MULTIOBJECTIVE FRACTIONAL PROGRAMMING INVOLVING ρ-SEMILOCALLY PREINVEX AND RELATED FUNCTIONS

MULTIOBJECTIVE FRACTIONAL PROGRAMMING INVOLVING ρ-SEMILOCALLY PREINVEX

AND RELATED FUNCTIONS

CRISTIAN NICULESCU

For a multiobjective fractional programming problem involving ρ-semilocally preinvex and related functions, we prove some optimality conditions and a weak duality theorem for a general Mond-Weir dual.

AMS 2000 Subject Classification: 90C29, 90C32.

Key words: multiobjective fractional programming, semilocally preinvex function, optimality, duality, weak minimum.

1. INTRODUCTION

Many optimality conditions and approaches to duality for the nonlinear multiple objective optimization problem have been of much interest in the recent past and many contributions have been made to this development, e.g.

Bector et al. [1], Bitran [3], Cambini and Martein [4, 5], Corley [6], Craven [7, 8], Elster and Nehse [9], Geoffrion [11], Heal [14], Ivanov and Nehse [15], Jeyakumar [16–20], Singh [32], Tanino and Sawaragi [34], Weir [35], Weir and Mond [36], White [38]. These studies differ in their approaches and/or in the sense in which the optimality concept is defined for a multiple objective programming problem. Some approaches to duality include the use of vector valued Lagrangians and Lagrangians (e.g., [8, 16, 17, 34, 35, 37]), incorporating matrix Lagrange multipliers (e.g., [3, 6, 7, 15]).

Also, it is well known (see Mangasarian [26]) that, under a convexity assumption and a regularity hypothesis, there exists an equivalence between saddle-points of the Lagrangian and optima for an inequality constrained min- imization problem (for discussions and extensions of this result see Heal [14], Ben-Israel and Mond [2], and Jeyakumar [16]). Jeyakumar [17] discussed a class of nonsmooth non-convex problems in which functions are locally Lip- schitz and satisfy some invex type conditions. Further, it was shown that duality theorems of Wolfe type [39] hold for this class of problems. Also, in

REV. ROUMAINE MATH. PURES APPL.,52(2007),1, 69–85

Cambini and Martein [4, 5], a new approach to optimality conditions in vector and scalar optimization was given.

Elster and Nehse [9] considered a class of convexlike functions and ob- tained a saddlepoint optimality condition for mathematical programs involving such functions. Hayashi and Komiya [13] also considered Lagrangian duality for convexlike programs. Ben-Israel and Mond [2] and Hanson and Mond [12]

considered a class of functions called preinvex. Jeyakumar and Mond [19]

introduced new classes of generalized convex vector functions, called v-invex, and some results relative to Lagrangian sufficiency, weak duality and global optimality were given.

Craven [8] gave Lagrangian necessary conditions for optimality, of both Fritz-John and Kuhn-Tucker types for a constrained minimization problem, where the functions are locally Lipschitz and the directional derivatives are assumed to have some convexity properties as functions of direction. Further, in Craven [8], some sufficient Kuhn-Tucker conditions and a criterion for the locally solvable constraint qualification were obtained.

In [18] and [20], some classes of nonsmooth programming problems were given. The concept of semilocally convex functions was introduced by Ewing [10] and was further extended to semilocally quasiconvex, semilocally pseudo- convex functions by Kaul and Kaur [21–23]. In Suneja and Gupta [33] the (strict) semilocally pseudoconvexity is defined at a point with respect to a set.

A number of properties of these functions were given by Kaul and Kaur [22, 23], Suneja and Gupta [33] (see Mahajanm and Vartak [25] for the semilocally convex case). By using these concepts for a scalar valued nonlinear program- ming problem in Kaul and Kaur [21–23] and Suneja and Gupta [33], some optimality conditions and duality results were obtained. These results were extended in [28] for a multiple objective programming problem.

Preda and Stancu-Minasian [31] stated the Fritz John and Karush-Kuhn- Tucker optimality conditions for veak vector minima usingη-semidifferentials and functions satisfying generalized semilocally preinvex properties. These results were then used to extend the Wolfe [39] and Mond-Weir [27] duals.

Also, some results of Preda [28], Preda and Stancu-Minasian and Batatorescu [30] and Suneja and Sudha Gupta [33] were generalized.

In Lyall, Suneja and Aggarwal [24], some necessary and sufficient op- timality conditions for a fractional programming problem with semilocally convex, semilocally quasiconvex and semilocally pseudoconvex fucntions were stated. Also, a dual program and duality results of weak and strong duality were proved for the pair of primal and dual programs.

Preda [29] considered necessary and sufficient optimality conditions for a nonlinear fractional multiple objective programming problem involving η- semidifferentiable functions. Also, a general dual was formulated and duality

results were proved using concepts of generalized semilocally preinvex func- tions. Thus, many results of Lyall, Suneja and Aggarwal [24], Preda [28], Preda, Stancu-Minasian and Batatorescu [30], Preda and Stancu-Minasian [31] and Suneja and Gupta [33] were generalized.

In this paper we give necessary and sufficient optimality conditions for a nonlinear fractional multiple objective programming problem involving η- semidifferentiable functions. Also, a general dual is formulated and duality results are proved using concepts of generalized ρ-semilocally preinvex func- tions. Thus, many results of Preda [29] are generalized.

2. DEFINITIONS AND PRELIMINARIES Forx, y ∈Rⁿ, byx <

= y we mean xi <

= yi for all i,x≤y meansxi <

= yi

for all iand x_j < y_j for at least one j, 1 <

= j <

= n. By x < y we meanx_i < y_i for alliand by x≤y the negation ofx≤y.

LetX₀⊆Rⁿ be a set andη:X₀×X₀ →Rⁿ a vectorial application.

We say that the set X₀ is η-vex at x∈X₀ ifx+λη(x, x) ∈X₀ for any x∈ X₀ and λ∈[0,1]. We say that the set X⁰ is η-vex if X₀ is η-vex at any x∈X₀.

We remark that ifη(x, x) =x−xfor any x∈X₀, thenX₀ isη-vex atx iffX₀ is a convex set atx.

Definition 2.1. We say that the set X₀ ⊆Rⁿ is an η-locally starshaped set at x, x ∈ X₀, if for any x ∈ X₀ there exists 0 < aη(x, x) <

= 1 such that x+λη(x, x)∈X₀ for any λ∈[0, aη(x, x)].

Letρ∈Rⁿ andd:X₀×X₀→R such thatd(x, y)= 0 if x=y.

Definition 2.2. Let f :X₀ → Rⁿ be a function, where X₀ ⊆ Rⁿ is an η-locally starshaped set at x∈X₀. We say thatf is:

(i₁) ρ-semilocally preinvex (ρ-slpi) at x if, corresponding to x and each x ∈ X₀, there exists a positive number dη(x, x) <

= aη(x, x) such that f(x+ λη(x, x))<

= λf(x) + (1−λ)f(x)−ρλd²(x, x) for 0< λ < dη(x, x);

(i₂)ρ-semilocally quasi-preinvex (ρ-slqpi) atxif, corresponding toxand each x ∈ X₀, there exists a positive number d_η(x, x) <

= a_η(x, x) such that f(x)<

= f(x) and 0< λ < d_η(x, x) impliesf(x+λη(x, x))<

= f(x)−ρλd²(x, x).

Ifρ= 0 in the above definition,f is said to be semilocally preinvex (slpi) at x, respectively semilocally quasi-preinvex (slqpi) atx [28].

Definition 2.3. Let f : X₀ → Rⁿ be a function, where X₀ ⊆ Rⁿ is an η-locally starshaped set at x ∈ X₀. We say that f is η-semidifferentiable

at x if (df)⁺(x, η(x, x)) exists for each x ∈ X₀, where (df)⁺(x, η(x, x)) =

λlim→0+

1λ[f(x+λη(x, x))−f(x)] (the right derivative at x along the direction η(x, x))).

If f is η-semidifferentiable at any x ∈ X₀, then f is said to be η- semidifferentiable onX₀.

Remark2.1. Ifη(x, x) =x−x,theη-semidifferentiability is the semidiffer- entiability notion. As shown in [24], if a function is directionally differentiable, then it is semidifferentiable but the converse is not true.

Theorem 2.1. Let f : X₀ → Rⁿ be an η-semidifferentiable function at x ∈ X₀. If f is ρ-slqpi at x and f(x) <

= f(x), then (df)⁺(x, η(x, x)) <

−ρd²(x, x). =

Definition 2.4. We say that f isρ-semilocally pseudo-preinvex (ρ-slppi) at xif for anyx∈X₀,(df)⁺(x, η(x, x))>

= −ρd²(x, x)⇒f(x)>

=f(x).

Iff is ρ-slppi at any x∈X₀, thenf is said to be ρ-slppi on X₀.

If ρ = 0 in the above definition, f is said to be semilocally pseudo- preinvex (slppi) [28].

Definition 2.5. Let X and Y be two subsets of X₀ and y ∈ Y. We say that Y is η-locally starshaped at y with respect to X if for any x ∈X there exists 0< a_η(x, x)<

=1 such thaty+λη(x, y)∈Y for any 0<

= λ <

= a_η(x, y).

Definition2.6. Let beη-locally starshaped aty with respect toX andf be anη-semidifferentiable function at y. We say thatf is:

(i₁)ρ-slppi aty∈Y with respect toXif, for anyx∈X,(df)⁺(y, η(x, y))>

−ρd²(x, x)⇒f(x)> =

= f(y);

(i₂)ρ-strictly semilocally pseudo-preinvex (ρ-sslppi) atywith respect to X,if for each x∈X, x=y,(df)⁺(y, η(x, y))>

=−ρd²(x, x)⇒f(x)> f(y).

We say thatf is (ρ-slppi)ρ-sslppi onY with respect toXiff is (ρ-slppi) ρ-sslppi at any point ofY with respect toX.

Definition 2.7 (Elster, Nehse [9]). A functionf :X₀ →R^k is said to be convexlikeif for any x, y∈X₀ and 0<

= λ <

= 1 there isz∈X₀ such that f(z)<

=λf(x) + (1−λ)f(y).

Remark. The convex and the preinvex functions are convexlike functions.

Lemma2.1 (Hayashi, Komiya [13]). Let S be a nonempty set inRⁿ and ψ:S →R^k a convexlike function. Then either

ψ(x)<0 has a solution x∈S

or

λ^Tψ(x)>

= 0 for allx ∈S, for some λ∈R^k, λ≥0, but both alternatives are never true. (Here the symbol ^T stands for the transpose of a matrix.)

Using our Lemma 2.1 instead of Lemma 2.9 from [31], we get that Theo- rems 3.4 and 3.5 stated there are still true. Thus, in the next section we’ll use the following version of Theorem 3.5 from [31].

Theorem 2.2. Let x ∈ X be a (local) weak minimum solution for the problem

minimize (ϕ₁(x), . . . , ϕ_p(x)) subject to

hj(x)<

=0, j∈M x∈X₀,

where ϕ = (ϕ₁, . . . , ϕ_p) : X₀ → R^p and h₁, . . . , h_m are η-semidifferentiable at x. Also, assume that h_j, j ∈ N(x) is a continuous function at x while (dϕ)⁺(x, η(x, x)) and (d)⁺(x, η(x, x)) are convexlike functions of x on X₀. If hsatisfies a regularity condition atx(see[31], Definition3.2), then there exist λ⁰ ∈R^p, u⁰ ∈R^m such that

λ^0T(dϕ)⁺(x, η(x, x)) +u^0T(dh)⁺(x, η(x, x))>

= 0 for allx∈X₀, u^0Th(x) = 0, h(x)<

= 0, λ^0Te= 1, λ⁰ ≥0, u⁰ >

= 0, where e= (1, . . . ,1)^T ∈R^p.

3. NECESSARY OPTIMALITY CONDITIONS

In this paper we consider the multiobjective nonlinear fractional pro- gramming problem

(VFP)

minimize _f

1(x)

g₁(x), . . . ,^f_g^p(x)

p(x)

subject to:

h_j(x)<

= 0, j= 1,2, . . . , m x∈X₀,

whereX₀ ⊆Rⁿ is a nonempty set and g_i(x) >0 for all x ∈X₀ and each i= 1, . . . , p.Let f = (f₁, . . . , fp), g= (g₁, . . . , gp), h= (h₁, . . . , hm). We putX = {x∈X₀ |h_j(x)<

= 0, j= 1,2, . . . , m} for the feasible set of problem (VFP).

Definition3.1. A pointx∈Xis said to be a weak minimum for problem (VFP) if there exists no other feasible pointx for which ^f_g(⁽_x^x₎⁾ > ^f_g(^(x_x)⁾.

For x ∈ X we putM(x) ={j ∈ M |hj(x) = 0}, h⁰ = (hj)_j∈M(x) and N(x) =M \M(x),where M ={1,2, . . . , m}.

Definition 3.2. We say that (VFP) satisfies the generalized Slater’s con- straint qualification (GSCQ) atx∈Xifh⁰ is slppi atxand there existsx∈X such thath⁰(x)<0.

Lemma 3.1 [29]. Let x ∈ X be a (local) weak minimum solution to (VFP). Further, assume that hj is continuous at x for any j∈N(x) and that f, g, h⁰ areη-semidifferentiable at x. Then the system

(3.1)







(df)⁺(x, η(x, x))<0 (dg)⁺(x, η(x, x))>0 (dh⁰)⁺(x, η(x, x))<0 has no solutionx∈X₀.

Theorem3.1 (Fritz-John Type Necessary Optimality Criteria) [29]. Let us suppose that h_j is continuous at x for j ∈ N(x), (df)⁺(x, η(x, x)), (dg)⁺(x, η(x, x)) and (dh⁰)⁺(x, η(x, x)) are convexlike functions of x on X₀. If x is a (local) weak minimum solution to (VFP), then there exist λ⁰ ∈R^p, u⁰ ∈R^p, v⁰ ∈R^m such that

(3.2) λ^0T(df)⁺(x, η(x, x))−u^0T(dg)⁺(x, η(x, x))+

+v^0T(dh⁰)⁺(x, η(x, x))>

= 0for all x∈X₀,

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

(3.4) (λ⁰, u⁰, v⁰)= 0, (λ⁰, u⁰, v⁰)>

= 0.

The next theorem is a Karush-Kuhn-Tucker type necessary optimality criterium. In this theorem, the above defined generalized constraint qualifica- tion is very important.

For each u= (u₁, . . . , u_p) ∈R^p₊, whereR^p₊ denotes the positive orthant ofR^p,consider the problem

(VFP_u)

minimize (f₁(x)−u₁g₁(x), . . . , f_p(x)−u_pg_p(x)) subject to:

hj(x)<

= 0, j∈M x∈X₀.

The following result can be proved without difficulty.

Lemma 3.1 [29]. If x is a (local) weak minimum for (VFP), then x is a (local) weak minimum for (VFP_u0), where u⁰ = ^f_g⁽₍^x_x⁾₎.

Using Lemma 3.1 we can derive a Karush-Kuhn-Tucker type necessary optimality criterium for the problem (VFP).

Theorem 3.2 (Karush-Kuhn-Tucker Type Necessary Optimality Cri- terium) [29]. Let x be a (local) weak minimum solution for (VFP), let h_j be continuous at x for j ∈N(x) and let (dfi)⁺(x, η(x, x)), (dgi)⁺(x, η(x, x)), i∈P and (dh⁰)⁺(x, η(x, x))be convexlike functions of x on X₀.If g satisfies (GSQ) atx, then there exist λ⁰∈R^p₊, u⁰ ∈R^p₊, v⁰∈R^m such that

p i=1

λ⁰_i

(dfi)⁺(x, η(x, x))−u⁰_i(dgi)⁺(x, η(x, x)) + +v^0T(dh)⁺(x, η(x, x))>

= 0for all x∈X₀, v^0Th(x) = 0, h(x)<

= 0, λ^0Te= 1, λ⁰ ≥0, u⁰>

= 0, v⁰ >

= 0, where e= (1, . . . ,1)^T ∈R^p.

Remark. In Theorem 3.1 we can assume, for any i ∈ P, that (df_i)⁺(x, η(x, x))−u⁰_i(dgi)⁺(x, η(x, x)) is convexlike onX₀, whereu⁰_i = ^f_g^i(x)

i(x) instead of assuming that (df_i)⁺(x, η(x, x)) and (dg_i)⁺(x, η(x, x)) are convexlike on X₀, for any i∈P.

4. SUFFICIENT OPTIMALITY CRITERIA

In this section, using the concept of (local) weak optimality, we give some sufficient optimality conditions for problem (VFP).

Theorem 4.1. Let x ∈ X. Assume f is ρ-η-semilocally convex at x, g is σ-η-semilocally concave at x, and h is τ-η-semilocally convex at x. Also, assume that there existλ⁰∈R^p, u⁰ ∈R^p and v⁰ ∈R^m such that

p i=1

λ⁰_i

(df_i)⁺(x, η(x, x))

+v^0T(dh)⁺(x, η(x, x))>

=0 for all x∈X, (4.1)

(dg_i)⁺(x, η(x, x)) +σ_id²(x, x)<

= 0, ∀x∈X, ∀i∈P, (4.2)

v^0Th(x) = 0, (4.3)

h(x)<

= 0, (4.4)

λ^0Te= 1, (4.5)

λ⁰ ≥0, u⁰ >

= 0, v⁰ >

=0, (4.6)

p i=1

λ⁰_iρi+

m j=1

v⁰_jτj >

= 0.

(4.6)

Thenx is a weak minimum solution for (VFP).

Proof. We proceed by contradiction. Hence there existsx∈X such that

(4.7) fi(x)

g_i(x) < fi(x) g_i(x)

for any i ∈ P. Since f is ρ-η-semilocally convex at x, g is σ-η-semilocally concave atx,and h isτ-η-semilocally convex at x, we get

fi(x) −fi(x)>

= (dfi)⁺(x, η(x, x)) + ρid²(x, x), i∈P, (4.8)

gi(x)−gi(x)<

= (dgi)⁺(x, η(x, x)) +σid²(x, x), i∈P, (4.9)

hj(x) −hj(x)>

= (dhj)⁺(x, η(x, x)) +τjd²(x, x), j∈M.

(4.10)

Multiplying (4.8) by λ⁰_i >

= 0, i∈ P, λ⁰ ∈R^p₊, (4.10) by v⁰_j >

= 0, j ∈M, and then summing the relations obtained, we get

p

i=1λ⁰_i (fi(x)−fi(x))+

m

j=1v⁰_j(hj(x)−hj(x))>

=

p

i=1λ⁰_i(dfi)⁺(x, η(x, x))+

+ m

j=1v_j⁰(dhj)⁺(x, η(x, x)) + p

i=1λ⁰_iρi+ m j=1v_j⁰τj

d²(x, x) >

=0, where the last inequality holds by (4.1) and (4.6). Hence

(4.11)

p i=1

λ⁰_i (f_i(x)−f_i(x)) +

m j=1

v⁰_jh_j(x)−v^0Th(x)>

= 0.

Since x∈X, v⁰ >

= 0, by (4.3) and (4.11) we get (4.12)

p i=1

λ⁰_i (fi(x)−fi(x))>

=0.

Using (4.5), (4.6) and (4.12), we obtain that there existsi₀ ∈P such that

(4.13) fi₀(x)>

=fi₀(x).

It follows by (4.2) and (4.9) that

(4.14) g_i(x)<

=g_i(x), i∈P.

Now, using (4.13), (4.14) and taking into account that f >

= 0, g > 0, we obtain

fi₀(x) g_i0(x) >

=

fi₀(x) g_i0(x),

which contradicts (4.7). Thus, the theorem is proved andxis a weak minimum solution to (VFP).

Corollary 4.1. Let x ∈ X and assume that there exist λ⁰ ∈ R^p, u⁰ ∈ R^p and v⁰ ∈ R^m such that

p

i=1λ⁰_ifi(·) + ^m

j=1v_j⁰hj(·) is ρ₀-η-semilocally convex at x, ρ₀ ∈R₊,g is σ-η-semilocally concave at x,and (4.2)–(4.6) hold.

Thenx is a weak minimum solution to (VFP).

Theorem4.2. Let x ∈X and u⁰_i = ^f_g^i(x)

i(x),i∈P.Assume f isρ-η-semi- locally convex atx, g isσ-η-semilocally concave atx, andh isτ-η-semilocally convex at x. Also, assume that there existλ⁰∈R^p, and v⁰ ∈R^m such that

p i=1λ⁰_i

(df_i)⁺(x, η(x, x))−u⁰_i(dg_i)⁺(x, η(x, x)) + +v^0T(dh)⁺(x, η(x, x))>

= 0 for allx∈X, (4.15)

v^0Th(x) = 0, (4.16)

h(x)<

= 0, (4.17)

λ^0Te= 1, (4.18)

λ⁰ ≥0, u⁰ >

= 0, v⁰ >

=0, (4.19)

p i=1

λ⁰_iρ_i−

p i=1

λ⁰_iu⁰_iσ_i+

m j=1

v_j⁰τ_j >

=0.

(4.19)

Thenx is a weak minimum solution to(VFP).

Proof.We proceed by contradiction. Ifxis not a weak minimum solution to (VFP), then there exists x∈X such that

fi(x)

gi(x) < fi(x) gi(x) for any i∈P,i.e., f_i(x) < u⁰_ig_i(x) for any i∈P.

By theρ-η-semilocally convexity off at x, theτ-η-semilocally convexity ofh at x and the σ-η-semilocally concavity of gat x, we have

fi(x) −fi(x)>

= (dfi)⁺(x, η(x, x)) + ρid²(x, x), i∈P, g_i(x)−g_i(x)<

= (dg_i)⁺(x, η(x, x)) +σ_id²(x, x), i∈P, hj(x)−hj(x)>

= (dhj)⁺(x, η(x, x)) +τid²(x, x), j∈M.

Using these inequalities and (4.19), we get

p i=1

λ⁰_i (f_i(x) −f_i(x))−

p i=1

λ⁰_iu⁰_i (g_i(x)−g_i(x)) +

m j=1

v⁰_j(h_j(x)−h_j(x))>

=

>= p i=1

λ⁰_i

(dfi)⁺(x, η(x, x))−u⁰_i(dgi)⁺(x, η(x, x)) +

m j=1

v_j⁰(dhj)⁺(x, η(x, x))+

+



 ^p

i=1

λ⁰_iρi−

p i=1

λ⁰_iu⁰_iσi+

m j=1

v_j⁰τj



d²(x, x)>

= 0, where the last inequality holds by (4.15) and (4.19). Therefore,

p i=1

λ⁰_i





f_i(x)−u⁰_ig_i(x)

−

f_i(x)−u⁰_ig_i(x)

=0



+

m j=1

v⁰_jh_j(x) −v^0Th(x)>

= 0.

Sinceu⁰_i = ^f_g^i(x)

i(x),i∈P,we obtain

p i=1

λ⁰_i

f_i(x) −u⁰_ig_i(x)

+v^0Th(x) −v^0Th(x)>

= 0.

Now,x∈X, (4.16) and (4.19) yield

p i=1

λ⁰_i

fi(x)−u⁰_igi(x)

>= 0.

Sinceλ⁰_i >

= 0, λ^0Te= 1,there existsi₀ ∈P such thatfi₀(x)−u⁰_i0gi₀(x)>

0,i.e., =

f_i0(x) g_i0(x) >

=

f_i0(x) g_i0(x)

which contradicts (4.20). Hencex is a weak minimum solution to (VFP) and the proof is complete.

Corollary 4.2. Let x ∈ X and u⁰_i = ^f_g^i(x)

i(x). Assume that there exist λ⁰ ∈ R^p and v⁰ ∈R^m such that conditions (4.15)–(4.19) of Theorem 4.2 are satisfied and

p

i=1λ⁰_i(fi(·)−u⁰_igi(·)) + m

j=1v⁰_jhj(·) isρ₀-η-semilocally convex atx, andρ₀∈R₊. Then x is a weak minimum solution to (VFP).

Theorem4.3. Letx∈X, u⁰_i = ^f_g^i(x)

i(x), i∈P,andλ⁰ ∈R^p, v⁰ ∈R^m such that conditions (4.15)−(4.19) of Theorem 4.2 hold. Moreover, assume that for anyj∈M, fi(·)−u⁰_igi(·) isρi-η-semilocally pseudoconvex, for any i∈P,

h_j(·) is τ_j-η-semilocally quasiconvex at x and p i=1λ⁰_iρ_i+

m

j=1v⁰_jτ_j >

= 0. Then x is a weak minimum solution for (VFP).

Proof. Suppose thatxis not a weak minimum solution for (VFP). Then there exists x∈X such that

fi(x)

gi(x) < fi(x) gi(x)

for any i∈P, i.e., fi(x) −u⁰_igi(x)<0 for any i∈P,which is equivalent to f_i(x)−u⁰_ig_i(x) < f_i(x)−u⁰_ig_i(x)

for any i∈P.

Now, by the ρ_i-η-semilocally pseudoconvexity of f_i(·) −u⁰_ig_i(·) at x we have

(dfi)⁺(x, η(x, x)) −u⁰_i(dgi)⁺(x, η(x, x)) <−ρid²(x, x) for any i∈P. Asλ⁰_i ∈R^p₊,e^Tλ₀ = 1, we obtain

(4.20)

p i=1

λ⁰_i

(df_i)⁺(x, η(x, x)) −u⁰_i(dg_i)⁺(x, η(x, x))

<−

p i=1

λ⁰_iρ_id²(x, x).

For x ∈ X we have h(x) <

= 0. But for j ∈ M(x), h_j(x) = 0. Hence h_j(x)<

= h_j(x) for anyj∈M(x).Now, by theτ_j-η-semilocally quasiconvexity of h_j at x we have (dh_j)⁺(x, η(x, x)) <

= −τ_jd²(x, x) for any j ∈ M(x). But v⁰ ∈R^m₊ and v_j⁰ = 0 forj∈N(x), so that we have

(4.21)

m j=1

v_j⁰(dhj)⁺(x, η(x, x))<

=−

m j=1

v⁰_jτjd²(x, x).

Now, by (4.20), (4.21) and p i=1λ⁰_iρ_i+

m j=1v_j⁰τ_j >

= 0 we have p

i=1λ⁰_i

(df_i)⁺(x, η(x, x))−u⁰_i(dg_i)⁺(x, η(x, x)) + +

m

j=1v⁰_j(dh_j)⁺(x, η(x, x)) <− p

i=1λ⁰_iρ_i+ m j=1v_j⁰τ_j

d²(x, x) <0

which contradicts (4.15). Hence x is a weak minimum for (VFP) and the theorem is proved.

Corollary 4.3. Let x ∈X, u⁰_i = ^f_g^i(x)

i(x),i∈ P, and λ₀ ∈R^p, v⁰ ∈R^m such that conditions(4.15)–(4.19) hold. If

p

i=1λ⁰_i(fi(·)−u⁰_igi(·)) +^m

j=1v_j⁰hj(·) is ρ₀-η-semilocally pseudoconvex at x and ρ₀ ∈R₊,then x is a weak minimum solution for (VFP).

5. DUALITY

Consider, for (VFP) a general Mond-Weir dual (FMWD) defined as maximizeψ(y, λ, u, v) =u−v^T_I0h_I0(y)e subject to (FMWD)

p

i=1λi((dfi)⁺(y, η(x, y))−ui(dgi)⁺(y, η(x, y))) + +v^T(dh)⁺(y, η(x, y))>

=0 for all x∈X, (5.1)

fi(y)−uigi(y)>

= 0 for any i∈P, (5.2)

v_I^T_Sh_IS(y)>

= 0 (1<

=s <

= γ), (5.3)

λ^Te= 1, λ≥0, λ∈R^p, u >

=0, u∈R^p, v >

= 0, y∈X₀, (5.4)

where γ >

= 1, Is ∩It = ∅ for s = t and γ

s=0Is = M. (Here vI_s = (vj)j∈I_s, h_Is = (h_j)_j∈Is.)

Let W denote the set of all feasible solutions to (FMWD). Also, define the sets

A={(λ, u, v)∈R^p×R^p×R^m |(y, λ, u, v)∈W for somey∈X₀} andB(λ, u, v) = {y ∈ X₀ | (y, λ, u, v) ∈ W} for (λ, u, v) ∈ A. Put B =

(λ,u,v)∈A

B(λ, u, v) and note that B ⊂ X₀. Also, note that if (y, λ, u, v) ∈ W then (λ, u, v)∈A and y∈B(λ, u, v).

Now, we establish duality results between (VFP) and (FMWD). Assume thatf, g andh are η-semidifferentiable on X.

Theorem 5.1 (Weak Duality). Assume that for all feasible solutions x∈X and(y, λ, u, v)∈W to(VFP) and (FMWD), respectively, we have

(i₁) v_I^T_shI_s(y), for 1 <

= s <

= γ, are τs-slppi on B(λ, u, v). Assume either of (i₂), (i₃) and (i₄) below holds:

(i₂) f_i(·)−u_ig_i(·) +v^T_I0h_I0(·) isρ_i-η-sslppi at y on B(λ, u, v) for alli∈P, and

p i=1λiρi+

γ s=1τs>

= 0;

(i₃) λ > 0 and p

i=1λ_i(f_i(·)−u_ig_i(·)) + v_I^T₀h_I0(·) is ρ₀-slppi on B(λ, u, v), and ρ₀+

γ s=1τ_s >

= 0;

(i₄) p

i=1λ_i(f_i(·)−u_ig_i(·)) +v_I^T₀h_I0(·) is ρ₀-η-sslppi on B(λ, u, v), and ρ₀+ γ

s=1τ_s>

=0.

Then(5.5) and(5.6) below cannot hold:

f_i(x)−u_ig_i(x)<

= v_I^T₀h_I0(y) for any i∈P, (5.5)

f_i0(x)−u_i0g_i0(x)< v^T_I0h_I0(y) for some i₀ ∈P.

(5.6)

Proof.Using the feasibility ofxfor (VFP) and of (y, λ, u, v) for (FMWD), we obtain

(5.7) v_I^T_shI_s(x)<

= v^T_IshI_s(y), for all s, 1<

= s <

= γ.

By (5.7) and (i₁) we have (5.8)

j∈I_s

vj(dhj)⁺(y, η(x, y))<

=−τsd²(x, y), 1<

= s <

= γ.

Now, assume for a contradiction that (5.5) and (5.6) hold. Hence if (5.5) and (5.6) hold for some feasible x for (VFP) and (y, λ, u, v) feasible for (FMWD), we have

(5.9) fi(x)−uigi(x)<

= v_I^T₀hI₀(y) for any i∈P and

(5.10) fi₀(x)−ui₀gi₀(x)< v_I^T₀hI₀(y) for somei₀ ∈P.

According to (5.2), (5.4) and the feasibility ofx for (VFP), we have (5.11) v_I^T₀h_I0(x)<

=0<

=f_i(y)−u_ig_i(y) for alli∈P.Combining (5.9)–(5.11) we get

(5.12) fi(x)−uigi(x) +v_I^T₀hI₀(x)<

= fi(y)−uigi(y) +v^T_I0hI₀(y) for alli∈P and

(5.13) f_i0(x)−u_i0g_i0(x) +v_I^T₀h_I0(x)< f_i0(y)−u_i0g_i0(y) +v^T_I0h_I0(y) for somei₀ ∈P.

Now, if (i₂) holds, then by (5.12) and (5.13) we have (5.14)

(df_i)⁺(y, η(x, y))−u_i(dg_i)⁺(y, η(x, y))+

+

j∈I₀

v_j(dh_j)⁺(y, η(x, y))<−ρ_id²(x, y) for any i∈P. By (5.14) and (5.4) we get

p i=1

λi

(dfi)⁺(y, η(x, y))−ui(dgi)⁺(y, η(x, y)) + +

j∈I₀

vj(dhj)⁺(y, η(x, y))<−

p i=1

λiρid²(x, y).

Now, by (5.1) and p i=1λ_iρ_i+

γ s=1τ_s>

= 0 we have

γ s=1j∈I_s

vj(dhj)⁺(y, η(x, y))>

p i=1

λiρid²(x, y)>

= −

γ s=1

τsd²(x, y), which contradicts (5.8). Thus, in case (i₂) holds the theorem is proved.

Now, assume (i₃) holds. Sinceλ_i>0 for anyi,if (5.5) and (5.6) hold for some feasiblex for (VFP) and (y, λ, u, v) feasible for (FMWD), then

(5.15)

p i=1

λ_i(f_i(x)−u_ig_i(x))< v^T_I0h_I0(y).

Also, we have (5.16)

p i=1

λ_i(f_i(y)−u_ig_i(y))>

= 0 and

(5.17) v_I^T₀hI₀(x)<

= 0.

From (5.15)–(5.17) we have

p i=1

λi(fi(x)−uigi(x)) +v_I^T₀hI₀(x)<

p i=1

λi(fi(y)−uigi(y)) +v_I^T₀hI₀(y).

Now, by (i₃) we have

(5.18)

p i=1

λ_i

(df_i)⁺(y, η(x, y))−u_i(dg_i)⁺(y, η(x, y)) + +

j∈I₀

vj(dhj)⁺(y, η(x, y))<−ρ₀d²(x, y).