –Hi(i∈ γ+_∪_β+

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γ–_∪_β–

G∪βH–=φ,then,for any z feasible for theMPEC,we have

f(z)≥f(u).

Proof The proof follows the lines of the proof of Theorem ..

Theorem .(Strong duality) Assume¯z is a locally optimal solution of MPECwhere X is an Asplund space,such thatNNAMCQis satisﬁed atz and the index sets I¯ g,α,β,γ deﬁned

accordingly.Let f,gi(i∈Ig),hi(i∈J+), –hi(i∈J–),Gi(i∈α–∪βH–), –Gi(i∈α+∪βH+∪β+),

Hi(i∈γ–∪βG–), –Hi(i∈γ+∪βG+∪β+)satisfy the assumption of the Theorem..Then

there existsλ¯,such that(z,¯ λ¯)is an optimal solution ofMWDMPEC(z),¯ and the respective objective values are equal.

Proof z¯is a locally optimal solution of MPEC and the NNAMCQ is satisﬁed atz, by Theo-¯ rem .,∃¯λ= (λ¯g_,_λ¯h_,_λ¯G_,_λ¯H₎_∈_Rk+p+l_{, such that the nonsmooth M-stationarity conditions}

for MPEC are satisﬁed, that is, there existξ¯∈∂cf(z),¯ ξ¯ig∈∂cgi(z),¯ ξ¯ih∈∂chi(z),¯ ξ¯iG∈∂cGi(¯z)

andξ¯_iH∈∂cHi(z), such that¯

 =ξ¯+

i∈Ig

¯

λg_iξ¯_ig+

p

i= ¯

λh_iξ¯_ih–

l

i=

_¯

λG_iξ¯_iG+λ¯H_i ξ¯_iH,

¯

λg_I_g≥, λ¯_γG= , λ¯H_α = , eitherλ¯_iG> ,λ¯H_i >  orλ¯_iGλ¯H_i = ,∀i∈β.

Since¯zis an optimal solution for MPEC, we have

i∈Ig

¯

λg_igi(z) = ,¯ p

i= ¯

λh_ihi(z) = ,¯ l

i= ¯

λG_iGi(z) = ,¯ l

i= ¯

λH_i Hi(z) = .¯

Therefore, (z,¯ λ¯) is feasible forMWDMPEC(z). Also, by Theorem ., for any feasible (u,¯ λ), we have

f(¯z)≥f(u).

Thus, (z,¯ λ¯) is an optimal solution forMWDMPEC(¯z) and the respective objective values

are equal. This completes the proof.

Example . Consider the following MPEC problem inR_:

MPEC min|z|+z

subject to|z|+z≥,

z–|z| ≥,

|z|+zz–|z|= .

The Mond-Weir-type dual problemMWDMPEC(z) for the MPEC is¯

max

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subject to

 

=

ξ



–λG

ξ 

–λH

η



, (.)

whereξ,ξ,η∈[–, ],

λG|u|+u

≤, (.)

λHu–|u|≤, (.)

ifβis non-empty, then either

λG> , λH> , or λGλH= .

From (.),λGξ+λHη=ξ, andλG +λH= , we getλH= ξ_ξ_–ξ_–η andλG =ξ_ξ_–η_–η, where

ξ=η. Ifz¯= (, ), then the index setsα(, ) andγ(, ) are empty sets, butβ(, ) is non-empty. It is clear that the assumptions of Corollary . are satisﬁed. So, Corollary . holds between MPEC andMWDMPEC(, ).

Also, we can see that the NNAMCQ is satisﬁed at¯z. Then by Theorem . there exists ¯

λ= (λ¯G,λ¯H) such that (¯z,λ¯) is an optimal solution ofMWDMPEC(, ) and the optimal values are equal.

Now, we establish weak and strong duality theorems for the MPEC and its Mond-Weir-type dual problem under generalized convexity assumptions.

Theorem .(Weak duality) Letz be feasible for¯ MPECwhere X is a Banach space, (u,λ)

be feasible forMWDMPEC(z),¯ and the index sets Ig,α,β,γare deﬁned accordingly.Suppose

that f is pseudoconvex atz,¯ gi(i∈Ig),hi (i∈J+),Gi(i∈α–∪βH–),Hi (i∈γ–∪βG–)are

quasiconvex at u and radially nonconstant.Also,assume that–hi(i∈J–), –Gi(i∈α+∪ β_H+ ∪β+_{), –H}

i(i∈γ+∪βG+∪β+)are directionally Lipschitzian,quasiconvex at u,and

radially nonconstant.Ifα–∪γ–∪β_G–∪β_H– =φ,then,for any z feasible for theMPEC,we have

f(z)≥f(u).

Proof Suppose that, for some feasible pointz, such thatf(z) <f(u), then, by pseudocon-vexity off atu, we have

ξ,z–u< , ∀ξ∈∂cf(u). (.)

From (.), there exist ξ¯_ig∈∂cgi(u) (i∈Ig),ξ¯ih∈∂chi(u) (i= , . . . ,p), ξ¯iG∈∂cGi(u) (i∈ α∪β) andξ¯_iH∈∂cHi(u) (i∈β∪γ), such that

–

i∈Ig λg_iξ¯_ig–

p

i=

λh_iξ¯_ih+ α∪β

λG_iξ¯_iG+ β∪γ

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By (.), we get

–

p

i=

λH_i ξ¯_iH

,z–u

< . (.)

For eachi∈Ig,gi(z)≤≤gi(u). Hence, by Theorem . in [], we have

ξ,z–u ≤, ∀ξ∈∂cgi(u),∀i∈Ig. (.)

Similarly, we have

ξ,z–u ≤, ∀ξ∈∂chi(u),∀i∈J+. (.)

Now, for any feasible pointuofMWDMPEC(z), and for each¯ i∈J–_{,  = –h}

i(u) =hi(z). On

the other hand, –Gi(z)≤–Gi(u),∀i∈α+∪βH+, and –Hi(z)≤–Hi(u),∀i∈γ+∪βG+. Since

all of these functions are directionally Lipschitzian, by Theorem ., we get

ξ,z–u ≥, ∀ξ∈∂chi(u),∀i∈J–, (.)

ξ,z–u ≥, ∀ξ∈∂cGi(u),∀i∈α+∪βH+, (.)

ξ,z–u ≥, ∀ξ∈∂cHi(u),∀i∈γ+∪βG+. (.)

From equation (.)-(.), it is clear that

_¯

ξ_ig,z–u≤ (i∈Ig), _¯

ξ_ih,z–u≤ i∈J+, ξ¯_ih,z–u≥ i∈J–,

_¯

ξ_iG,z–u≥, ∀i∈α+∪β_H+, ξ¯_iH,z–u≥, ∀i∈γ+∪β_G+.

Sinceα–_∪_γ–_∪_β–

G∪βH– =φ, we have

α∪β

λG_i ξ¯_iG,z–u

≥,

β∪γ

λH_i ξ¯_iH,z–u

≥,

i∈Ig

λg_iξ¯_ig,z–u

≥,

p

i=

λh_iξ¯_ih,z–u

≥.

Therefore,

–

p

i=

λH_i ξ¯_iH

,z–u

≥,

which contradicts (.). Hence,f(z)≥f(u). This completes the proof.

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Suppose that f is pseudoconvex at¯z,gi(i∈Ig),hi(i∈J+),Gi(i∈α–∪βH–),Hi(i∈γ–∪βG–)

are quasiconvex at u and radially nonconstant.Also, assume that–hi(i∈J–), –Gi(i∈ α+∪β_H+∪β+), –Hi(i∈γ+∪βG+∪β+)are directionally Lipschitzian,quasiconvex at u,and

radially nonconstant.Ifα–_∪_γ–_∪_β–

G∪βH– =φ,then,for any z feasible for theMPEC,we

have

f(z)≥f(u).

Proof The proof follows the lines of the proof of Theorem ..

Theorem .(Strong duality) Assumez is a locally optimal solution of¯ MPECwhere X is an Asplund space,such thatNNAMCQis satisﬁed atz,¯ and the index sets Ig,α,β, γ are deﬁned accordingly.Let f,gi(i∈Ig),hi(i∈J+), –hi(i∈J–),Gi (i∈α–∪βH–), –Gi

(i∈α+∪β_H+ ∪β+),Hi (i∈γ–∪βG–), –Hi (i∈γ+∪βG+ ∪β+)satisfy the assumption of

Theorem..Then there existsλ¯,such that(z,¯ λ¯)is an optimal solution ofMWDMPEC(z),¯ and the respective objective values are equal.

Proof The proof follows the lines of the proof of Theorem ., invoking Theorem ..

4 Results and discussion

We have studied mathematical programs with equilibrium constraints (MPECs). The ob-jective function and functions in the constraint part are assumed to be lower semicontinu-ous. We studied the Wolfe-type dual problem for the MPEC under the convexity assump-tion. A Mond-Weir-type dual problem was also formulated and studied for the MPEC under convexity and generalized convexity assumptions. Conditions for weak duality the-orems were given to relate the MPEC and two dual programs in Banach space, respectively. Also conditions for strong duality theorems were established in an Asplund space. We also discussed the cases when all the constraint functions are aﬃne. Two numerical examples were given to illustrate the Wolfe-type duality and the Mond-Weir-type duality with our MPECs, respectively.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YP conceived of the study and drafted the manuscript initially. S-MG participated in its design, coordination and ﬁnalized the manuscript. SKM outlined the scope and design of the study. All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Management, Chang Gung University and Research Division, Chang Gung Memorial Hospital, Taoyuan,}

Taiwan.2_{Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India.}

Acknowledgements

The authors would like to express their deep appreciation to the helpful comments of reviewers. The research of S-M Guu was partially supported by NSC 102-2221-E-182-040-MY3 of the National Science Council, Taiwan and BMRPD017 of Chang Gung Memorial Hospital, Taiwan. The research of Yogendra Pandey was supported by the Council of Scientiﬁc and Industrial Research, New Delhi, Ministry of Human Resource Development, Government of India. Grant

09/013(0388)2011-EMR-1 02.05.2011.

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