AND RELATED FUNCTIONS

PROGRAMMING INVOLVING GENERALIZED D-TYPE-I RELATIVE TO BIFUNCTIONS

AND RELATED FUNCTIONS

VASILE PREDA, COSTEL B ˘ALC ˘AU and CRISTIAN NICULESCU

Optimality conditions are obtained for a nonlinear fractional multiobjective pro- gramming problem involving generalized d-type-I relative to bifunctions and re- lated functions. The place of the derivatives is taken by bifunctions having certain properties. Also some examples are given.

AMS 2010 Subject Classification: 90C29, 90C32, 90C56.

Key words: fractional multiobjective programming, optimality conditions, bifunc- tion, convexlike.

1. INTRODUCTION

Optimality conditions for nonlinear singleobjective or multiobjective op- timization problems involving generalized convex functions have been of much interest and many contributions have been made to this development (e.g., [7, 8]).

We consider the following multiobjective nonlinear fractional program- ming problem:

(VFP)

min^f_g(x)^(x) =_f

1(x)

g1(x), . . . ,^f_g^p(x)

p(x)

subject to

h_j(x)50, j= 1,2, . . . , m, x∈X0,

where X₀ ⊆Rⁿ is a nonempty open set, f = (f₁, . . . , f_p), g = (g₁, . . . , g_p) : X0 → R^p, h = (h1, . . . , hm) : X0 → R^m, gi(x) > 0 for all x ∈ X0 and each i = 1, . . . , p. We denote X = {x ∈ X₀ | h_j(x) 5 0, j = 1,2, . . . , m}, the feasible set of problem (VFP).

In this paper, we obtain optimality conditions result for (VFP) involving general classes of functions. The place of the derivatives is taken by bifunctions having certain properties. In a final section are examples and comments. Thus,

MATH. REPORTS14(64),1 (2012), 95–106

we extend our work [5] and generalize results obtained in the literature on this topic [4, 6].

2. DEFINITIONS AND PRELIMINARIES

Throughout this paper we use the following conventions for x, y∈Rⁿ: x < y iffx_i< y_i for any i= 1, n,

x5y iffx_i5y_i for any i= 1, n, x≤y iffx5y and x6=y, x≮y is the negation ofx < y.

We denote Rⁿ₊={x∈Rⁿ|x=0}.

Definition 2.1. A point x ∈ X is said to be a weak Pareto solution or weak minimum for (VFP) if ^f(x)_g(x) ≮ ^f(x)_g(x) for all x∈X.

Definition 2.2. A pointx∈X is said to be a local weak Pareto solution or local weak minimum for (VFP) if there is a neighborhood V(x) aroundx, such that

f(x)

g(x) ≮ f(x)

g(x) for all x∈X∩V(x).

Letfbi,bgi :X0×Rⁿ→Rfori∈P ={1,2, . . . , p},andbhj :X0×Rⁿ→R for j∈M ={1,2, . . . , m}, bifunctions.

Let ρ¹ = (ρ¹₁, . . . , ρ¹_p)^T, ρ² = (ρ²₁, . . . , ρ²_p)^T ∈ R^p, ρ³ = (ρ³₁, . . . , ρ³_m)^T ∈ R^m, andd:X₀×X₀→R₊.

Definition 2.3. (f, g, h) is said to be of general (ρ¹, ρ², ρ³, d)-type I at x ∈ X₀ relative to (fb_i,bg_i,bh_j), i ∈ P, j ∈ M if there exist the functions η :X0 ×X0 → Rⁿ, αi, βi, γj :X0×X0 → R+\{0}, i∈ P, j ∈ M such that for all x∈X₀, we have

fi(x)−fi(x)=αi(x, x)fbi(x, η(x, x)) +ρ¹_id(x, x), ∀i∈P, g_i(x)−g_i(x)5β_i(x, x)bg_i(x, η(x, x))−ρ²_id(x, x), ∀i∈P,

−h_j(x)=γ_j(x, x)bh_j(x, η(x, x)) +ρ³_jd(x, x), ∀j∈M.

3. SUFFICIENT OPTIMALITY CRITERIA

In this section we obtain some sufficient conditions for a feasible solution x to be weak minimum for (VFP). The proofs of Theorems 3.1 and 3.2 follow the lines, for example, of the proofs of Theorems 1, respectively 2 from [5], and are therefore omitted.

Theorem 3.1.Let x∈X such that f(x)=0,and(f, g, h) be of general (ρ¹, ρ², ρ³, d)-type I at x.Also, we assume that there exists λ⁰ ∈R^p, v⁰ ∈R^m such that λ^0Tρ¹+v^0Tρ³ =0,ρ² =0and

p

X

i=1

λ⁰_iαi(x, x)fbi(x, η(x, x)) +

m

X

j=1

v_j⁰γj(x, x)bhj(x, η(x, x))=0, ∀x∈X,

gb_i(x, η(x, x))50, ∀x∈X, ∀i∈P, v^0Th(x) = 0, λ⁰ ≥0,v⁰=0.

Thenx is a weak minimum solution for (VFP).

Theorem 3.2. Let x∈X, u⁰_i = ^f_g^i(x)

i(x),∀i∈P, and(f, g, h)be of general (ρ¹, ρ², ρ³, d)-type I at x. Also, we assume that there exists λ⁰ ∈R^p,v⁰ ∈R^m such that λ^0Tρ¹−

p

P

i=1

λ⁰_iu⁰_iρ²_i +v^0Tρ³ =0,

p

X

i=1

λ⁰_i α_i(x, x)e fb_i(x, η(x, x))−u⁰_iβ_i(ex, x)bg_i(x, η(x, x)) + +

m

X

j=1

v_j⁰γ_j(x, x)e bh_j(x, η(x, x))=0, ∀x∈X, v^0Th(x) = 0,λ⁰ ≥0,u⁰ =0,v⁰ =0.

Thenx is a weak minimum solution for (VFP).

4. NECESSARY OPTIMALITY CONDITIONS

Let ϕ : X₀ → R^k, ϕ = (ϕ₁, ϕ₂, . . . , ϕ_k), ϕb : X₀ ×Rⁿ → R^k, ϕb = (ϕb1,ϕb2, . . . ,ϕbk) and we consider thatϕbis a bifunction associated toϕ.

We say thatϕand ϕbsatisfy the hypothesis H1 at x∈X if (H1) for any y ∈Rⁿsuch thatϕ(x, y)b <0, there existstr&0 such thatϕ(x+try)< ϕ(x), for any r∈N^∗.

We say thatϕand ϕbsatisfy the hypothesis H2 at x∈X if (H2) for any y ∈Rⁿ such that ϕ(x, y)b <0, there exists t0 >0 such thatϕ(x+ty)< ϕ(x), for any t∈(0, t₀).

Proposition 4.1. Ifϕandϕbsatisfy H2 atx, then they satisfy also H1 atx. Converse is not true.

Proof. Ifϕ and ϕb satisfy H2 atx, lety ∈ Rⁿ such that ϕ(x, y)b < 0. It follows that there exists t0>0 such that ϕ(x+ty)< ϕ(x), for any t∈(0, t0).

Let tr = _r+1^t0 , for any r∈N^∗. Then tr ∈(0, t0), for anyr∈N^∗, and therefore ϕ(x+t_ry)< ϕ(x), for anyr∈N^∗. Sincet_r&0,ϕandϕbsatisfy H1 atx.

To see that converse is not true, consider the following example.

LetX0=R,ϕ:R→R,ϕ(x) =

cos¹_x ifx6= 0,

0 ifx= 0, andϕb:R×R→R, ϕ(x, y) =b x−y. Letx= 0,andy ∈R such thatϕ(x, y)b <0, i.e., y >0.

Lettr= _(2r+1)πy¹ , for anyr∈N^∗. It followsϕ(x+try) = cos(2r+ 1)π=

−1<0 =ϕ(x), for anyr ∈N^∗, andt_r&0. Thereforeϕandϕbsatisfy H1 atx.

For any t0 >0, let r ∈N^∗ such that r > _2πt¹

0y and t= _2rπy¹ . It follows t ∈ (0, t0), and ϕ(x+ty) = cos 2rπ = 1> 0 =ϕ(x). Therefore, ϕ and ϕb do not satisfy H2 at x.

Definition 4.1 [2]. A function f : X0 → R^k is convexlike if for any x, y∈X0 and 05λ51,there existsz∈X0 such that

f(z)5λf(x) + (1−λ)f(y).

Remark4.1. The convex and the preinvex functions are convexlike func- tions.

Lemma 4.1 [3]. Let S be a nonempty set in Rⁿ and ψ : S → R^k be a convexlike function. Then eitherψ(x)<0 has a solutionx∈Sorλ^Tψ(x)=0 for all x∈S,for some λ∈R^k, λ≥0, but both alternatives are never true.

DenoteM(x) ={j∈M |h_j(x) = 0}.

Lemma 4.2. Let x be a (local) weak minimum of (VFP). We suppose that f(x) = 0, there exists lim

x→xhj(x) < 0, for j ∈ M\M(x), and one of the following properties holds:

a)fi,fbi,−g_i,−bgi, (i∈P), hj,bhj, (j∈M(x)) satisfy H1 at x, and there exists r⁰ ∈ N^∗ large enough such that tⁱ_r0 = tⁱ⁰_r0 = t^j00_r0, for any i ∈ P and j ∈ M(x), where (tⁱ_r)r∈N^∗, (tⁱ⁰_r)r∈N^∗, (i ∈ P), (t^j00r )r∈N^∗, (j ∈ M(x)) are the sequences from H1 for f_i,fb_i, −g_i,−bg_i, (i ∈ P), and h_j,bh_j, (j ∈ M(x)), respectively;

b) fi,fbi, −g_i,−bgi, (i ∈ P), hj,bhj, (j ∈ M(x)) satisfy H1 at x, and tⁱ_r =tⁱ⁰_r =t^j00r ,for anyr ∈N^∗,i∈P andj ∈M(x);

c) there exists j₀ ∈ M(x) such that f_i,fb_i, −g_i,−bg_i, (i ∈ P), h_j,bh_j, (j∈M(x)\{j₀})satisfy H2 at x, and hj0,bhj0 satisfy H1 at x;

d) there exists i₀ ∈P such that f_i,fb_i , (i ∈P), −g_i,−bg_i, (i ∈P\{i₀}), h_j,bh_j, (j∈M(x))satisfy H2 at x, and−g_i0,−bg_i0 satisfy H1 at x;

e) there exists i₀ ∈ P such that f_i,fb_i , (i∈ P\{i₀}), −g_i,−bg_i, (i ∈ P), h_j,bh_j, (j∈M(x))satisfy H2 at x, andf_i0,fb_i0 satisfy H1 at x;

f)fi,fbi,−g_i,−bgi, (i∈P),andhj,bhj, (j∈M(x))satisfy H2 at x ;

g) ^f_g,^fb

bg, hj,bhj, (j ∈ M(x)) satisfy H1 at x, and there exists r⁰ ∈ N^∗ large enough such that t_r⁰ = t^j_r0, for any j ∈ M(x), where (t_r)r∈N^∗,(t^jr)r∈N^∗, (j ∈ M(x)) are the sequences from H1 for ^f_g,^fb

bg, and h_j,bh_j, (j ∈ M(x)), respectively;

h) ^f_g,^fb

bg,hj,bhj, (j∈M(x))satisfy H1 at x, and tr=t^jr,for any r∈N^∗, and j∈M(x);

i)there exists j₀ ∈M(x) such that ^f_g,^fb

bg,h_j,bh_j, (j ∈M(x)\{j₀}) satisfy H2 at x, and h_j0,bh_j0 satisfy H1 at x;

j) ^f_g,^fb

bg,hj,bhj, (j ∈M(x))satisfy H2 at x.

Then the system

(4.1)







fbi(x, y)<0, i∈P, bg_i(x, y)>0, i∈P, bhj(x, y)<0, j∈M(x), has no solution y∈Rⁿ.

Proof. We proceed by contradicting. We assume that (4.1) has a solution y ∈Rⁿ. Ify= 0, we get a contradiction to H1 or H2.

Now, lety6= 0. Forj ∈M\M(x) we have lim

x→xh_j(x)<0, therefore there existset >0 such that

(4.2) hj(x+ty)<0, for anyt∈(0,et ).

a) According to H1, we have

f_i(x+tⁱ_ry)< f_i(x), i∈P, (4.3)

gi(x+tⁱ⁰_ry)> gi(x), i∈P, (4.4)

and

(4.5) h_j(x+t^j00_r y)< h_j(x), j∈M(x), for any r ∈N^∗.

There existsr⁰∈N^∗large enough such thattⁱ_r0 =tⁱ⁰_r0 =t^j00_r0, for anyi∈P and j ∈ M(x). Let t_r⁰ = tⁱ_r0 = tⁱ⁰_r0 = t^j00_r0. Since tⁱ_r & 0, tⁱ⁰_r & 0, t^j00r & 0, for any i∈P and j∈M(x), andr⁰ is large enough, we havet_r⁰ ∈(0,et).

Sinceg(x)>0 for allx∈X₀ and f(x)=0, from (4.3), (4.4) written for r =r⁰, we get

(4.6) f(x+t_r⁰y)

g(x+t_r⁰y) < f(x) g(x).

Since x ∈ X, we have h(x) 50, and now from (4.5) written for r =r⁰ we get

(4.7) hj(x+tr⁰y)<0, for any j∈M(x).

Now, from (4.2), t_r⁰ ∈ (0,et), and (4.7) we obtain that for any j ∈ M, h_j(x+t_r⁰y)<0, therefore,x+t_r⁰y∈X.

This and (4.6) contradicts the fact that x is a (local) weak minimum of (VFP).

b) We have b)⇒ a).

c) From H2, for anyi∈P and j ∈M(x)\{j₀}, there existtⁱ₀,tⁱ⁰₀,t^j00₀ >0 such that

f_i(x+ty)< f_i(x), for any t∈(0, tⁱ₀), gi(x+ty)> gi(x), for any t∈(0, tⁱ⁰₀), and

h_j(x+ty)< h_j(x), for any t∈(0, t^j00₀ ).

Let t₀ = min

i∈P, j∈M(x)\{j₀}{tⁱ₀, tⁱ⁰₀, t^j00₀ }. It follows t₀ > 0 and, for any t ∈ (0, t₀),

fi(x+ty)< fi(x), i∈P, g_i(x+ty)> g_i(x), i∈P, h_j(x+ty)< h_j(x), j∈M(x)\{j₀}.

From H1, there existst^j_r⁰ &0 such that h_j0(x+t^j_r⁰y) < h_j0(x), for any r ∈N^∗.

Sincet^jr⁰ &0 andt₀ >0, there existsr₀∈N^∗ such thatt^jr⁰ ∈(0, t₀), for any r > r0.

Lett_r=t^j_r−r⁰ ₀, for any r ∈N^∗. It followst_r&0 and, for anyr ∈N^∗, fi(x+try)< fi(x), i∈P,

g_i(x+t_ry)> g_i(x), i∈P, h_j(x+t_ry)< h_j(x), j ∈M(x).

Now we apply b).

d), e) Similar to c).

f) It follows from Proposition 4.1 and c), d) or e).

g) Sincey is solution for (4.1), we get ( _f(x,y)b

bg(x,y) <0,

bhj(x, y)<0, j ∈M(x).

According to H1, we have for any r ∈ N^∗, ^f_g(x+t^(x+try)

ry) < ^f(x)_g(x) and hj(x+ t^jry)< hj(x),j∈M(x).

Further the proof is similar to a).

h) We have h)⇒ g).

i) Similar to c), applying h) instead of b).

j) It follows from Proposition 4.1 and i).

Theorem 4.1(Fritz John type necessary optimality conditions).Letxbe a (local) weak minimum of (VFP). We suppose that there exists lim

x→xh_j(x)<0 forj∈M\M(x),f(x)=0,

fbi(x, y)

i∈P,(−bgi(x, y))_i∈P, bhj(x, y)

j∈M(x)

is a convexlike function ofy onRⁿ,and one of propertiesa)–j) from Lemma4.2 holds. Then there existλ⁰,u⁰ ∈R^p,v⁰ ∈R^m such that

(4.8)

p

X

i=1

λ⁰_ifb_i(x, y)−

p

X

i=1

u⁰_ibg_i(x, y) +

m

X

j=1

v_j⁰bh_j(x, y)=0 for ally∈Rⁿ,

(4.9) v^0Th(x) = 0,

(4.10) (λ⁰, u⁰, v⁰)≥0.

Proof. From Lemma 4.2, the system (4.1) has no solution y ∈ Rⁿ. From Lemma 4.1, there exists λ⁰, u⁰ ∈ R^p, v_j⁰ ∈ R (j ∈ M(x)), such that (λ⁰, u⁰,(v⁰_j)_j∈M(x))≥0, and

(4.11)

p

X

i=1

λ⁰_ifbi(x, y)−

p

X

i=1

u⁰_ibgi(x, y) + X

j∈M(x)

v_j⁰bhj(x, y)=0 for ally∈Rⁿ. We putv⁰_j = 0 forj∈M\M(x). From (4.11) we get (4.8). Finally, (4.9) and (4.10) follow obviously.

We consider the following multiobjective programming problem:

(VP)

minϕ(x) = (ϕ1(x), . . . , ϕp(x)) subject to

( h_j(x)50, j= 1,2, . . . , m, x∈X₀.

In the same way as Lemma 4.2, one can prove the following lemma.

Lemma 4.3.Let xbe a (local) weak minimum of (VP). We suppose that forj∈M\M(x), there exists lim

x→xhj(x)<0and one of the following properties holds:

a) ϕi,ϕbi, (i ∈ P) , and hj,bhj, (j ∈ M(x)) satisfy H1 at x, and there exists r⁰ ∈N^∗ large enough such thattⁱ_r0 =t^j0_r0,for any i∈P and j ∈M(x), where (tⁱ_r)r∈N^∗(i ∈ P),(t^j0r)r∈N^∗, (j ∈ M(x)) are the sequences from H1 for ϕ_i,ϕb_i , (i∈P), and h_j,bh_j, (j∈M(x)), respectively;

b) ϕi,ϕbi, (i∈P) , and hj,bhj, (j ∈M(x)) satisfy H1 at x, and tⁱ_r =t^j0r, for any r∈N^∗,i∈P and j∈M(x);

c)there existsj₀ ∈M(x)such thatϕ_i,ϕb_i, (i∈P) ,h_j,bh_j, (j∈M(x)\{j₀}) satisfy H2 at x, and h_j0,bh_j0 satisfy H1 at x;

d) there exists i0 ∈P such that ϕi,ϕbi, (i∈ P\{i₀}), hj,bhj, (j ∈M(x)) satisfy H2 at x, and f_i0,fb_i0 satisfy H1 at x;

e)ϕi,ϕbi, (i∈P), andhj,bhj, (j∈M(x))satisfy H2 at x;

f) ϕ,ϕ,b h_j,bh_j, (j ∈ M(x)) satisfy H1 at x, and there exists r⁰ ∈ N^∗ large enough such that tr⁰ =t^j_r0,for any j∈M(x), where (tr)r∈N^∗, (t^jr)r∈N^∗, (j ∈ M(x)) are the sequences from H1 for ϕ,ϕ, andb hj,bhj, (j ∈ M(x)), respectively;

g)ϕ,ϕ,b hj,bhj, (j ∈M(x)) satisfy H1 at x, and tr =t^jr,for any r ∈N, and j∈M(x);

h) there existsj0 ∈M(x) such that ϕ,ϕ,b hj,bhj, (j ∈M(x)\{j₀}) satisfy H2 at x, and h_j0,bh_j0 satisfy H1 at x;

i)ϕ,ϕ,b hj,bhj, (j ∈M(x)) satisfy H2 at x.

Then the system (

ϕb_i(x, y)<0, i∈P, bhj(x, y)<0, j ∈M(x) has no solution y∈Rⁿ.

Lemma 4.4. Let xbe a (local) weak minimum of(VP). We suppose that there exists lim

x→xhj(x)<0forj∈M\M(x), ((ϕbi(x, y))i∈P, (bhj(x, y))j∈M(x))is a convexlike function of y onRⁿ, and one of properties a)–i)from Lemma4.3 holds. Then there exist λ⁰ ∈R^p,v_j⁰ ∈R, (j∈M(x))such that

p

X

i=1

λ⁰_iϕbi(x, y) + X

j∈M(x)

v_j⁰bhj(x, y)=0 for all y∈Rⁿ, (λ⁰,(v_j⁰)j∈M(x))≥0.

Proof. It follows from Lemma 4.3 and Lemma 4.1.

Now we define the H1 constraint qualification (H1CQ).

Definition4.2. We say thathsatisfy the H1CQ atxifhj,bhj, (j ∈M(x)) satisfy H1 at x, and there exists y ∈ Rⁿ such that bh_j(x, y) < 0, for any j ∈M(x).

Lemma 4.5. Let x be a (local) weak minimum of (VP). We suppose that there exists lim

x→xhj(x) < 0 for j ∈ M\M(x), h satisfy the H1CQ at x, ((ϕb_i(x, y))i∈P,(bh_j(x, y))_j∈M(x)) is a convexlike function of y on Rⁿ, and one of properties a)–i)from Lemma4.3holds. Then there existλ⁰ ∈R^p₊,v⁰ ∈R^m₊, such that

p

X

i=1

λ⁰_iϕbi(x, y) +

m

X

j=1

v_j⁰bhj(x, y)=0 for any y∈Rⁿ, v^0Th(x) = 0, λ^0Te= 1,where e= (1,1, . . . ,1)^T ∈R^p.

Proof. Using Lemma 4.4, there exist eλ∈R^p, evj ∈R, (j ∈ M(x)) such that (λ,e (ve_j)_j∈M(x))≥0, and

(4.12)

p

X

i=1

λeiϕbi(x, y) + X

j∈M(x)

evjbhj(x, y)=0 for any y∈Rⁿ. Now we prove thateλ6= 0. We proceed by contradicting.

Ifeλ= 0, from (4.12) we have

(4.13) X

j∈M(x)

ev_jbh_j(x, y)=0 for any y∈Rⁿ.

From (eλ,(evj)_j∈M(x))≥0 andeλ= 0 it follows (evj)_j∈M(x) ≥0. Using this and the fact thathsatisfy the H1CQ atx, we get P

j∈M(x)

evjbhj(x, y)<0, which contradicts (4.13).

Thereforeeλ6= 0 and from (eλ,(evj)j∈M(x))≥0 we geteλ^Te >0.

Let λ⁰ = ¹

eλ^Teλe and v⁰_j = ^vej

λe^Te, (j ∈ M(x)). It follows λ⁰=0, v_j⁰ = 0, (j∈M(x)), λ^0Te= 1, and from (4.12), we get

(4.14)

p

X

i=1

λ⁰_iϕbi(x, y) + X

j∈M(x)

v⁰_jbhj(x, y)=0 for any y∈Rⁿ.

We put v_j⁰ = 0, (j ∈ M\M(x)). It follows v⁰ ∈ R^m₊, v^0Th(x) = 0, and from (4.14) we get

p

X

i=1

λ⁰_iϕbi(x, y) +

m

X

j=1

v_j⁰bhj(x, y)=0 for anyy ∈Rⁿ.

For eachu= (u1, . . . , up)^T ∈R^p, we consider (VFP_u)

min(f1(x)−u1g1(x), . . . , fp(x)−upgp(x)) subject to

hj(x)50, j∈M, x∈X0.

Lemma 4.6.Ifxis a(local)weak minimum for(VFP), thenxis a(local) weak minimum for (VFP_u⁰), where u⁰= ^f_g(x)^(x).

Proof. Ifx is a local weak minimum for (VFP), there is a neighborhood V(x) around x, such that ^f_g(x)^(x) ≮ u⁰ for all x ∈ X∩V(x). Since g(x) > 0 for all x ∈X∩V(x), it follows (f₁(x)−u⁰₁g₁(x), . . . , f_p(x)−u⁰_pg_p(x))≮0 = (f1(x)−u⁰₁g1(x), . . . , fp(x)−u⁰_pgp(x)) for all x ∈ X∩V(x) and thus x is a (local) weak minimum for (VFP_u⁰).

Ifx is a weak minimum for (VFP), we repeat this proof takingV(x) = Rⁿ.

Theorem 4.2 (Karush-Kuhn-Tucker type necessary optimality condi- tions). Let xbe a(local)weak minimum of(VFP)andu⁰= ^f(x)_g(x). We suppose thatf(x)=0,there exists lim

x→xh_j(x)<0for j∈M\M(x),hsatisfy the H1CQ at x, ((fb_i(x, y)−u⁰_ibg_i(x, y))i∈P,(bh_j(x, y))_j∈M(x)) is a convexlike function of y on Rⁿ, and for ϕ_i=f_i−u⁰_ig_i,ϕb_i =fb_i−u⁰_ibg_i, (i∈P), andh_j,bh_j, (j ∈M(x)) one of properties a)–i) from Lemma 4.3 holds. Then there exist λ⁰ ∈ R^p₊, v⁰ ∈R^m₊,such that

p

X

i=1

λ⁰_i(fbi(x, y)−u⁰_ibgi(x, y)) +

m

X

j=1

v_j⁰bhj(x, y)=0 for any y∈Rⁿ,v^0Th(x) = 0,λ^0Te= 1,u⁰=0.

Proof. From Lemma 4.6 we have that x is a (local) weak minimum of (VFP_u⁰). Sincef(x)=0,x∈X⊆X₀ and g(x)>0 for anyx∈X₀, it follows u⁰=0. Finally, we apply Lemma 4.5 for (VFP_u⁰).

Remark 4.2. Lemmas 4.2, 4.3, 4.4, 4.5 and Theorems 4.1, 4.2 also hold when we replace the assumption “there exists lim

x→xh_j(x)<0 forj∈M\M(x)”

with “h_j is a continuous function atx forj ∈M\M(x)”.

5. SOME EXAMPLES AND COMMENTS

Now we give some examples.

Example5.1. Ifϕis a differentiable function atx, then we takeϕ(x, y) =b y^T∇ϕ(x) = lim

t→0t⁻¹[ϕ(x+ty)−ϕ(x)].

Example 5.2. If ϕ is a semidifferentiable function at x, then we take ϕ(x, y) =b ϕ⁺(x, y) = lim

t→0+t⁻¹[ϕ(x+ty)−ϕ(x)].

Example 5.3. We suppose that there exists D⁺ϕ(x, y) and ϕ(x, y)b = D⁺ϕ(x, y) = lim sup

t→0+

t⁻¹[ϕ(x+ty)−ϕ(x)].

In Examples 5.1, 5.2, 5.3,ϕand ϕbsatisfy both H1 and H2 at x.

Example5.4. Also we can take ϕ(x, y)b =D⁻ϕ(x, y) = lim inf

t→0+ t⁻¹[ϕ(x+ ty)−ϕ(x)].

In this case,ϕand ϕbsatisfy H1, but, generally, not H2 at x.

Definition 5.1 [1]. A function f : Rⁿ → R is said to be subinvex at x ∈Rⁿ with respect to a given η:Rⁿ×Rⁿ→Rⁿ if there exists an element ξ ∈ Rⁿ such that for all y ∈ Rⁿ, f(y)−f(x) = ξ^Tη(y, x). This ξ is called an η-subgradient of f at x. The set of η-subgradients at x is called the η- subdifferential off atxand is denoted by∂^ηf(x). Hence a function is subinvex atx∈Rⁿwith respect toηif∂^ηf(x)6=∅. The functionf is said to be subinvex if it is subinvex at all x∈Rⁿ.

Letf :Rⁿ→R be subinvex with respect to a givenη:Rⁿ×Rⁿ→Rⁿ at x∈Rⁿ, i.e.,∂^ηf(x)6=∅. The function f is said to satisfy the assumption (B) at x if there exists a v∈∂^ηf(x) such thatD⁺f(x, y)5v^Ty,∀y ∈Rⁿ.

Lemma 5.1 [1, Lemma 2.1]. Let f : Rⁿ → R be subinvex such that 0∈/ ∂^ηf(x), then the setF₀(v) ={y|v^Ty <0} 6=∅ for every v∈∂^ηf(x).

Now, we give an example ofϕand ϕbfor which there existsy∈Rⁿ such that ϕ(x, y)b <0.

Example5.5. Letϕ:Rⁿ→Rbe subinvex with respect toη :Rⁿ×Rⁿ→ Rⁿ atx∈Rⁿ such that ϕsatisfies the assumption (B) at x and 0 ∈/ ∂^ηf(x).

Let ϕ(x, y) =b D⁺ϕ(x, y). From the assumption (B) and Lemma 5.1 it follows that there exist v∈∂^ηf(x) andy∈Rⁿ such thatϕ(x, y)b 5v^Ty <0.

Remark5.1. Instead ofM(x) ={j∈M |h_j(x) = 0}, we may define M(x) =n

j∈M | lim

x→xh_j(x) does not exist or lim

x→xh_j(x)=0o

and Lemmas 4.2, 4.3, 4.4, 4.5 and Theorems 4.1, 4.2 also hold without the assumption “there exists lim

x→xh_j(x)<0 for j∈M\M(x)”.

REFERENCES

[1] J. Dutta and V. Vetrivel, Mathematical programming with a class of non-smooth func- tions. J. Syst. Sci. Compl.15(2002), 52–60.

[2] K.H. Elster and R. Nehse,Optimality Conditions for Some Nonconvex Problems. Springer Verlag, New York, 1980.

[3] M. Hayashi and H. Komiya, Perfect duality for convexlike programs. J. Optim. Theory Appl.38(1982) 179–189.

[4] S.K. Mishra, S.Y. Wang and K.K. Lai,Multiple objective fractional programming involv- ing semilocally type I-preinvex and related functions. J. Math. Anal. Appl.310 (2005), 626–640.

[5] C. Niculescu, Optimality and duality in multiobjective fractional programming involving ρ-semilocally type I-preinvex and related functions. J. Math. Anal. Appl. 335 (2007), 7–19.

[6] V. Preda,Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions. J. Math. Anal. Appl.288 (2003), 365–382.

[7] V. Preda, I.M. Stancu-Minasian and A. B˘at˘atorescu,Optimality and duality in nonlinear programming involving semilocally preinvex and related functions. J. Inf. Optim. Sci.17 (3)(1996), 585–596.

[8] V. Preda and I.M. Stancu-Minasian,Duality in multiple objective programming involving semilocally preinvex and related functions. Glas. Mat.32 (52)(1997), 153–165.

Received 11 June 2010 University of Bucharest

Faculty of Mathematics and Computer Science 14 Academiei Street

Bucharest 70109, Romania University of Pite¸sti

Faculty of Mathematics and Computer Science 1 Tˆargul din Vale Street

Pite¸sti 110040, jud. Arge¸s, Romania