( F , G ) -CONVEXITY G LOBAL J OURNALOF E NGINEERING S CIENCEAND R ESEARCHESSECOND-ORDERMULTIOBJECTIVESYMMETRICPROGRAMMINGPROBLEMANDDUALITYRELATIONSUNDER

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

G LOBAL J OURNAL OF E NGINEERING S CIENCE AND R ESEARCHES SECOND-ORDER MULTIOBJECTIVE SYMMETRIC PROGRAMMING PROBLEM

AND DUALITY RELATIONS UNDER (F, G

_f

)-CONVEXITY Ramu Dubey

¹

, Vandana

^∗2

and Vishnu Narayan Mishra

³

1

Department of Mathematics, Central University of Haryana, Jant-Pali-123029, India

∗2

Department of Management Studies, Indian Institute of Technology, Madras, Chennai -600 036, Tamil Nadu, India

3

Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh-484887, India

ABSTRACT

In this paper, we introduce the definition (F, G_f)-convexity and give a nontrivial example existing such type of func- tions. Further, a generalization of convexity, namely (F, G_f)-convexity, is introduced in the case of non-linear multi- objective programming problems where the functions constituting vector optimization problems are differentiable.

Further, second-order Wolfe type primal-dual pair has been formulated and proved weak, strong and converse duality theorems under (F, G_f)-convexity assumptions.

Keywords: Symmetric duality, Non-linear programming, (F, G_f)-convexity; Wolfe type model

I. INTRODUCTION

An Optimization problem is the problem of finding the best solution from all feasible solutions. Generally, all problems to be optimized should be able to be formulated as a system with its status controlled by one or more input variables and its performance specified by a well defined Objective Function.The goal of optimization is to find the best value for each variable in order to achieve satisfactory performance.Optimization is an active and fast growing research area and has a great impact on the real world.

One practical advantage of second order duality is that it provides tighter bounds for the value of the objective function of the primal problem when approximations are used because there are more parameters involved. Duality is one of the most important topic in operation research as it helps us in generating useful insights about the optimization problem. Mangasarian [11] is introduced the concept of second-order duality for nonlinear programming . Two dis- tinct pairs of second-order symmetric dual problems under generalized bonvexity/ boncavity assumptions studied in Gulati et al. [7]. Furthermore, Gulati and Gupta [8] have been introduced the concept of η1-bonvexity/ η2-boncavity and derived duality results for a Wolfe type model.

The concept of G-invex function has been introduced by Antczak [1] and further derived some optimality conditions for constrained optimization problem. Extended the above notion by defining a vector valued G_f- invex function, Antzcak [2] proved necessary and sufficient optimality conditions for a multiobjective nonlinear program- ming problem. In last several years, various optimality and duality results have been obtained for multiobjective fractional programming problems. In Chen [3], multiobjective fractional problem and its duality theorems have been considered under higher-order (F, α, ρ, d)- convexity. Later on, Suneja et al. [13] discussed higher-order Mond-Weir and Schaible type nondifferentiable dual programs and their duality theorems under higher-order (F, ρ, σ ) -type I- assumptions. Recently, several researchers like (see[5, 6, 14]) have also worked in the same direction.

In this article, we have introduce the definition of (F, G_f)-convex function. Further, we construct a nontrivial numerical example which is (F, G_f)-convex but it is neither second-order F-convex nor F-convex. Also, we have consider Wolfe type multiobjective second-order symmetric dual program and establish the duality relations under

187

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

(F, G_f)- convexity assumptions.

II. NOTATIONS AND PRELIMINARIES

Consider the following vector minimization problem:

(MP) Minimize f (x) =



f₁(x), f₂(x), ..., f_k(x)

T

Subject to X⁰= {x ∈ X ⊂ Rⁿ: g_j(x) ≤ 0, j = 1, 2, ..., m}

where f = { f₁, f₂, ..., f_k} : X → R^kand g = {g₁, g₂, ..., g_m} : X → R^mare differentiable functions defined on X . Definition 2.1 A point ¯x∈ X⁰ is said to be an efficient solution of (MP) if there exists no other x ∈ X⁰such that

f_r(x) < f_r( ¯x), for some r = 1, 2, ..., k and fi(x) ≤ f_i( ¯x), for all i = 1, 2, ..., k.

Definition 2.2. A functional F : X × X × Rⁿ→ R is said to be sublinear with respect to the third variable if for all (x, u) ∈ X × X ,

(i) F_x,u(a₁+ a₂) ≤ F_x,u(a₁) + F_x,u(a₂), for all a1, a₂∈ Rⁿ, (ii) F_x,u(αa) = αFx,u(a), for all α ∈ R+ and a ∈ Rⁿ.

Many generalizations of the definition of a convex function have been introduced in optimization theory in order to weak the assumption of convexity for establishing optimality and duality results for new classes of nonconvex opti- mization problems, including vector optimization problems. One of such a generalization of convexity in the vectorial case is the G-invexity notion introduced by Antczak for differentiable scalar and vector optimization problems (see [1, 2] respectively). We now generalize and extend it to the nondifferentiable vectorial case namely, motivated by Jeyakumar and Mond [12] and Antczak [2], we introduce the concept of (F, G_f)-convex.

Definition 2.3. Let f : X → R^k be a vector-valued differentiable function. If there exist a sublinear functional F and differentiable function G_f = (G_f1, G_f2, ..., G_fk) : R → R^ksuch that every component G_fi : I_fi(X ) → R is strictly increasing on the range of I_fisuch that ∀ x ∈ X and p ∈ Rⁿ,

G_fi( f_i(x)) − G_fi( f_i(u)) ≥ F_x,u[G⁰_f

i( f_i(u))∇xf_i(u) + {G⁰⁰_f

i( f_i(u))∇xf_i(u)(∇xf_i(u))^T+ G⁰_f

i( f_i(u))∇xxf_i(u)}p]

−1 2p^T[G⁰⁰_f

i( f_i(u))∇xf_i(u)(∇xf_i(u))^T+ G⁰_f

i( f_i(u))∇xxf_i(u)]p, for all i = 1, 2, ..., k, then f is called (F, G_f)-convex at u ∈ X .

If the above inequality sign changes to ≤, then f is called (F, G_f)-concave at u ∈ X with respect to η.

If the function f_i, i = 1, 2, ..., k satisfies above inequality, then we will say that fiis (F, G_fi)-convex at u ∈ X . If the above inequalities sign changes to ≤, then f is called (F, G_f)-concave at u ∈ X .

Remark 2.1 If Fx,u(a) = η^T(x, u)a, then Definition 2.3 becomes Gf-bonvex given by [9].

Now, we give a nontrivial example which is (F, G_f)-convex function but not F-convex function.

Example 2.1. Let f : [−1, 1] → R²be defined as

f(x) = { f₁(x), f₂(x)},

188

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

Figure 1:

189

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

Figure 2:

where f₁(x) = x³, f₂(x) = x⁴and G_f = {G_f1, G_f2} : R → R²be defined as:

G_f1(t) = t²+ 1, Gf₂(t) = t⁴. Let F : X × X × R²→ R be given as:

F_x,u(a) = |a|(x²− u²).

For showing that f is (F, G_f)-convex at u = 0, for this we have to claim that πi= G_fi( f_i(x)) − G_fi( f_i(u)) − F_x,u[G⁰_f

i( f_i(u))∇xf_i(u) + {G⁰⁰_f

i( f_i(u))∇xf_i(u)(∇xf_i(u))^T+ G⁰_f

i( f_i(u))∇xxf_i(u)}p_i] +1 2p^T_i[G⁰⁰_f

i( f_i(u))∇xf_i(u)(∇xf_i(u))^T+ G⁰_f

i( f_i(u))∇xxf_i(u)]p_i≥ 0, for i = 1, 2.

Putting the values of f₁, f₂, G_f1 and G_f2 in the above expressions, we have π1= (x⁶+ 1) − (u⁶+ 1) − Fx,u[6u⁵+ {18u⁴+ 6u⁵}p₁] +1

2p²₁{18u⁴+ 6u⁵},

190

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

Figure 3:

and

π₂= x¹⁶− u¹⁶− −Fx,u(16u¹⁵+ {12 × 16u¹⁷+ 48u¹⁴}p₂) +1

2p²₂{12 × 16u¹⁷+ 48u¹⁴}, At u = 0 ∈ [−1, 1], the above expressions reduces:

π1= x⁶and π2= x¹⁶for all x ∈ [−1, 1]

and hence π1≥ 0 and π2≥ 0,



from figures (1) and (2)



, for all x ∈ [−1, 1].

Therefore, f is (F, G_f)-convex at u = 0.

Now, suppose

π3= f₁(x) − f₁(u) − F_x,u[∇xf₁(u) + ∇xxf₁(u)p₁] +1

2p^T₁[∇xxf₁(u)]p₁. or

π3= x³− u³− F_x,u[3u²+ 6up1] + 3up²₁ which at u = 0 yields

π3= x³, for all x ∈ [−1, 1].

191

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

Obviously,

π3 0,



from figure (3)

 .

Therefore, f₁is not second-order F-convex at u = 0 with respect to p. Hence, f = ( f₁, f₂) is not second-order F- convex at u = 0 with respect to p.

Finally, consider

ξ = f₁(x) − f₁(u) − Fx,u(∇xf₁(u)) or

ξ = x³− u³− F_x,u(3u²) which at u = 0 and using sublinearity of functional F, we get

ξ = x³. At the point x =⁻¹₃ ∈ [−1, 1], obtain

ξ =−1 27  0.

Therefore, f₁is not F-convex at u = 0. Hence, f = ( f₁, f₂) is not F-convex at u = 0.

III. WOLFE TYPE SYMMETRIC DUAL PROGRAM Consider the following pair of Wolfe type dual program:

Primal problem (WP):

Minimize R(x, y, λ , p) =



R₁(x, y, λ1, p), R₂(x, y, λ2, p), ...R_k(x, y, λk, p)

T

Subject to

k i=1

∑

λi



G⁰_fi( f_i(x, y))∇yfi(x, y) + {G⁰⁰_f

i( f_i(x, y))∇yfi(x, y)(∇yfi(x, y))^T+ G⁰_f

i( f_i(x, y))∇yyfi(x, y)}p



≤ 0, (1)

λ > 0, λ^Te_k= 1. (2)

Dual problem (WD):

Maximize S(u, v, λ , q) =



S₁(u, v, λ1, q), S₂(u, v, λ2, q), ..., S_k(u, v, λk, q)

T

Subject to

k

∑

i=1

λ_i

 G⁰_f

i( f_i(u, v))∇xf_i(u, v) + {G⁰⁰_f

i( f_i(u, v))∇xf_i(u, v)(∇xf_i(u, v))^T+ G⁰_f

i( f_i(u, v))∇xxf_i(u, v))}q



≥ 0, (3)

λ > 0, λ^Te_k= 1, (4)

192

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

where for all i = 1, 2, ..., k,

R_i(x, y, λ , p) = Gf_i( f_i(x, y))−y^T

k i=1

∑

λi

 G⁰_f

i( f_i(x, y))∇yf_i(x, y)+[G⁰⁰_f

i( f_i(x, y))∇yf_i(x, y)(∇yf_i(x, y))^TG⁰_f

i( f_i(x, y))∇yyf_i(x, y)]p



−1 2

k i=1

∑

λip^T



G⁰⁰_fi( f_i(x, y))∇yfi(x, y)(∇yfi(x, y))^T+ G⁰_f

i( f_i(x, y))∇yyfi(x, y)

 p,

S_i(u, v, λ , q) = G_fi( f_i(u, v))−u^T

k

∑

i=1

λ_i

 G⁰_f

i( f_i(u, v))∇_xf_i(u, v)+[G⁰⁰_f

i( f_i(u, v))∇_xf_i(u, v)(∇_xf_i(u, v))^T+G⁰_f

i( f_i(u, v))∇_xxf_i(u, v)]q



−1 2

k

∑

i=1

λ_iq^T

 G⁰⁰_f

i( f_i(u, v))∇xf_i(u, v)(∇xf_i(u, v))^T+ G⁰_f

i( f_i(u, v))∇xxf_i(u, v)

 q,

(i) f_iand G_fi are differentiable functions in x and y, e_k= (1, 1, ..., 1)^T ∈ R^k, (ii) p_iand q_iare vectors in R^mand Rⁿ, respectively, λ ∈ R^k.

The following example shows the feasibility of the (WP) and (WD) problem discussed above:

Example 3.1 Let k = 2. Let fi: X ×Y → R be defined as

f₁(x, y) = x³, f₂(x, y) = y³. Suppose G_fi(t) = t, i = 1, 2.

(EWP) Minimize

R(x, y) = (x³− yλ2[3y²+ 6yp] − 3λ2yp², y³− yλ2[3y²+ 6yp] − 3λ2yp²) Subject to

λ₂(3y²+ 6yp) ≤ 0, λ1, λ2> 0, λ1+ λ2= 1.

(EWD) Maximize

S(u, v) = (u³− uλ1[3u²+ 6uq] − 3λ1q²u, v³− uλ1[3u²+ 6uq] − 3λ1q²u) Subject to

λ1[3u²+ 6uq] ≥ 0, λ1, λ2> 0, λ1+ λ2= 1.

One can easily verify that x = 3, y = 0, λ1=1 2, λ2=1

2, p = 2 is a feasible solution of primal problem and u = 0, v = 1

3, λ1= 1

2, λ2= 1

2, q = 1 is a feasible solution of dual problem. This shows that such primal-dual pair (EWP) and (EWD) exist.

Next, we prove duality theorems for the pair (WP) and (WD).

Theorem 3.1 (Weak duality theorem). Let (x, y, λ , p) and (u, v, λ , q) be feasible solutions of primal and dual problem, respectively. Let for all i= 1, 2, ..., k,

193

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

(i) f_i(., v) be (F, Gfi)-convex at u ,

(ii) f_i(x, .) be (H, G_fi)- concave at y,

where the sublinear functionals F : Rⁿ× Rⁿ× Rⁿ→ R and H : R^m× R^m× R^m→ R satisfy the following conditions:

(iii) F_x,u(a) + a^Tu≥ 0, ∀ a ∈ Rⁿ₊, (iv) H_v,y(b) + b^Ty≥ 0, ∀ b ∈ R^m₊. Then, the following cannot hold:

Ri(x, y, λ , p) ≤ Si(u, v, λ , q), for all i = 1, 2, ..., k (5) and

R_r(x, y, λ , p) < Sr(u, v, λ , q), for some r = 1, 2, ..., k. (6) Proof. Suppose on the contrary that (5) and (6) hold. Then, using λ > 0, we obtain

k

∑

i=1

λ_i



G_fi( f_i(x, y)) − y^T

k

∑

i=1

λ_i

 G⁰_f

i( f_i(x, y))∇yf_i(x, y) + G⁰⁰_f

i( f_i(x, y))∇yf_i(x, y)(∇yf_i(x, y))^T+ G⁰_f

i( f_i(x, y))

∇_yyf_i(x, y)]p



−1 2

k

∑

i=1

λ_i

 p^T[G⁰⁰_f

i( f_i(x, y))∇yf_i(x, y)(∇yf_i(x, y))^T+ G⁰_f

i( f_i(x, y))∇yyf_i(x, y)]p



<

k

∑

i=1

λ_i



G_fi( f_i(u, v)) − u^T

k

∑

i=1

λ_i

 G⁰_f

i( f_i(u, v))∇xf_i(u, v) + G⁰⁰_f

i( f_i(u, v))∇xf_i(u, v)(∇xf_i(u, v))^T+ G⁰_f

i( f_i(u, v))

∇_xxf_i(u, v)]q



−1 2

k

∑

i=1

λ_i

 q^T[G⁰⁰_f

i( f_i(u, v))∇xf_i(u, v)(∇xf_i(u, v))^T+ G⁰_f

i( f_i(u, v))∇xxf_i(u, v)]q



. (7)

Since f_i(., v) is (F, Gfi)-convex at u, we get G_fi( f_i(x, v)) − G_fi( f_i(u, v)) ≥ F_x,u[G⁰_f

i( f_i(u, v))∇_xf_i(u, v) + {G⁰⁰_f

i( f_i(u, v))∇_xf_i(u, v)(∇_xf_i(u, v))^T+ G⁰_f

i( f_i(u, v))

∇xxfi(u, v)}q] −1 2q^T[G⁰⁰_f

i( f_i(u, v))∇xfi(u, v)(∇xfi(u, v))^T+ G⁰_f

i( f_i(u, v))∇xxfi(u, v)]q.

Since λ > 0 and using sublinearity of F at the third position, the above inequality yields

k i=1

∑

λiGf_i( f_i(x, v)) − G_fi( f_i(u, v)) ≥ Fx,u

 k i=1

∑

λiG⁰_fi( f_i(u, v))∇xf_i(u, v) + {G⁰⁰_f

i( f_i(u, v))∇xf_i(u, v)(∇xf_i(u, v))^T

+ G⁰_f

i( f_i(u, v))∇xxf_i(u, v)}q



−1 2

k i=1

∑

λiq^T[G⁰⁰_f

i( f_i(u, v))∇xf_i(u, v)(∇xf_i(u, v))^T+ G⁰_f

i( f_i(u, v))∇xxf_i(u, v)]q.

Further, using hypothesis (iii) and dual constraint (3), we get

k

∑

i=1

λ_iG_fi( f_i(x, v)) − G_fi( f_i(u, v)) ≥ −u^T

∑

^k

i=1

λ_iG⁰_f

i( f_i(u, v))∇xf_i(u, v) + {G⁰⁰_f

i( f_i(u, v))∇xf_i(u, v)(∇xf_i(u, v))^T

+ G⁰_f

i( f_i(u, v))∇_xxf_i(u, v)}q −1 2

k

∑

i=1

λ_iq^T[G⁰⁰_f

i( f_i(u, v))∇_xf_i(u, v)(∇_xf_i(u, v))^T+ G⁰_f

i( f_i(u, v))∇_xxf_i(u, v)]q. (8)

194

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

Similarly, using hypotheses (i), (iv) and dual constraint (1), we get

k i=1

∑

λi − Gf_i( f (x, v)) + G_fi( f_i(x, y)) ≥ y^T

k i=1

∑

λiG⁰_fi( f_i(x, y))∇yf_i(x, y) + {G⁰⁰_f

i( f_i(x, y))∇yf_i(x, y)(∇yf_i(x, y))^T

+ G⁰_f

i( f_i(x, y))∇yyf_i(x, y)}p +1 2

k i=1

∑

λip^T[G⁰⁰_f

i( f_i(x, y))∇yf_i(x, y)(∇yf_i(x, y))^T+ G⁰_f

i( f_i(x, y))∇yyf_i(x, y)]p. (9) Finally, adding inequalities (8), (9) and λ^Te_k= 1, we get

k

∑

i=1

λi



G_fi( f_i(x, y)) − y^T

k

∑

i=1

λi

 G⁰_f

i( f_i(x, y))∇yf_i(x, y) + G⁰_f

i( f_i(x, y))∇yf_i(x, y)(∇yf_i(x, y))^T+ G⁰_f

i( f_i(x, y))

∇_yyf_i(x, y)]p



−1 2

k

∑

i=1

λ_i

 p^T[G⁰⁰_f

i( f_i(x, y))∇yf_i(x, y)(∇yf_i(x, y))^T+ G⁰_f

i( f_i(x, y))∇yyf_i(x, y)]p



≥

k i=1

∑

λ_i



G_fi( f_i(u, v)) − u^T

k i=1

∑

λ_i



G⁰_fi( f_i(u, v))∇_xf_i(u, v) + G⁰_f

i( f_i(u, v))∇_xf_i(u, v)(∇_xf_i(u, v))^T+ G⁰_f

i( f_i(u, v))

∇xxfi(u, v)]q



−1 2

k i=1

∑

λi

 q^T[G⁰⁰_f

i( f_i(u, v))∇xfi(u, v)(∇xfi(u, v))^T+ G⁰_f

i( f_i(u, v))∇xxfi(u, v)]q



.

This contradicts (7). Hence, the result.

Theorem 3.2 (Strong duality). Let ( ¯x, ¯y, ¯λ , ¯p) be an efficient solution of (WP); fix λ = ¯λ in (WD) such that (i) for all i = 1, 2, ..., k, [G⁰⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)(∇yf_i( ¯x, ¯y))^T+ G⁰_f

i( f_i( ¯x, ¯y))∇yyf_i( ¯x, ¯y)] is nonsingular, (ii)

k i=1

∑

λ¯i∇y

 {G⁰⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)(∇yf_i( ¯x, ¯y))^T+ G⁰_f

i( f_i( ¯x, ¯y))∇yyf_i( ¯x, ¯y)} ¯p



¯ p

∈ span/

 G⁰_f

1( f₁( ¯x, ¯y))∇yf₁( ¯x, ¯y), ..., G⁰_f

k( f_k( ¯x, ¯y))∇yf_k( ¯x, ¯y)



\ {0},

(iii) the vectors

 G⁰_f

1( f₁( ¯x, ¯y))∇_yf1( ¯x, ¯y), G⁰_f

2( f₂( ¯x, ¯y))∇_yf2( ¯x, ¯y), ..., G⁰_f

k( f_k( ¯x, ¯y))∇_yf_k( ¯x, ¯y)



are linearly indepen- dent,

(iv)

k

∑

i=1

λ¯i∇y

 {G⁰⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)(∇yf_i( ¯x, ¯y))^T+ G⁰_f

i( f_i( ¯x, ¯y))∇yyf_i( ¯x, ¯y)} ¯p



¯

p= 0 ⇒ ¯p= 0.

Then, ¯q= 0 such that ( ¯x, ¯y, ¯λ , ¯q= 0) is feasible solution for (WD) and the value of objective functions are equal. Also, if the assumptions of weak duality theorem hold, then ( ¯x, ¯y, ¯λ , ¯q= 0) is an efficient solution for (WD).

Proof. Since ( ¯x, ¯y, ¯λ , ¯p) is an efficient solution of (WP), using Fritz -John necessary conditions [4], then there ex- ist α ∈ R^k, β ∈ R^mand η ∈ R such that

k

∑

i=1

αi[G⁰_f

i( f_i( ¯x, ¯y))∇xf_i( ¯x, ¯y)] +

k

∑

i=1

λ¯i

 G⁰⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)∇xf_i( ¯x, ¯y) + G⁰_f

i( f_i( ¯x, ¯y))∇xyf_i( ¯x, ¯y)



β − (α^Te_k) ¯y



+

k

∑

i=1

λ¯_i∇_x

 (G⁰⁰_f

i( f_i( ¯x, ¯y))∇_yf_i( ¯x, ¯y)(∇_yf_i( ¯x, ¯y))^T+ G⁰_f

i( f_i( ¯x, ¯y))∇_yyf_i( ¯x, ¯y)) ¯p



β − (α^Te_k)( ¯y+1 2p)¯



= 0, (10)

195

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

k

∑

i=1



αi− (α^Te_k) ¯λi



G⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)

 +

k

∑

i=1

λ¯i



G⁰⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)(∇yf_i( ¯x, ¯y))^T + G⁰_f

i( f_i( ¯x, ¯y))∇yyfi( ¯x, ¯y)



β − (α^Te_k)( ¯y+ ¯p)

 +

k i=1

∑

λ¯i



∇y{G⁰⁰_f

i( f_i( ¯x, ¯y))∇yfi( ¯x, ¯y)(∇yfi( ¯x, ¯y))^T

+ G⁰_f

i( f_i( ¯x, ¯y))∇_yyf_i( ¯x, ¯y)) ¯p}]



β − (α^Te_k)( ¯y+1 2p)¯



= 0, (11)

 G⁰⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)(∇yf_i( ¯x, ¯y))^T+ G⁰_f

i( f_i( ¯x, ¯y))∇yyf_i( ¯x, ¯y)



β − (α^Te_k)( ¯p+ ¯y)

 λ¯i



= 0, i = 1, 2, ..., k, (12)

G⁰_f( f ( ¯x, ¯y))∇yf( ¯x, ¯y)



β − (α^Te_ky)¯



+ ηe_k+



β − (α^Te_k)( ¯y+¹₂p¯₁)

T G⁰⁰_f

1( f₁( ¯x, ¯y))∇yf₁( ¯x, ¯y)(∇yf₁( ¯x, ¯y))^T + G⁰_f

1( f₁( ¯x, ¯y))∇yyf₁( ¯x, ¯y)) ¯p₁

 , ...,



β − (α^Te_k)( ¯y+¹₂p¯_k)

T G⁰⁰_f

k( f_k( ¯x, ¯y))∇yf_k( ¯x, ¯y)(∇yf_k( ¯x, ¯y))^T + G⁰_f

k( f_k( ¯x, ¯y))∇_yyf_k( ¯x, ¯y)) ¯p_k



= 0, (13)

β^T

k i=1

∑

λ¯_i[G⁰_f

i( f_i( ¯x, ¯y))∇_yf_i( ¯x, ¯y) + {G⁰⁰_f

i( f_i( ¯x, ¯y))∇_xf_i( ¯x, ¯y)(∇_yf_i( ¯x, ¯y))^T+ G⁰_f

i( f_i( ¯x, ¯y))∇_yyf_i( ¯x, ¯y)} ¯p] = 0, (14)

η^T[ ¯λ^Te_k− 1] = 0, (15)

(α, β ) ≥ 0, (α, β , η) 6= 0. (16)

Equation (13) can be written as

G⁰_f

i( f_i( ¯x, ¯y))∇yfi( ¯x, ¯y)



β − (α^Te_k) ¯y

 +



β − (α^Te_k)( ¯y+¹₂p)¯

T (G⁰⁰_f

i( f_i( ¯x, ¯y))∇yfi( ¯x, ¯y)(∇yfi( ¯x, ¯y))^T + G⁰_f

i( f_i( ¯x, ¯y))∇yyf_i( ¯x, ¯y)) ¯p



+ η = 0, i = 1, 2, ..., k. (17)

By hypothesis (i) and ¯λi> 0, for i = 1, 2, ..., k, (12) gives

β = (α^Te_k)( ¯p+ ¯y), i = 1, 2, ..., k. (18)

If α = 0, then (18) implies that β = 0. Further, equation (17) gives η = 0. Consequently, (α, β , η) = 0, which contra- dicts (16). Hence, α 6= 0, or α^Te_k> 0.

Using (18) and α^Te_k> 0 in (11), we get

k i=1

∑

λ¯i



∇y

 (G⁰⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)(∇yf_i( ¯x, ¯y))^T+ G⁰_f

i( f_i( ¯x, ¯y))∇yyf_i( ¯x, ¯y)) ¯p



¯ p



= − 2 α^Te_k

k

∑

i=1

[G⁰_f

i( f_i( ¯x, ¯y))∇_yf_i( ¯x, ¯y)]



α_i− (α^Te_k) ¯λ_i



. (19)

196

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

It follows from hypothesis (ii) that

k i=1

∑

λ¯i



∇y

 (G⁰⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)(∇yf_i( ¯x, ¯y))^T+ G⁰_f

i( f_i( ¯x, ¯y))∇yyf_i( ¯x, ¯y)) ¯p



¯ p



= 0. (20)

Hence, by hypothesis (iv), we obtain

¯

p= 0. (21)

Therefore, the inequality (18) implies

β = (α^Te_k) ¯y. (22)

Now, using (21) in (19), we obtain

k

∑

i=1



αi− (α^Te_k) ¯λi

 [G⁰_f

i( f_i( ¯x, ¯y))∇yf_i( ¯x, ¯y)] = 0.

From hypothesis (iii), it yields

α_i= (α^Te_k) ¯λ_i, i = 1, 2, ..., k. (23)

Using α^Te_k> 0, (21)-(23) in (10), we get

¯ x^T

k i=1

∑

λ¯i

 G⁰_f

i( f_i( ¯x, ¯y))∇xf_i( ¯x, ¯y)]



= 0.

Further, using (14), α^Te_k> 0, (21) and (22), we have

¯ y^T

k

∑

i=1

λ¯i

 G⁰_f

i( f_i( ¯x, ¯y))∇xf_i( ¯x, ¯y)]



= 0. (24)

Hence, ( ¯x, ¯y, ¯λ , ¯q= 0) satisfies the constraints (3) and (4) of (WD) and clearly a feasible solution for the dual problem (WD). Hence, the result.

Theorem 3.3 (Converse duality). Let ( ¯u, ¯v, ¯λ , ¯q) be an efficient solution of (WD); fix λ = ¯λ in (WP) such that (i) for all i = 1, 2, ..., k, [G⁰⁰_f

i( f_i( ¯u, ¯v))∇xfi( ¯u, ¯v)(∇xfi( ¯u, ¯v))^T+ G⁰_f

i( f_i( ¯u, ¯v))∇xxfi( ¯u, ¯v)] is nonsingular, (ii)

k

∑

i=1

λ¯_i∇_x

 {G⁰⁰_f

i( f_i( ¯u, ¯v))∇_xf_i( ¯u, ¯v)(∇_xf_i( ¯u, ¯v))^T+ G⁰_f

i( f_i( ¯u, ¯v))∇_xxf_i( ¯u, ¯v)} ¯q



¯ q

∈ span/

 G⁰_f

1( f₁( ¯u, ¯v))∇xf₁( ¯u, ¯v), ..., G⁰_f

k( f_k( ¯u, ¯v))∇xf_k( ¯u, ¯v)



\ {0},

(iii) the vectors

 G⁰_f

1( f₁( ¯u, ¯v))∇xf₁( ¯u, ¯v), G⁰_f

2( f₂( ¯u, ¯v))∇xf₂( ¯u, ¯v), ..., G⁰_f

k( f_k( ¯u, ¯v))∇xf_k( ¯u, ¯v)



are linearly indepen- dent,

(iv)

k

∑

i=1

λ¯_i∇_x

 {G⁰⁰_f

i( f_i( ¯u, ¯v))∇_xf_i( ¯u, ¯v)(∇_xf_i( ¯u, ¯v))^T+ G⁰_f

i( f_i( ¯u, ¯v))∇_xxf_i( ¯u, ¯v)} ¯q



¯

q= 0 ⇒ ¯q= 0.

197

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

Then, ¯p= 0 such that ( ¯u, ¯v, ¯λ , ¯p= 0) is feasible solution of (WP) and R( ¯u, ¯v, ¯λ , ¯p) = S( ¯u, ¯v, ¯λ , ¯p). Also, if the hypothe- ses of Theorem 3.1 hold, then ( ¯u, ¯v, ¯λ , ¯p= 0) is an efficient solution for (WP).

Proof. It follows on the lines of Theorem 3.2.

Addresses:

Ramu Dubey¹

1Department of Mathematics, Central University of Haryana, Jant-Pali-123029, India Email: [email protected]

Vandana^∗2

∗2Department of Management Studies, Indian Institute of Technology, Madras, Chennai -600 036, Tamil Nadu, India Email: [email protected]

Vishnu Narayan Mishra^∗

3Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh-484887, India

Email: [email protected]

∗2 denotes for corresponding author.

REFERENCES

[1] Antczak, T.: New optimality conditions and duality results of G-type in differentiable mathematical program- ming, Nonlinear Anal. 66, 1617-1632 (2007).

[2] Antczak, T.: On G-invex multiobjective programming, Part I. Optimality. J. Global Optim, 43 , 97-109 (2009).

[3] Chen, X.H.: Higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math.

Anal. Appl. 290, 423-435 (2004).

[4] Craven, B.D.: Lagrangian condition and quasiduality, Bull. Aust. Math. Soc. 16, 587-592 (1977).

[5] Dubey, R., Gupta, S.K. and Khan, M. A.: Optimality and duality results for a nondifferentiable multiobjective fractional programming , Journal of Inequalities and Applications 354, 1-18 (2015), DOI 10.1186/s13660-015- 0876-0.

[6] Dubey, R. and Gupta, S.K.: Duality for a nondifferentiable multiobjective higher-order symmetric fractional programming problems with cone constraints, Journal of Non-linear Analysis and Optimization 7, 1-15 (2016).

[7] Gulati, T.R., Ahmad, I. and Husain, I., Second-order symmetric duality with generalized convexity, Opsearch.

38, 210-222 (2001).

[8] Gulati, T.R. and Gupta, S.K.: Wolfe type second-order symmetric duality in nondifferentiable programming, J.

Math. Anal. Appl. 310, 247-253 (2005).

[9] Gupta, S. K., Dubey, R. and Debnath, I.P.: Second-order multiobjective programming problems and symmetric duality relations with G_f -bonvexity, Opsearch, (2016).

[10] Jeyakumar, V. and Mond, B: On generalized convex mathematical programming, J. Aust. Math. Soc. Ser. B. 34, 43-53 (1992).

198

DOI: 10.5281/zenodo.1341853 Impact Factor- 5.070

[11] Mangasarian, O.L.: Second and higher order duality in nonlinear programming, J. Math. Anal. Appl. 51, 607-620

(1975).

[12] Jeyakumar, V. and Mond, B: On generalized convex mathematical programming, J. Aust. Math. Soc. Ser. B. 34, 43-53 (1992).

[13] Suneja, S.K., Srivastava, M.K. and Bhatia, M.: Higher order duality in multiobjective fractional programming with support functions, J. math. anal. appl. (2009).

[14] Vandana, Dubey, R., Deepmala, Mishra, L.N. and Mishra, V.N.: Duality relations for a class of a multiobjective fractional programming problem involving support functions, American Journal of Operations Research 8, 294- 311 (2018), DOI:10.4236/ajor.2018.84017.

199