Journal of the Egyptian Mathematical Society (2015) xxx, xxx–xxx
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
ORIGINAL ARTICLE
Wolfe type higher-order duality in non-smooth multi-objective programming involving cones with generalized (F, a, q, d)-convex functions Arun Kumar Tripathy Department of Mathematics, Trident Academy of Technology F2/A, Chandaka Industrial Estate, Bhubaneswar, Odisha 751 024, India Received 13 February 2014; revised 11 June 2014; accepted 2 August 2014
KEYWORDS Higher-order (F, a, q, d)convex function; Efficient solution; Schwartz Inequality
Abstract In this paper, we consider the non-differentiable multi-objective problem involving cone constraints, where every component of the objective function contains square root term of positive semidefinite quadratic form. For this problem, Wolfe type higher-order duality is proposed and duality results are established for efficient solution under suitable higher-order (F, a, q, d)-convexity assumptions. 2000 MATHEMATICS SUBJECT CLASSIFICATION:
90C29; 90C30; 90C32
ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction The concept of duality, in mathematical programming has received special attention of researchers in solving different real life problem and mathematical models that require the relative comparison of two magnitudes. Higher-order duality in non-linear programming has been studied in last few years by many researchers. Mangasarian [1] first formulated a class E-mail address:
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of second-order and higher-order for non-linear programming problem involving twice differentiable functions. Higher-order duality has been studied by many researchers like Chen [2], Mishra [3], Mishra and Rueda [4], Mond and Zhang [5] and Yang et al. [6]. One practical advantage of second-order and higher-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. On the other hand, to relax convexity assumption imposed on the functions in theorems on optimality and duality, various generalized convexity concept have been proposed. Hanson and Mond [7] introduced F-convex function. Preda [8] extended the concept of F-convexity to (F, q)-convexity, which was further extended to second-order (F, q)-convexity and its natural generalization by Mishra and Rueda [4]. Zhang [9] introduced higher-order (F, q)-convexity and established
http://dx.doi.org/10.1016/j.joems.2014.08.004 1110-256X ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. Please cite this article in press as: A.K. Tripathy, Wolfe type higher-order duality in non-smooth multi-objective programming involving cones with generalized (F, a, q, d)-convex functions, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.004
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higher-order duality for multi-objective programming problems, there by extending the results of Gulati [10], Mangasarian [1], Preda [8] Mishra and Rueda [4] and Mond Weir [11]. Mond [12] considered the following class of non-differentiable mathematical programming problem: 1
ðNDPÞ Minimize fðxÞ þ ðxT BxÞ2 Subject to gðxÞ P 0; where f:Rn fi R and g:Rn fi Rm are twice differentiable functions and B is n · n positive semidefinite (symmetric) matrix. Zhang [9] introduced Mangasarian type [1] higher-order dual (MHD) and Mond-Weir type [11] higher-order dual (MWHD) for (NDP) as follows: ðMHDÞ Maximize fðuÞ þ hðu;pÞ þ ðu þ pÞT Bw yT gðuÞ yT kðu; pÞ Subject to rp hðu; pÞ þ Bw ¼ rp ðyT kðu; pÞÞ wT Bw 6 1; y P 0;
where u, w, p 2 Rn and y 2 Rm. T
T
ðMWHDÞ Maximize fðuÞ þ hðu;pÞ þ u Bw p rp hðu;pÞ
Subject to rp hðu; pÞ þ Bw ¼ rp ðyT kðu; pÞÞ yT gðuÞ þ yT kðu;pÞ þ pT rp ðyT kðu;pÞÞ 6 0; wT Bw 6 1; y P 0;
where u, w, p 2 R ,y 2 Rm and h:Rn · Rn fi R,k:Rn · Rn fi Rm are differentiable functions and B is n · n positive semidefinite (symmetric) matrix. In [9], Zhang established various duality results between (NDP), (MHD) and (MWHD) under higher-order invexity assumptions. Later on, Yang et al. [6] discussed higher-order duality results under generalized convexity assumptions for multi-objective programming problems involving support functions. Kim and Lee [13] introduced the non-differentiable multi-objective problem involving support function with cone constraints and for this problem, he proposed Wolfe and Mond-Weir type higher-order duals and established the duality results under suitable higher-order generalized invexity conditions. Recently, Jayswal and Stancu-Minasian [14] introduced higher-order duality for non-differentiable minimax programming problem with generalized convexity. Based on the various generalized convex functions, Liang et al. [15] introduced a unified formulation of generalized convex function, called (F, a, q, d)-convex function. Ahmad and Husain [16] generalized (F, a, q, d)-convex functions to second-order (F, a, q, d)-convex functions and discussed duality theorems for second-order Mond-Weir type multi-objective dual model under (F, a, q, d)-convexity/pseudo convexity assumption. Megahed [17] introduced a class of second-order (F, a, q, d, E)-convex functions and their generalization on functions and established converse duality theorems for a second-order Mond-Weir type dual problem. Ahmad et al. [18] introduced a new generalized higher-order (F, a, q, d)-type 1 function and proposed a general Mond-Weir type higher-order dual and established the duality results under higher-order (F, a, q, d)-type 1 function. Motivated by Kim and Lee [13], Yang et al. [6], Ahmad and Husain [16], Ahmad et al. [18] and Zhang [9], we proposed Wolfe type higher-order dual programs for the non-differentiable multi-objective problems involving cone constraints, where every component of objective functions contain a square root term of positive semidefinite quadratic form and established the duality results under suitable higher-order (F, a, q, d)convexity conditions. n
2. Preliminaries and definitions Let Rn be n-dimensional Euclidean space and Rnþ be the nonnegative orthant. The following convention for inequalities will be used in this paper. For vectors x and u in Rn, we denote x < u ) u x 2 int Rnþ ; x 6 u () u x 2 Rnþ ; x 6 u () u x 2 Rnþ nf0g; xiu is the negation of x 6 u. Definition 2.1. A nonempty set C in Rn is said to be a cone with vertex zero, if x 2 C implies that kx 2 C for all k=0. If, in addition, C is convex, then C is called a convex cone. Definition 2.2. The polar cone C* of C is defined by C* = {z 2 Rn ŒxTz 5 0, "x 2 C}. Consider the following non-smooth multi-objective programming problem: 1 1 ðMPÞ Minimize fðxÞ þ ðxT BxÞ2 ¼ f1 ðxÞ þ ðxT B1 xÞ2 ; f2 ðxÞ 1 1 þðxT B2 xÞ2 ; . . . ; fk ðxÞ þ ðxT Bk xÞ2 Subject to gðxÞ 2 C2 ; x 2 C1 ;
where f:Rn fi Rk and g:Rn fi Rm are continuously differentiable functions, Bi, i = 1, 2, . . . , k is n · n positive semidefinite symmetric matrix, C1 and C2 are closed convex cone with nonempty interior in Rn and Rm respectively. C1 and C2 are polar cones of C1 and C2 respectively. Let X0 ¼ fx 2 C1 : gðxÞ 2 C2 g, be the set of all feasible solutions of (MP). Since the objectives in multi-objective programming problems generally conflict with one another, an optimal solution is chosen from the set of efficient or weak efficient solutions. Definition 2.3. A point x 2 X0 is an efficient solution of (MP) if there exists no other x 2 X0 such that fð xÞ fðxÞ P 0. That is there exists no other x 2 X0 such that fi ð xÞ fi ðxÞ P 0; 8i ¼ 1; 2; . . . ; k and fr ð xÞ fr ðxÞ > 0; for some r 2 f1; 2; . . . ; kg: Definition 2.4. Let x, w 2 Rn and B 2 Rn · Rn be a positive semidefinite matrix, then 1
1
xT Bw 6 ðxT BxÞ2 ðwT BwÞ2 . Equality holds if for some k P 0, Bx = kBw. Definition 2.5. A function F:X · X · Rn fi R (where X ˝ Rn) is said to be sub-linear in its third component if for all x, u 2 X (i) F(x, u; a1 + a2) 6 F(x, u; a1) + F(x, u; a2) for any a1, a2 2 Rn, and (ii) F(x, u; aa) = aF(x, u; a),"a 2 R+ and for all a 2 Rn. Clearly F(x, u; 0) = 0. Let X ˝ Rn. f = (f1,f2, . . . ,fk):X fi Rk, g = (g1,g2, . . . ,gm): X fi Rm, h = (h1, h2, . . . , hk):X · X fi Rk, k = (k1, k2, . . . , km):X · X fi Rm be differentiable functions and pi 2 Rn, qj 2 Rn, where i = 1, 2, . . . , k; j = 1, 2, . . . , m.
Please cite this article in press as: A.K. Tripathy, Wolfe type higher-order duality in non-smooth multi-objective programming involving cones with generalized (F, a, q, d)-convex functions, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.004
Wolfe type duality in multi-objective programming cones with (F, a, q, d)-convex functions Suppose that d:X · X fi R is a pseudo metric, that is for all x, y, z 2 X; (i) d(x, y) P 0 and x = y ) d(x, y) = 0. (ii) d(x, y) = d(y, x). (iii) d(x, z) 6 d(x, y) + d(y, z). Definition 2.6. A twice differentiable function fi:X fi R is called higher-order (F, a, q, d)-convex function at u 2 D with respect to hi:X · X fi R if there exist real valued function a:X · X fi R+n{0}, d:X · X fi R and q 2 R such that "x 2 X, fi ðxÞ fi ðuÞ hi ðu; pi Þ þ pTi rpi hi ðu; pi Þ
P Fðx; u; aðx; uÞðrfi ðuÞ þ rpi hi ðu; pi ÞÞÞ þ qd2 ðx; uÞ:
Definition 2.7. A twice differentiable function fi:X fi R is called higher-order (F, a, q, d)-pseudo convex function at u 2 D with respect to hi:X · X fi R if there exist real valued function a:X · X fi R+n{0}, d:X · X fi R and q 2 R such that "x 2 X, Fðx; u; aðx; uÞðrfi ðuÞ þ rpi hi ðu; pi ÞÞÞ þ qd2 ðx; uÞ P 0
Theorem 3.1 (Weak duality theorem). Let x and (u, y, k, w, p) be the feasible solution for (MP) and (WHMD) respectively. Assume that (a) kT[f(.) + (.)TBw] is higher-order (F, a, q, d)-convex at u with respect to kTh(u, p) and yTg(.) is higher-order (F, a, r, d)-convex at u with respect to yTk(u, p) along with q + r P 0. (b) kT[f(.) + (.)TBw] yTg(.) is higher-order (F, a, q, d)convex at u with respect to kTh(u, p) yTk(u, p) along with q P 0. (c) kT[f(.) + (.)TBw] yTg(.) is higher-order (F, a, q, d)pseudo convex at u with respect to kTh(u, p) yTk(u, p) along with q P 0. 1
Then fðxÞ þ ðxT BxÞ2 ifðuÞ þ uT Bw ðkT hðu; pÞÞe þ pT rp ðk hðu; pÞÞe þ yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞe T
Proof. Suppose that contradiction holds. 1
That is fðxÞ þ ðxT BxÞ2 6 ðfðuÞ þuT Bw þ ðkT hðu; pÞÞe T p rp ðkT hðu; pÞÞe yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞeÞ.
) fi ðxÞ fi ðuÞ hi ðu; pi Þ þ pTi rpi hi ðu; pi ÞÞ P 0:
Definition 2.8. A twice differentiable function fi:X fi R is called higher-order (F, a, q, d)-quasi convex function at u 2 D with respect to hi:X · X fi R if there exist real valued function a:X · X fi R+n{0}, d:X · X fi R and q 2 R such that "x 2 X, fi ðxÞ fi ðuÞ hi ðu; pi Þ þ pTi rpi hi ðu; pi Þ 6 0
Since k > 0, we obtain 1 kT ½fðxÞ þ ðxT BxÞ2 < kT fðuÞ þ uT Bw
þðkT hðu; pÞÞe pT rp ðkT hðu; pÞÞe yT ½gðuÞ þkðu; pÞ pT rp kðu; pÞe ; 1
) kT ½fðxÞ þ ðxT BxÞ2 < kT ½fðuÞ þ uT Bw þ ðkT hðu; pÞÞ
2
) Fðx; u; aðx; uÞðrfi ðuÞ þ rpi hi ðu; pi ÞÞÞ þ qd ðx; uÞ 6 0:
pT rp ðkT hðu; pÞÞ yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞ:
3. Wolfe type multi-objective higher-order duality In this section, we have introduced the Wolfe type multiobjective higher-order dual of (MP) and established weak and strong duality theorems under generalized higher-order (F, a, q, d)-convex functions. We consider the following Wolfe type higher-order dual for (MP). ðWHMDÞ : Maximize Subject to
3
Now by Schwartz inequality and (3.2), we obtain 1 xT Bw 6 ðxT BxÞ2 . So (3.5) becomes kT ½fðxÞ þ xT Bw < kT ½fðuÞ þ uT Bw þ ðkT hðu; pÞÞ pT rp ðkT hðu; pÞÞ yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞ:
fðuÞ þ uT Bw þ ðkT hðu; pÞÞe pT rp ðkT hðu; pÞÞe yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞe
kT ½rfðuÞ þ Bw þ rp hðu; pÞ yT ½rgðuÞ þ rp kðu; pÞ ¼ 0;
wTi Bi wi 6 1; i ¼ 1; 2; . . . ; k; y 2 C2 ; T
k > 0; k e ¼ 1;
!
ð3:1Þ ð3:2Þ ð3:3Þ
ð3:4Þ
where (i) f:X fi R and g:X fi R , are continuously differentiable functions, (ii) C1 and C2 are closed convex cones in Rn and Rm with nonempty interior respectively, (iii) C1 and C2 are polar cones of C1 and C2 respectively, (iv) e = (1, 1, . . . , 1)T is a vector in Rk, (v) wi 2 Rn, (vi) Bi, i = 1, 2, . . . , k; are positive semidefinite symmetric matrix of order n · n, and (vii) h:X · X fi Rk and k:X · X fi Rm are differentiable functions and pi 2 Rn, qj 2 Rn, where i = 1, 2, . . . , k; j = 1, 2, . . . , m. k
m
ð3:5Þ
ð3:6Þ
(a) Since kTf(.) + (.)TBw is higher-order (F, a, q, d)- convex at u with respect to kTh(u, p) and yTg(.) is higher-order (F, a, r, d)- convex at u with respect to yTk(u, p), we have kT ½fðxÞ þ xT Bw kT ½fðuÞ þ uT Bw kT hðu; pÞ þ pT rp ðkT hðu; pÞÞ
P Fðx; u; aðx; uÞ½kT ðrfðuÞ þ BwÞ þ rp ðkT hðu; pÞÞÞ þ qd2 ðx; uÞ
ð3:7Þ
and yT gðxÞ þ yT gðuÞ þ yT kðu; pÞ pT rp ðyT kðu; pÞÞ P Fðx; u; aðx; uÞ½yT rgðuÞ rp ðyT kðu; pÞÞÞ þ rd2 ðx; uÞ:
Since gðxÞ 2
ð3:8Þ C2 ,
T
for y 2 C2, we get y g(x) 5 0.
So (3.8) can be written as
Please cite this article in press as: A.K. Tripathy, Wolfe type higher-order duality in non-smooth multi-objective programming involving cones with generalized (F, a, q, d)-convex functions, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.004
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A.K. Tripathy
yT gðuÞ þ yT kðu; pÞ pT rp ðyT kðu; pÞÞ T
From (3.1), (3.11) and (3.12) along with F(x, u;0) = 0, we get q < 0, which contradicts the assumption that q P 0. Hence we proved.
T
P Fðx; u; aðx; uÞ½y rgðuÞ rp ðy kðu; pÞÞÞ þ rd2 ðx; uÞ:
ð3:9Þ
(c) From (3.1), we obtain
Now adding (3.7) and (3.9), we get T
T
k ½fðxÞ þ xT Bw k ½fðuÞ þ uT Bw k hðu; pÞ T
T
Fðx; u; aðx; uÞ½kT ðrfðuÞ þ BwÞ þ rp ðkT hðu; pÞÞ yT rgðuÞ
T
T
T
T
T
þ p rp ðk hðu; pÞÞ þ y gðuÞ þ y kðu; pÞ p rp ðy kðu; pÞÞ
P Fðx; u; aðx; uÞ½kT ðrfðuÞ þ BwÞ þ rp ðkT hðu; pÞÞÞ
þ qd2 ðx; uÞ þ Fðx; u; aðx; uÞ½yT rgðuÞ rp ðyT kðu; pÞÞÞ
þ rd2 ðx; uÞ:
Since a(x, u) > 0 and F is sub-linear, the above inequality implies that
kT ½fðxÞ þ xT Bw yT gðxÞ kT ½fðuÞ þ uT Bw kT hðu; pÞ
þ pT rp ðkT hðu; pÞÞ þ yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞ P 0;
) kT ½fðxÞ þ xT Bw yT gðxÞ
P kT ½fðuÞ þ uT Bw þ kT hðu; pÞ pT rp ðkT hðu; pÞÞ Since gðxÞ 2
þ pT rp ðkT hðu; pÞÞ þ yT gðuÞ þ yT kðu; pÞ
C2 ,
ð3:13Þ
T
for y 2 C2, we get y g(x) 5 0.
So, (3.13) implies kT[f(x) + xTBw] P kT[f(u) + uTBw] + kTh(u, p) pTrp (kTh(u, p))
pT rp ðyT kðu; pÞÞ
P Fðx; u; aðx; uÞ½kT ðrfðuÞ þ BwÞ þ rp ðkT hðu; pÞÞ
yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞ:
yT rgðuÞ rp ðyT kðu; pÞÞÞ þ qd2 ðx; uÞ
ð3:10Þ
In view (3.1) and F(x, u;0) = 0, d(x, u) P 0 and q + r P 0, (3.10) implies kT ½fðxÞ þ xT Bw kT ½fðuÞ þ uT Bw kT hðu; pÞ
þ pT rp ðkT hðu; pÞÞ þ yT gðuÞ þ yT kðu; pÞ pT rp ðyT kðu; pÞÞ
P 0: ) kT ½fðxÞ þ xT Bw P kT ½fðuÞ þ uT Bw þ kT hðu; pÞ
This is a contradiction to (3.6). Hence we proved.
þ pT rp ðyT kðu; pÞÞ:
Lemma 3.1. If that x 2 X0 is an efficient solution of (MP) and a constraint qualification [22] is satisfied at x, then there exist i 2 Rn ; i ¼ 1; 2; . . . ; k and y 2 C2 with ð k 2 Rk ; w k; yÞ–0 such that yT rgð kT ðrfð ðx xÞ ½ xÞ þ BwÞ xÞ=0; for all x 2 C1 ; xÞ ¼ 0; ygð
ð3:14Þ ð3:15Þ
ð xT Bi xÞ2 ¼ xT Bi wi ; i ¼ 1; 2; . . . ; k;
ð3:16Þ
1
wTi Bi wi 6 1; i ¼ 1; 2; . . . ; k:
This is a contradiction to (3.6). Hence we proved.
h
We obtain the following Lemma from Mond [12], Bazarra and Goode [20] and Mond and Schechter [21] in order to prove Strong duality theorem.
T
þ pT rp ðkT hðu; pÞÞ yT gðuÞ yT kðu; pÞ
ð3:17Þ
Theorem 3.2 (Strong Duality). Let x be an efficient solution of (MP) at which a constraint qualification [22] is satisfied and
(b) Using yTg(x) 5 0 in (3.6) we obtain kT ½fðxÞ þ xT Bw yT gðxÞ < kT ½fðuÞ þ uT Bw þ ðkT hðu; pÞÞ pT rp ðkT hðu; pÞÞ yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞ;
) kT ½fðxÞ þ xT Bw yT gðxÞ kT ½fðuÞ þ uT Bw ðkT hðu; pÞÞ þ pT rp ðkT hðu;pÞÞ þ yT ½gðuÞ þ kðu;pÞ pT rp kðu; pÞ < 0:
So by assumption (c), we get
yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞ:
kT ½fðxÞ þ xT Bw kT ½fðuÞ þ uT Bw kT hðu; pÞ
þ rd2 ðx; uÞ:
rp ðyT kðu; pÞÞÞ ¼ 0:
ð3:11Þ
By hypothesis (b), we have kT ½fðxÞ þ xT Bw yT gðxÞ kT ½fðuÞ þ uT Bw ðkT hðu; pÞÞ þ pT rp ðkT hðu; pÞÞ þ yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞ P F x; u; aðx; uÞ kT ðrfðuÞ þ BwÞ þ rp ðkT hðu; pÞÞ yT rgðuÞ rp ðyT kðu; pÞÞ þ qd2 ðx; uÞ: ð3:12Þ
hð x; 0Þ ¼ 0; kð x; 0Þ ¼ 0; rp hð x; 0Þ ¼ 0; rp kð x; 0Þ ¼ 0:
ð3:18Þ
i 2 Rn Then there exist k 2 Rkþ ; y 2 C2 with ð k; yÞ ¼ 0 and w p ¼ 0Þ is a feasible solution ði ¼ 1; 2; . . . ; kÞ such that ð k; w; x; y; of (WHMD) and the corresponding value of objective functions are equal. Further, if the conditions of weak duality Theorems p ¼ 0Þ is an efficient solution 3.1 are satisfied then ð x; y; k; w; of (WHMD). Proof. Since x be an efficient solution of (MP) at which a constraint qualification [22] is satisfied, then by Lemma 3.1, there exist k 2 Rk ; y 2 C2 with ð k; yÞ ¼ 0 and i 2 Rn ; i ¼ 1; 2; . . . ; k such that w yT rgð ðx xÞT ½ xÞ þ BwÞ xÞ = 0; for all x 2 C1 kT ðrfð xÞ ¼ 0; ygð
ð3:19Þ
xT Bi xÞ ¼ xT Bi wi ; i ¼ 1; 2;. . .; k; ð wTi Bi wi 6 1; i ¼ 1;2; . .. ; k:
ð3:21Þ ð3:22Þ
1 2
ð3:20Þ
Please cite this article in press as: A.K. Tripathy, Wolfe type higher-order duality in non-smooth multi-objective programming involving cones with generalized (F, a, q, d)-convex functions, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.004
Wolfe type duality in multi-objective programming cones with (F, a, q, d)-convex functions Since x 2 C1 ; x 2 C1 and C1 is a closed convex cone, we have x þ x 2 C1 . T
So replacing x by x þ x in (3.19), we get
yT rgð xÞ þ BwÞ xÞ = 0; for all x 2 C1 x ½ kT ðrfð T T T y rgð i:e: x ½k ðrfð xÞ þ BwÞ xÞ ¼ 0:
ð3:23Þ
So, by relation (3.18), (3.20), (3.21), (3.22) along with (3.23) k; p ¼ 0Þ is feasible for (WHMD) and the imply that ð x; y; w; corresponding value of objective functions are equal. The proof of the remaining part follows from the weak duality Theorem 3.1. 4. Numerical example Let n = 2, k = 2, C1 = {(x1, x2): x1 x2 = 0, x1, x2 P 0} and C2 = {(y1, y2): y1 + y2 = 0, y1 6 0,y2 P 0}. Then C1 ¼ fðx1 ; x2 Þ : x1 þ x2 6 0g and C2 ¼ fðy1 ; y2 Þ : y1 y2 P 0g. Let f = (f1, f2):R2 fi R2, g = (g1, g2):R2 fi R2 defined by f1 ðx1 ; x2 Þ ¼ x21 þ x22 þ 1 and f2 ðx1 ; x2 Þ ¼ x21 þ x1 x2 þ 3; g1 ðx1 ; x2 Þ ¼ x1 x2 þ 3 and g2 ðx1 ; x2 Þ ¼ x21 þ x2 þ 1. 1 0 Also, let B1 ¼ B2 ¼ . 0 1 So the primal (MP) reduces to
ðMPÞ 1
Minimize fðxÞ þ ðxT BxÞ2 ¼
x22
Subject to g ¼ ðg1 ; g2 Þ 2 C2 ; x ¼ ðx1 ; x2 Þ 2 C1 :
ð4:1Þ
h1 ðu; pÞ ¼ ðp1 þ p2 Þðu1 u2 1Þ; h2 ðu; pÞ ¼ 2ðp1 þ p2 Þðu1 þ u2 þ 1Þ; k1 ðu; pÞ ¼ ðp1 þ p2 Þðu1 u2 þ 1Þ and k2 ðu; pÞ ¼ 2ðp1 þ p2 Þðu1 þ u2 þ 1Þ:
u21 þ u22 þ 1 þ u1 w01 þ u2 w001 þ y1 ðu1 þ u2 3Þ B C Maximize @ þy2 u21 u2 1 ; u21 þ u1 u2 þ 3 þ u1 w02 A þu2 w002 þ y1 ðu1 þ u2 3Þ þ y2 u21 u2 1 k1 3u1 u2 þ w01 1 þ k2 4u1 þ 3u2 þ w001 þ 2 Subject to y1 ðu1 u2 Þ y2 ðu2 u1 þ 1Þ ¼ 0; ð4:2Þ k1 u1 þ u2 þ w02 1 þ k2 2u2 þ 3u1 þ w002 þ 2 y1 ðu1 u2 Þ y2 ðu1 þ u2 þ 2Þ ¼ 0; ð4:3Þ 002 02 002 w02 1 þ w1 6 1; w2 þ w2 6 1; y ¼ ðy1 ; y2 Þ 2 C2 ; k1 þ k2 ¼ 1; k1 ; k2 P 0:
1 2
ð4:7Þ
Case (a): Now, let F:X0 · X0 · R fi R define by Fðx; u; aÞ ¼ jajððx1 u1 Þ2 þ ðx2 u2 Þ2 Þ; aðx; uÞ ¼ 101 ; q ¼ 32 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dðx; uÞ ¼ ðx1 u1 Þ2 þ ðx2 u2 Þ2 . 1 2 0 00 0 00 1 If we take k1 ¼ 3; k2 ¼ 3 ; w1 ¼ w1 ¼ w2 ¼ w2 ¼ 2 ; p1 ; p2 2 R at u1 ¼ 14 ; u2 ¼ 54 , we have 2
kT ½fðxÞ þ xT Bw kT ½fðuÞ þ uT Bw kT hðu; pÞ
þ pT rp ðkT hðu; pÞÞ Fðx; u; aðx; uÞ½kT ðrfðuÞ þ BwÞ þ rp ðkT hðu; pÞÞÞ qd2 ðx; uÞ
4 2 19 2 37 ¼ x21 2x22 x1 þ x2 þ x1 x2 þ P 0; 3 3 3 3 24
8x 2 X0 :
1 2 0 00 Also, if we take aðx; uÞ ¼ 101 ; r ¼ 32 ; k1 ¼ 3 ; k2 ¼ 3 ; w1 ¼ w1 0 00 1 1 5 ¼ w2 ¼ w2 ¼ 2 ; p1 ; p2 2 R; u1 ¼ 4 ; u2 ¼ 4 , we obtained
’ 1:764x21 þ 1:098x22 0:118x1 4:078x2 þ 1:625 P 0;
2 X0 :
8x
T T (F, a, q, d)-convex at Hence k1 [f(.) +5(.) Bw] is higher-order u1 ¼ 4 ; u2 ¼ 4 with respect to kTh(u, yTg(.) is p) and 1 higher-order (F, a, q, d)-convex at u1 ¼ 4 ; u2 ¼ 54 with respect to yTk(u, p) along with q + r P 0. Case (b): Also, if we take aðx; uÞ ¼ 3; q ¼ 2; k1 ¼ 13 ;k2 ¼ 2 1 1 5 0 00 0 00 ; w 1 ¼ w1 ¼ w2 ¼ w2 ¼ 2 ; p1 ; p2 2 R, at u1 ¼ 4 ; u2 ¼ 4 , we 3 obtained
kT ½fðxÞ þ xT Bw yT gðxÞ kT ½fðuÞ þ uT Bw þ yT gðuÞ
kT hðu; pÞ þ pT rp ðkT hðu; pÞÞ þ yT kðu; pÞ pT rp ðyT kðu; pÞÞ
Then our proposed dual (WHMD) reduces to 0
! fðuÞ þ uT Bw þ ðkT hðu; pÞÞe pT rp ðkT hðu; pÞÞe fðxÞ þ ðx BxÞ i : yT ½gðuÞ þ kðu; pÞ pT rp kðu; pÞe T
Fðx; u; aðx; uÞ½yT rgðuÞ rp ðyT kðu; pÞÞÞ rd2 ðx; uÞ
Here the feasible region is X0 ¼ fðx1 ; x2 Þjx1 x2 ¼ 0; x21 þ x1 þ 2x2 P 2g. Clearly the minimum value of the objective functions of MP are (4.41, 6.41), which are obtained at x1 = 1, x2 = 1. To find the corresponding higher-order dual (WHMD), consider h = (h1, h2):R2 · R2 fi R2 and k = (k1, k2):R2 · R2 fi R2 defined by
ðWHMDÞ
From we find that (4.2)–(4.6), u1 ¼ 14 ; u2 ¼ 54 ; k1 ¼ 13 ; k2 ¼ 23 ; w01 ¼ w001 ¼ w02 ¼ w002 ¼ 12 ; y1 ¼ 23 ; y2 ¼ 23 ; p1 ; p2 2 RÞ is a feasible solution of (WHMD) and the values of the objective functions are (3, 4.16). From the above discussion, we observe that for any x 2 X0,
yT gðxÞ þ yT gðuÞ þ yT kðu; pÞ pT rp ðyT kðu; pÞÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! þ þ 1 þ x21 þ x22 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x1 x2 þ 3 þ x21 þ x22 x21
5
1
ð4:4Þ ð4:5Þ ð4:6Þ
Fðx; u; aðx; uÞ½kT ðrfðuÞ þ BwÞ þ rp ðkT hðu; pÞÞ yT rgðuÞ
rp ðyT kðu; pÞÞÞ qd2 ðx; uÞ
1 1 7 11 2 16 ¼ x21 þ x22 þ x1 þ x2 þ x1 x2 þ 3 3 6 6 3 3 ! pffiffiffiffiffi 13 1 5 26 aðx; uÞ þ q x21 þ x22 þ x1 x2 þ 2 2 16 3
P 0;
8x 2 X0 :
T T Hence kT[f(.) + (.) 1Bw] y5g(.) is higher-order T(F, a, q, d)convex at u1 ¼ 4 ; u2 ¼ 4 with respect to k h(u, p) yTk(u, p) along with q P 0. 1 Case (c): Again, if we take aðx; uÞ > 0; q P 0; k1 ¼ 3 ; k2 ¼ 23 ; w01 ¼ w001 ¼ w02 ¼ w002 ¼ 12 ; p1 ; p2 2 R, at u1 ¼ 14 ; u2 ¼ 54 , we obtained
Please cite this article in press as: A.K. Tripathy, Wolfe type higher-order duality in non-smooth multi-objective programming involving cones with generalized (F, a, q, d)-convex functions, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.004
6
A.K. Tripathy
kT ½fðxÞ þ xT Bw yT gðxÞ kT ½fðuÞ þ uT Bw þ yT gðuÞ T
T
T
T
T
T
k hðu; pÞ þ p rp ðk hðu; pÞÞ þ y kðu; pÞ p rp ðy kðu; pÞÞ
1 1 7 11 2 16 ¼ x21 þ x22 þ x1 þ x2 þ x1 x2 þ > 0; 8x 2 X0 : 3 3 6 6 3 3 And Fðx; u; aðx; uÞ½kT ðrfðuÞ þ BwÞ þ rp ðkT hðu; pÞÞ yT rgðuÞ rp ðyT kðu; pÞÞÞ þ qd2 ðx; uÞ ! pffiffiffiffiffi 13 1 5 26 aðx; uÞ þ q x21 þ x22 þ x1 x2 þ ¼ 3 2 2 16 8x 2 X0 :
P 0; T
Hence k [f(.) + (.)TBw] yTg(.) is higher-order (F, a, q, d)pseudo convex at u1 ¼ 14 ; u2 ¼ 54 with respect to kTh(u, p) yTk(u, p) along with q P 0. Hence from (4.7), 4(a)–4(c) we observed that Theorem 3.1(Weak Duality) holds good. 5. Special cases (1) If C 1 ¼ Rnþ ; C 2 ¼ Rmþ ; k ¼ 1, then we get higher-order [4]. dual programs studied by Mishra and Rueda 1 (2) If C 1 ¼ Rnþ ; C 2 ¼ Rmþ ; sðxjDÞ ¼ ðxT BxÞ2 where D = {BwŒwTBw 6 1}, then we get higher order dual programs studied by Yang et al. [6]. (3) If C 1 ¼ Rnþ ; C 2 ¼ Rmþ ; Bi ¼ 0, i = 1, 2, . . . , k; hðu; pÞ ¼ pT rf ðuÞ and k(u, p) = pT$g(u), then our higher-order dual programs become first-order dual program in Wolfe [19]. (4) C 1 ¼ Rnþ ; C 2 ¼ Rmþ ; Bi ¼ 0, i = 1, 2, . . . , k; hðu; pÞ ¼ pT rf ðuÞ þ 12 pT r2 f ðuÞp 1 T 2 p r gðuÞp, 2
and
kðu; pÞ ¼ pT rgðuÞþ
then we obtain second-order dual programs
studied by Mangasarian [1].
Acknowledgements The authors wish to thank the referees for their valuable suggestions which improved the presentation of the paper. References [1] O.L. Mangasarian, Second- and higher-order duality in nonlinear programming, J. Math. Anal. Appl. 51 (1975) 607–620. [2] X. Chen, Higher-order symmetric duality in non-differentiable multi-objective programming problems, J. Math. Anal. Appl. 290 (2004) 423–435. [3] S.K. Mishra, Non-differentiable higher-order symmetric duality in mathematical programming with generalized invexity, European J. Oper. Res. 167 (2005) 28–34.
[4] S.K. Mishra, N.G. Rueda, Higher-order generalized invexity and duality in non-differentiable mathematical programming, J. Math. Anal. Appl. 272 (2002) 496–506. [5] B. Mond, J. Zhang, Higher-order invexity and duality in mathematical programming, in: J.P. Craizeix et al. (Eds.), Generalized Convexity, Generalized Monotoncity, Recent Results, Kluwer Academic, 1998, pp. 357–372. [6] X.M. Yang, K.L. Teo, X.Q. Yang, Higher-order generalized convexity and duality in non-differentiable multi-objective mathematical programming, J. Math. Anal. Appl. 297 (2004) 48–55. [7] M.A. Hanson, B. Mond, Necessary and sufficient conditions in constrained optimization, Math. Program. 37 (1987) 51–58. [8] V. Preda, On efficiency and duality for multi-objective programs, J. Math. Anal. Appl. 166 (1992) 365–377. [9] J. Zhang, Higher-order convexity and duality in multi-objective programming problems, in: A. Eberhaerd, R. Hill, D. Ralph, B.M. Glover (Eds.), Progress in Optimization, Contribution from Australasia, Appl. Optim., vol. 30, Kluwer Academic Publisher, Dordrecht–Boston–London, 1999, pp. 101–116. [10] T.R. Gulati, M.A. Islam, Sufficiency and duality in multiobjective programming involving generalized F-convex functions, J. Math. Anal. Appl. 183 (1994) 181–195. [11] B. Mond, T. Weir, Generalized convexity and higher order duality, J. Math. Sci. 16–18 (1981) 74–94 (1983). [12] B. Mond, A class of non-differentiable mathematical programming problems, J. Math. Anal. Appl. 46 (1974) 169– 174. [13] D.S. Kim, Y.J. Lee, Non-differentiable higher-order duality in multi-objective programming involving cones, Nonlinear Anal. 71 (2009) e2474–e2480. [14] A. Jayswal, I.M. Stancu-Minasian, Higher-order duality for non-differentiable minimax programming problem with generalized convexity, Nonlinear Anal. 74 (2011) 616–625. [15] Z.A. Liang, H.X. Huang, P.M. Pardalos, Efficiency conditions and duality for a class of multi-objective fractional programming problems, J. Glob. Opt. 27 (2003) 447–471. [16] I. Ahmad, Z. Husain, Second-order (F, a, q, d)-convexity and duality in multi-objective programming, Inf. Sci. 176 (2006) 3094–3103. [17] Abd El-Monem A. Megahed, Second-order (F, a, q, d, E)convex function and the duality problem, J. Egyptian Math. Soc. 22 (2014) 23–27. [18] I. Ahmad, Z. Husain, S. Sharma, Higher-order duality in nondifferentiable multi-objective programming, Numer. Funct. Anal. Opt. 28 (2007) 989–1002. [19] P. Wolfe, A duality theorem for non-linear programming, Quart. Appl. Math. 19 (1961) 239–244. [20] M.S. Bazarra, J.J. Goode, On symmetric duality in non-linear programming, Oper. Res. 21 (1) (1973) 1–9. [21] B. Mond, M. Schechter, Non-differentiable symmetric duality, Bull. Austral. Math. Soc. 53 (1996) 177–188. [22] O.L. Mangasarian, Non-Linear Programming, Mc. Graw Hill, New York, 1969.
Please cite this article in press as: A.K. Tripathy, Wolfe type higher-order duality in non-smooth multi-objective programming involving cones with generalized (F, a, q, d)-convex functions, Journal of the Egyptian Mathematical Society (2015), http://dx.doi.org/10.1016/j.joems.2014.08.004
