GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES

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1 [Dubey et al. 58: August 2018] GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES 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 , India 2 Department of Management Studies, Indian Institute of Technology, Madras, Chennai , Tamil Nadu, India 3 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amaranta, Anuppur, Madhya Pradesh , India ABSTRACT In this paper, we introduce the definition F,G f -convexity and give a nontrivial example existing such type of functions. Further, a generalization of convexity, namely F,G f -convexity, is introduced in the case of non-linear multiobjective programming problems where the functions constituting vector optimization problems are differentiable. Further, second-order Wolfe type primal-dual pair has been formulated and proved wea, 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 distinct 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 η 1 -bonvexity/ η 2 -boncavity and derived duality results for a Wolfe type model. The concept of G-invex function has been introduced by Antcza [1] and further derived some optimality conditions for constrained optimization problem. Extended the above notion by defining a vector valued G f - invex function, Antzca [2] proved necessary and sufficient optimality conditions for a multiobjective nonlinear programming 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 lie see[5, 6, 14] have also wored 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 CGlobal Journal Of Engineering Science And Researches

2 [Dubey et al. 58: August 2018] F,G f - convexity assumptions. II. NOTATIONS AND PRELIMINARIES Consider the following vector minimization problem: { MP Minimize f x = f 1 x, f 2 x,..., f x } T Subject to X 0 = {x X R n : g j x 0, j = 1,2,...,m} where f = { f 1, f 2,..., f } : X R and g = {g 1,g 2,...,g m } : X R m are differentiable functions defined on X. Definition 2.1 A point x X 0 is said to be an efficient solution of MP if there exists no other x X 0 such that f r x < f r x, for some r = 1,2,..., and f i x f i x, for all i = 1,2,...,. Definition 2.2. A functional F : X X R n R is said to be sublinear with respect to the third variable if for all x,u X X, i F x,u a 1 + a 2 F x,u a 1 + F x,u a 2, for all a 1, a 2 R n, ii F x,u αa = αf x,u a, for all α R + and a R n. Many generalizations of the definition of a convex function have been introduced in optimization theory in order to wea the assumption of convexity for establishing optimality and duality results for new classes of nonconvex optimization problems, including vector optimization problems. One of such a generalization of convexity in the vectorial case is the G-invexity notion introduced by Antcza 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 Jeyaumar and Mond [12] and Antcza [2], we introduce the concept of F,G f -convex. Definition 2.3. Let f : X R be a vector-valued differentiable function. If there exist a sublinear functional F and differentiable function G f = G f1,g f2,...,g f : R R such that every component G fi : I fi X R is strictly increasing on the range of I fi such that x X and p R n, G fi f i x G fi f i u F x,u [G f i f i u x f i u + {G f i f i u x f i u x f i u T + G f i f i u xx f i u}p] 1 2 pt [G f i f i u x f i u x f i u T + G f i f i u xx f i u]p, for all i = 1,2,...,, 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,..., satisfies above inequality, then we will say that f i is 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. Remar 2.1 If F x,u a = η T x,ua, then Definition 2.3 becomes G f -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] R 2 be defined as f x = { f 1 x, f 2 x}, 188 CGlobal Journal Of Engineering Science And Researches

3 [Dubey et al. 58: August 2018] Figure 1: 189 CGlobal Journal Of Engineering Science And Researches

4 [Dubey et al. 58: August 2018] Figure 2: where f 1 x = x 3, f 2 x = x 4 and G f = {G f1,g f2 } : R R 2 be defined as: Let F : X X R 2 R be given as: G f1 t = t 2 + 1, G f2 t = t 4. F x,u a = a x 2 u 2. For showing that f is F,G f -convex at u = 0, for this we have to claim that π i = G fi f i x G fi f i u F x,u [G f i f i u x f i u + {G f i f i u x f i u x f i u T + G f i f i u xx f i u}p i ] pt i [G f i f i u x f i u x f i u T + G f i f i u xx f i u]p i 0, for i = 1,2. Putting the values of f 1, f 2, G f1 and G f2 in the above expressions, we have π 1 = x u F x,u [6u 5 + {18u 4 + 6u 5 }p 1 ] p2 1{18u 4 + 6u 5 }, 190 CGlobal Journal Of Engineering Science And Researches

5 [Dubey et al. 58: August 2018] Figure 3: and π 2 = x 16 u 16 F x,u 16u 15 + {12 16u u 14 }p p2 2{12 16u u 14 }, At u = 0 [ 1,1], the above expressions reduces: π 1 = x 6 and π 2 = x 16 for all x [ 1,1] and hence π 1 0 and π 2 0, from figures 1 and 2, for all x [ 1,1]. Therefore, f is F,G f -convex at u = 0. Now, suppose or which at u = 0 yields π 3 = f 1 x f 1 u F x,u [ x f 1 u + xx f 1 up 1 ] pt 1 [ xx f 1 u]p 1. π 3 = x 3 u 3 F x,u [3u 2 + 6up 1 ] + 3up 2 1 π 3 = x 3, for all x [ 1,1]. 191 CGlobal Journal Of Engineering Science And Researches

6 [Dubey et al. 58: August 2018] Obviously, π 3 0, from figure 3. Therefore, f 1 is not second-order F-convex at u = 0 with respect to p. Hence, f = f 1, f 2 is not second-order F- convex at u = 0 with respect to p. Finally, consider ξ = f 1 x f 1 u F x,u x f 1 u or ξ = x 3 u 3 F x,u 3u 2 which at u = 0 and using sublinearity of functional F, we get At the point x = 1 3 [ 1,1], obtain ξ = x 3. ξ = Therefore, f 1 is not F-convex at u = 0. Hence, f = f 1, f 2 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: T Minimize Rx,y,λ, p = R 1 x,y,λ 1, p,r 2 x,y,λ 2, p,...r x,y,λ, p Subject to ] [G f i f i x,y y f i x,y + {G f i f i x,y y f i x,y y f i x,y T + G f i f i x,y yy f i x,y}p 0, 1 Dual problem WD: Maximize Su,v,λ,q = Subject to λ > 0, λ T e = 1. 2 T S 1 u,v,λ 1,q,S 2 u,v,λ 2,q,...,S u,v,λ,q ] [G f i f i u,v x f i u,v + {G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v}q 0, 3 λ > 0, λ T e = 1, CGlobal Journal Of Engineering Science And Researches

7 [Dubey et al. 58: August 2018] where for all i = 1,2,...,, R i x,y,λ, p = G fi f i x,y y T 1 2 G f i f i x,y y f i x,y+[g f i f i x,y y f i x,y y f i x,y T G f i f i x,y yy f i x,y]p p G T f i f i x,y y f i x,y y f i x,y T + G f i f i x,y yy f i x,y p, S i u,v,λ,q = G fi f i u,v u T G f i f i u,v x f i u,v+[g f i f i u,v x f i u,v x f i u,v T +G f i f i u,v xx f i u,v]q 1 2 q G T f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v q, i f i and G fi are differentiable functions in x and y, e = 1,1,...,1 T R, ii p i and q i are vectors in R m and R n, respectively, λ R. The following example shows the feasibility of the WP and WD problem discussed above: Example 3.1 Let = 2. Let f i : X Y R be defined as Suppose G fi t = t, i = 1,2. f 1 x,y = x 3, f 2 x,y = y 3. EWP Minimize Rx,y = x 3 yλ 2 [3y 2 + 6yp] 3λ 2 yp 2, y 3 yλ 2 [3y 2 + 6yp] 3λ 2 yp 2 Subject to λ 2 3y 2 + 6yp 0, EWD Maximize λ 1,λ 2 > 0,λ 1 + λ 2 = 1. Su,v = u 3 uλ 1 [3u 2 + 6uq] 3λ 1 q 2 u, v 3 uλ 1 [3u 2 + 6uq] 3λ 1 q 2 u Subject to λ 1 [3u 2 + 6uq] 0, λ 1,λ 2 > 0,λ 1 + λ 2 = 1. One can easily verify that x = 3,y = 0,λ 1 = 1 2,λ 2 = 1, p = 2 is a feasible solution of primal problem and u = 0,v = 2 1 3,λ 1 = 1 2,λ 2 = 1,q = 1 is a feasible solution of dual problem. This shows that such primal-dual pair EWP and 2 EWD exist. Next, we prove duality theorems for the pair WP and WD. Theorem 3.1 Wea 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,...,, 193 CGlobal Journal Of Engineering Science And Researches

8 [Dubey et al. 58: August 2018] i f i.,v be F,G fi -convex at u, ii f i x,. be H,G fi - concave at y, where the sublinear functionals F : R n R n R n R and H : R m R m R m R satisfy the following conditions: iii F x,u a + a T u 0, a R n +, iv H v,y b + b T y 0, b R m +. Then, the following cannot hold: and R i x,y,λ, p S i u,v,λ,q, for all i = 1,2,..., 5 R r x,y,λ, p < S r u,v,λ,q, for some r = 1,2,...,. 6 Proof. Suppose on the contrary that 5 and 6 hold. Then, using λ > 0, we obtain [G fi f i x,y y T G f i f i x,y y f i x,y + G f i f i x,y y f i x,y y f i x,y T + G f i f i x,y < yy f i x,y]p 1 2 ] p T [G f i f i x,y y f i x,y y f i x,y T + G f i f i x,y yy f i x,y]p [G fi f i u,v u T G f i f i u,v x f i u,v + G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v]q 1 ] 2 q T [G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v]q. 7 Since f i.,v is F,G fi -convex at u, we get G fi f i x,v G fi f i u,v F x,u [G f i f i u,v x f i u,v + {G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v}q] 1 2 qt [G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v]q. Since λ > 0 and using sublinearity of F at the third position, the above inequality yields [ G fi f i x,v G fi f i u,v ] { [ F x,u G fi f i u,v x f i u,v + {G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v}q ]} 1 2 Further, using hypothesis iii and dual constraint 3, we get [ G fi f i x,v G fi f i u,v ] u T + G f i f i u,v xx f i u,v}q ] 1 2 q T [G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v]q. [ G fi f i u,v x f i u,v + {G f i f i u,v x f i u,v x f i u,v T q T [G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v]q CGlobal Journal Of Engineering Science And Researches

9 [Dubey et al. 58: August 2018] Similarly, using hypotheses i, iv and dual constraint 1, we get [ G fi f x,v + G fi f i x,y ] y T + G f i f i x,y yy f i x,y}p ] Finally, adding inequalities 8, 9 and λ T e = 1, we get [ G fi f i x,y y f i x,y + {G f i f i x,y y f i x,y y f i x,y T p T [G f i f i x,y y f i x,y y f i x,y T + G f i f i x,y yy f i x,y]p. 9 [G fi f i x,y y T G f i f i x,y y f i x,y + G f i f i x,y y f i x,y y f i x,y T + G f i f i x,y yy f i x,y]p 1 2 ] p T [G f i f i x,y y f i x,y y f i x,y T + G f i f i x,y yy f i x,y]p [G fi f i u,v u T G f i f i u,v x f i u,v + G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v]q 1 ] 2 q T [G f i f i u,v x f i u,v x f i u,v T + G f i f i u,v xx f i u,v]q. This contradicts 7. Hence, the result. Theorem 3.2 Strong duality. Let x,ȳ, λ, p be an efficient solution of WP; fix λ = n WD such that i for all i = 1,2,...,, [G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ] is nonsingular, ii y {G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ} p p { } / span G f 1 f 1 x,ȳ y f 1 x,ȳ,...,g f f x,ȳ y f x,ȳ \ {0}, iii the vectors iv { } G f 1 f 1 x,ȳ y f 1 x,ȳ,g f 2 f 2 x,ȳ y f 2 x,ȳ,...,g f f x,ȳ y f x,ȳ are linearly independent, y {G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ} p p = 0 p = 0. Then, q = 0 such that x,ȳ, λ, q = 0 is feasible solution for WD and the value of objective functions are equal. Also, if the assumptions of wea duality theorem hold, then x,ȳ, λ, q = 0 is an efficient solution for WD. Proof. Since x,ȳ, λ, p is an efficient solution of WP, using Fritz -John necessary conditions [4], then there exist α R, β R m and η R such that α i [G f i f i x,ȳ x f i x,ȳ] + + ] [G f i f i x,ȳ y f i x,ȳ x f i x,ȳ + G f i f i x,ȳ xy f i x,ȳ β α T e ȳ x [G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ p ]β α T e ȳ + 12 p = 0, CGlobal Journal Of Engineering Science And Researches

10 [Dubey et al. 58: August 2018] [ ] α i α T e G f i f i x,ȳ y f i x,ȳ + } + G f i f i x,ȳ yy f i x,ȳ β α T e ȳ + p + [{G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T [ y {G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ p}] β α T e ȳ + 12 ] p = 0, 11 [ ][ ] G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ β α T e p + ȳ = 0, i = 1,2,...,, 12 { T G f f x,ȳ y f x,ȳ β α T e ȳ + ηe + β α T e ȳ p 1 G f 1 f 1 x,ȳ y f 1 x,ȳ y f 1 x,ȳ T T + G f 1 f 1 x,ȳ yy f 1 x,ȳ p 1,..., β α T e ȳ p G f f x,ȳ y f x,ȳ y f x,ȳ T + G f f x,ȳ yy f x,ȳ p } = 0, 13 β T [G f i f i x,ȳ y f i x,ȳ + {G f i f i x,ȳ x f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ} p] = 0, 14 η T [ λ T e 1] = 0, 15 Equation 13 can be written as α,β 0, α,β,η T G f i f i x,ȳ y f i x,ȳ β α T e ȳ + β α T e ȳ + 12 p G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ p + η = 0, i = 1,2,...,. 17 By hypothesis i and > 0, for i = 1,2,...,, 12 gives β = α T e p + ȳ, i = 1,2,...,. 18 If α = 0, then 18 implies that β = 0. Further, equation 17 gives η = 0. Consequently, α, β, η = 0, which contradicts 16. Hence, α 0, or α T e > 0. Using 18 and α T e > 0 in 11, we get [ } ] y {G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ p p = 2 α T e [G f i f i x,ȳ y f i x,ȳ] α i α T e CGlobal Journal Of Engineering Science And Researches

11 [Dubey et al. 58: August 2018] It follows from hypothesis ii that [ } ] y {G f i f i x,ȳ y f i x,ȳ y f i x,ȳ T + G f i f i x,ȳ yy f i x,ȳ p p = Hence, by hypothesis iv, we obtain Therefore, the inequality 18 implies Now, using 21 in 19, we obtain From hypothesis iii, it yields Using α T e > 0, in 10, we get Further, using 14, α T e > 0, 21 and 22, we have p = β = α T e ȳ. 22 α i α T e [G f i f i x,ȳ y f i x,ȳ] = 0. x T ȳ T α i = α T e, i = 1,2,...,. 23 ] [G f i f i x,ȳ x f i x,ȳ] = 0. ] [G f i f i x,ȳ x f i x,ȳ] = Hence, x,ȳ, 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 ū, v, λ, q be an efficient solution of WD; fix λ = n WP such that i for all i = 1,2,...,, [G f i f i ū, v x f i ū, v x f i ū, v T + G f i f i ū, v xx f i ū, v] is nonsingular, ii x {G f i f i ū, v x f i ū, v x f i ū, v T + G f i f i ū, v xx f i ū, v} q q { } / span G f 1 f 1 ū, v x f 1 ū, v,...,g f f ū, v x f ū, v \ {0}, iii the vectors { } G f 1 f 1 ū, v x f 1 ū, v,g f 2 f 2 ū, v x f 2 ū, v,...,g f f ū, v x f ū, v are linearly independent, iv x {G f i f i ū, v x f i ū, v x f i ū, v T + G f i f i ū, v xx f i ū, v} q q = 0 q = CGlobal Journal Of Engineering Science And Researches

12 [Dubey et al. 58: August 2018] Then, p = 0 such that ū, v, λ, p = 0 is feasible solution of WP and Rū, v, λ, p = Sū, v, λ, p. Also, if the hypotheses of Theorem 3.1 hold, then ū, v, λ, p = 0 is an efficient solution for WP. Proof. It follows on the lines of Theorem 3.2. Addresses: Ramu Dubey 1 1 Department of Mathematics, Central University of Haryana, Jant-Pali , India rdubeyjiya@gmail.com Vandana 2 2 Department of Management Studies, Indian Institute of Technology, Madras, Chennai , Tamil Nadu, India vdrai1988@gmail.com Vishnu Narayan Mishra 3 Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amaranta, Anuppur, Madhya Pradesh , India vishnunarayanmishra@gmail.com 2 denotes for corresponding author. REFERENCES [1] Antcza, T.: New optimality conditions and duality results of G-type in differentiable mathematical programming, Nonlinear Anal. 66, [2] Antcza, T.: On G-invex multiobjective programming, Part I. Optimality. J. Global Optim, 43, [3] Chen, X.H.: Higher-order symmetric duality in nondifferentiable multiobjective programming problems, J. Math. Anal. Appl. 290, [4] Craven, B.D.: Lagrangian condition and quasiduality, Bull. Aust. Math. Soc. 16, [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, , DOI /s [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, [7] Gulati, T.R., Ahmad, I. and Husain, I., Second-order symmetric duality with generalized convexity, Opsearch. 38, [8] Gulati, T.R. and Gupta, S.K.: Wolfe type second-order symmetric duality in nondifferentiable programming, J. Math. Anal. Appl. 310, [9] Gupta, S. K., Dubey, R. and Debnath, I.P.: Second-order multiobjective programming problems and symmetric duality relations with G f -bonvexity, Opsearch, [10] Jeyaumar, V. and Mond, B: On generalized convex mathematical programming, J. Aust. Math. Soc. Ser. B. 34, CGlobal Journal Of Engineering Science And Researches

13 [Dubey et al. 58: August 2018] [11] Mangasarian, O.L.: Second and higher order duality in nonlinear programming, J. Math. Anal. Appl. 51, [12] Jeyaumar, V. and Mond, B: On generalized convex mathematical programming, J. Aust. Math. Soc. Ser. B. 34, [13] Suneja, S.K., Srivastava, M.K. and Bhatia, M.: Higher order duality in multiobjective fractional programming with support functions, J. math. anal. appl [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, , DOI: /ajor CGlobal Journal Of Engineering Science And Researches