Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 8462602, 14 pages http://dx.doi.org/10.1155/2016/8462602
Research Article Sufficiency and Duality for Multiobjective Programming under New Invexity Yingchun Zheng and Xiaoyan Gao College of Science, Xi’an University of Science and Technology, Xi’an 710054, China Correspondence should be addressed to Yingchun Zheng;
[email protected] Received 24 March 2016; Accepted 28 August 2016 Academic Editor: Yakov Strelniker Copyright © 2016 Y. Zheng and X. Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A class of multiobjective programming problems including inequality constraints is considered. To this aim, some new concepts of generalized (𝐹, 𝑃)-type I and (𝐹, 𝑃)-type II functions are introduced in the differentiable assumption by using the sublinear function 𝐹. These new functions are used to establish and prove the sufficient optimality conditions for weak efficiency or efficiency of the multiobjective programming problems. Moreover, two kinds of dual models are formulated. The weak dual, strong dual, and strict converse dual results are obtained under the aforesaid functions.
1. Introduction The field of multiobjective programming, also called vector programming, has grown remarkably in different directions since the 1980s. Many researchers have been interested in the optimality conditions and duality results for the weak efficient solution and efficient solution of the multiobjective programming problems. A large literature was developed around the sufficiency and duality in multiobjective optimization [1]. In [2], Jayswal obtained the Kuhn-Tucker type sufficient optimality conditions for a feasible solution to be an efficient solution and the Mond-Weir type duality results are also presented. More specifically, Gao [3] considered the nonsmooth multiobjective semi-infinite programming and obtained several sufficient conditions and duality results. Also, Bae et al. [4] established duality theorems for nondifferentiable multiobjective programming problems under generalized convexity assumptions. Recently, Kim and Lee [5] introduced the nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. For more descriptions of the multiobjective programming, we refer to [6–9]. Furthermore, various types of generalizations of convexity theory have played an important role in the evolution of the multiobjective programming. During the past decades, the generalizations of invexity were enriched with
and without differentiability assumptions. For example, we can see [10–12]. In particular, Nahak and Mohapatra [13] introduced the concept of 𝜌-(𝜂, 𝜃)-invexity function and discussed a class of multiobjective programming problems by using the new generalized functions. Padhan and Nahak [14] introduced higher-order 𝜌-(𝜂, 𝜃)-invexity functions for studying two different pairs of higher-order symmetric dual programs. In [15], Antczak extended the concept of (𝜙, 𝜌)invexity for differentiable optimization problems to the case of mathematical programming problems with locally Lipschitz functions. In [16], Antczak and Stasiak introduced the concept of (𝜙, 𝜌)-invexity for strong compact Lipschitz mappings in Banach spaces. Sufficient optimality conditions and Mond-Weir duality theorems are derived by the assumption of generalized nonsmooth (𝜙, 𝜌)-invexity between Banach spaces. In [17], based upon the 𝐹-convexity and 𝜌-convexity, the authors defined the (𝜙, 𝜌)-V-type I functions to consider a class of nonsmooth multiobjective programming problems. The invexity of functions is more useful in the research of optimization. In this paper, we consider the multiobjective programming problems. The new class of generalized invexity functions, namely, pseudoinvex (𝐹, 𝑃)-type I (pseudoinvex (𝐹, 𝑃)type II, etc.) are introduced. The sufficient optimality conditions are obtained. Then weak, strong, and strict converse dual results are also established for two types of dual models
2
Mathematical Problems in Engineering
related to multiobjective programming problems involving the new generalized invex functions.
Definition 1. A feasible solution 𝑥 ∈ 𝑋0 of (MP) is said to be a weakly efficient solution for (MP), if there exists no other 𝑥 ∈ 𝑋0 , such that
2. Notations and Preliminaries
𝑓 (𝑥) < 𝑓 (𝑥) .
Throughout the paper, we use the following conventions for vectors in 𝑅𝑛 : 𝑥≦𝑦
if and only 𝑥𝑖 ≤ 𝑦𝑖 , ∀𝑖 = 1, 2, . . . , 𝑛;
𝑥≤𝑦
if and only 𝑥𝑖 ≦ 𝑦𝑖 , ∀𝑖 = 1, 2, . . . , 𝑛, 𝑥 ≠ 𝑦;
𝑥 0, the above inequality yields
𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥)) 𝑃) < 0.
(23)
𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥)) 𝑃)
from hypothesis (ii), which implies 2
𝐹 (𝑥, 𝑥; 𝛼𝑖 (𝑥, 𝑥) 𝜂 (𝑥, 𝑥) ∇𝑓𝑖 (𝑥) 𝑃) + 𝜌𝑖 𝑑 (𝑥, 𝑥) < 0, 𝑖 ∈ 𝐾, 𝑇
(28)
On the other hand, the hypothesis (i) implies
𝑏 (𝑥, 𝑥) (𝑓 (𝑥) − 𝑓 (𝑥)) < 0,
𝑇
(27)
from hypothesis (iii), which follows
Proof. Suppose contrary to the result that 𝑥 is not a weakly efficient solution to (MP). Then there exists 𝑥 ∈ 𝑋0 such that
2
(29)
= 0, (24)
𝐹 (𝑥, 𝑥; 𝛽𝑗 (𝑥, 𝑥) 𝜂 (𝑥, 𝑥) ∇𝑔𝑗 (𝑥) 𝑃) + 𝜏𝑗 𝑑 (𝑥, 𝑥) < 0,
𝜇𝑗 𝜏𝑗
which contradicts (28). Hence the conclusion of the theorem is established. Theorem 14. Let 𝑥 ∈ 𝑅𝑛 be a feasible solution in problem (MP). Suppose that
𝑗 ∈ 𝑀.
By sublinearity of 𝐹 with 𝛼𝑖 (𝑥, 𝑥) > 0 and 𝛽𝑗 (𝑥, 𝑥) > 0, inequalities (24) yield 𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) ∇𝑓𝑖 (𝑥) 𝑃) < −
𝜌𝑖 𝑑2 (𝑥, 𝑥) , 𝛼𝑖 (𝑥, 𝑥)
𝐹 (𝑥, 𝑥; 𝜂 (𝑥, 𝑥) ∇𝑔𝑗 (𝑥) 𝑃) < −
𝜏𝑗 𝛽𝑗 (𝑥, 𝑥)
(25) 2
𝑑 (𝑥, 𝑥) , 𝑗 ∈ 𝑀.
Using 𝜆 𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆 𝑖 = 1, and 𝜇𝑗 ≧ 0, 𝑗 ∈ 𝑀, along with the sublinearity of 𝐹, from inequality (25), we get 𝑘
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥)) 𝑃)
𝑘
𝑚
𝑖=1
𝑗=1
∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥) = 0;
𝑖 ∈ 𝐾, 𝑇
(i) there exists 𝜆 𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆 𝑖 = 1, 𝜇𝑗 ≧ 0, 𝑖 ∈ 𝐾, 𝑗 ∈ 𝑀, such that (30)
(ii) (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type II at 𝑥; (iii) ∑𝑘𝑖=1 (𝜆 𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑥)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 /𝛽𝑗 (𝑥, 𝑥)) ≧ 0. Then 𝑥 is an efficient solution for (MP). Proof. By the way of contradiction, suppose that 𝑥 is not an efficient solution for (MP). Then there exists 𝑥 ∈ 𝑋0 such that 𝑓 (𝑥) ≤ 𝑓 (𝑥) .
(31)
𝑖=1
𝑘
𝜆 𝑖 𝜌𝑖 𝑑2 (𝑥, 𝑥) , < −∑ 𝛼 𝑥) (𝑥, 𝑖 𝑖=1
With 𝑏(𝑥, 𝑥) > 0, the above inequality yields 𝑏 (𝑥, 𝑥) (𝑓 (𝑥) − 𝑓 (𝑥)) ≤ 0,
(32)
Mathematical Problems in Engineering
5 (ii) (𝑓, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type I at 𝑥;
by hypothesis (ii), which follows
(iii) ∑𝑘𝑖=1 (𝜆 𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑥)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 /𝛽𝑗 (𝑥, 𝑥)) ≧ 0.
𝐹 (𝑥, 𝑥; 𝛼𝑖 (𝑥, 𝑥) 𝜂𝑇 (𝑥, 𝑥) ∇𝑓𝑖 (𝑥) 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑥) < 0, 𝑖 ∈ 𝐾, 𝐹 (𝑥, 𝑥; 𝛽𝑗 (𝑥, 𝑥) 𝜂𝑇 (𝑥, 𝑥) ∇𝑔𝑗 (𝑥) 𝑃) + 𝜏𝑗 𝑑2 (𝑥, 𝑥) < 0,
Then 𝑥 is a weakly efficient solution for (MP). (33)
Proof. The proof follows the lines of Theorem 13. Theorem 16. Let 𝑥 ∈ 𝑅𝑛 be a feasible solution in problem (MP). Suppose that
𝑗 ∈ 𝑀.
Using 𝛼𝑖 (𝑥, 𝑥) > 0, 𝛽𝑗 (𝑥, 𝑥) > 0 and 𝜆 𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆 𝑖 = 1, 𝜇𝑗 ≧ 0, 𝑗 ∈ 𝑀, along with the sublinearity of 𝐹, inequality (33) yields
(i) there exists 𝜆 𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆 𝑖 = 1, 𝜇𝑗 ≧ 0, 𝑖 ∈ 𝐾, 𝑗 ∈ 𝑀, such that
𝑘
𝑇
𝐹 (𝑥, 𝑥; 𝜂 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥)) 𝑃) 𝑖=1
𝑘
𝜆 𝑖 𝜌𝑖 𝑑2 (𝑥, 𝑥) , 𝛼 𝑥) (𝑥, 𝑖=1 𝑖
𝑚
𝑖=1
𝑗=1
(39)
(ii) (𝑓, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type II at 𝑥;
< −∑
𝑚
𝑘
∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥) = 0;
(34)
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) ( ∑𝜇𝑗 ∇𝑔𝑗 (𝑥)) 𝑃)
(iii) ∑𝑘𝑖=1 (𝜆 𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑥)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 /𝛽𝑗 (𝑥, 𝑥)) ≧ 0. Then 𝑥 is an efficient solution for (MP).
𝑗=1
𝑚
≦ −∑
Proof. The proof follows the lines of Theorem 14.
𝜇𝑗 𝜏𝑗
𝑗=1 𝛽𝑗
(𝑥, 𝑥)
2
𝑑 (𝑥, 𝑥) .
Theorem 17. Let 𝑥 ∈ 𝑅𝑛 be a feasible solution in problem (MP). Suppose that
Summing inequalities (34) with the sublinearity of 𝐹, we obtain 𝑘
𝑚
𝑖=1
𝑗=1
(i) there exists 𝜆 𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆 𝑖 = 1, 𝜇𝑗 ≧ 0 (at least one 𝜇𝑗 > 0), 𝑖 ∈ 𝐾, 𝑗 ∈ 𝑀, such that
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑𝜇𝑗 ∇𝑔𝑗 (𝑥)) 𝑃) 𝑚 𝜇𝑗 𝜏𝑗 𝜆 𝑖 𝜌𝑖 +∑ ) 𝑑2 (𝑥, 𝑥) . 𝛼 𝑥) 𝛽 𝑥) (𝑥, (𝑥, 𝑖=1 𝑖 𝑗=1 𝑗 𝑘
(35)
< − (∑
𝑚 𝜇𝑗 𝜏𝑗 𝜆 𝑖 𝜌𝑖 +∑ ) 𝑑2 (𝑥, 𝑥) < 0; 𝛼 𝑥) 𝛽 𝑥) (𝑥, (𝑥, 𝑖=1 𝑖 𝑗=1 𝑗
𝑖=1
𝑗=1
(36)
Then 𝑥 is a weakly efficient solution for (MP). Proof. Suppose contrary to the result that 𝑥 is not a weakly efficient solution to (MP). Then there exists 𝑥 ∈ 𝑋0 such that
that is,
𝑓 (𝑥) < 𝑓 (𝑥) ,
𝑘
∑
(37)
Theorem 15. Let 𝑥 ∈ 𝑅𝑛 be a feasible solution in problem (MP). Suppose that (i) there exists 𝜆 𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆 𝑖 = 1, 𝜇𝑗 ≧ 0, 𝑖 ∈ 𝐾, 𝑗 ∈ 𝑀, such that 𝑚
𝑖=1
𝑗=1
∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥) = 0;
(38)
(41)
with 𝑏(𝑥, 𝑥) > 0; the above inequality yields 𝑏 (𝑥, 𝑥) (𝑓 (𝑥) − 𝑓 (𝑥)) < 0,
which contradicts hypothesis (iii). That completes the proof.
𝑘
(40)
(iii) ∑𝑘𝑖=1 (𝜆 𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑥)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 /𝛽𝑗 (𝑥, 𝑥)) ≧ 0.
𝑘
𝑚 𝜇𝑗 𝜏𝑗 𝜆 𝑖 𝜌𝑖 +∑ < 0, 𝑖=1 𝛼𝑖 (𝑥, 𝑥) 𝑗=1 𝛽𝑗 (𝑥, 𝑥)
𝑚
(ii) (𝑓, 𝑔) is quasipseudo-invex (𝐹, 𝑃)-type I at 𝑥;
From assumption (i), we have (∑
𝑘
∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥) = 0;
(42)
from hypothesis (ii), which implies 𝐹 (𝑥, 𝑥; 𝛼𝑖 (𝑥, 𝑥) 𝜂𝑇 (𝑥, 𝑥) ∇𝑓𝑖 (𝑥) 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑥) ≦ 0, 𝑖 ∈ 𝐾, 𝐹 (𝑥, 𝑥; 𝛽𝑗 (𝑥, 𝑥) 𝜂𝑇 (𝑥, 𝑥) ∇𝑔𝑗 (𝑥) 𝑃) + 𝜏𝑗 𝑑2 (𝑥, 𝑥) < 0,
𝑗 ∈ 𝑀.
(43)
6
Mathematical Problems in Engineering
By sublinearity of 𝐹 with 𝛼𝑖 (𝑥, 𝑥) > 0 and 𝛽𝑗 (𝑥, 𝑥) > 0, inequalities (43) yield 𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) ∇𝑓𝑖 (𝑥) 𝑃) ≦ −
𝜌𝑖 𝑑2 (𝑥, 𝑥) , 𝛼𝑖 (𝑥, 𝑥)
Minimize 𝑓 (𝑥) = 𝑥3 (44) subject to
𝑖 ∈ 𝐾, 𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) ∇𝑔𝑗 (𝑥) 𝑃) < −
𝜏𝑗 𝛽𝑗 (𝑥, 𝑥)
Example 18. We consider the following programming problem:
(45) 𝑗 ∈ 𝑀.
Using 𝜆 𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆 𝑖 = 1 along with the sublinearity of 𝐹, from inequality (44), we get 𝑘
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥)) 𝑃) (46)
𝑘
𝜆 𝑖 𝜌𝑖 ≦ −∑ 𝑑2 (𝑥, 𝑥) . 𝑖=1 𝛼𝑖 (𝑥, 𝑥)
The set of all feasible solutions of (P) can be given by 𝑋0 = {𝑥 | 𝑥 ≧ 0}. Again, let 𝐹(𝑥, 𝑥; 𝑎) be the function defined by 𝐹(𝑥, 𝑥; 𝑎) = 𝑎(2𝑥 + 1). It can be verified that (𝑓, 𝑔) is quasipseudo-invex (𝐹, 𝑃)type I at 𝑥 = 0 with 𝛼(𝑥, 𝑥) = 1/(𝑥2 + 1), 𝛽(𝑥, 𝑥) = 2/(𝑥2 + 1), 𝜂(𝑥, 𝑥) = 𝑥2 + 1, 𝑃 = 1, 𝑑(𝑥, 𝑥) = 𝑥 − 1, and 𝜌 = 0, 𝜏 = 2/3, ∇𝑓(0) = 0, ∇𝑔(0) = −1. Clearly, 𝑥 = 0 is a feasible solution for problem (P) and it satisfied the assumptions of Theorem 17, as there exist 𝜆 = 1, 𝜇 = 0, such that
With 𝜇𝑗 ≧ 0 (at least one 𝜇𝑗 > 0), 𝑗 ∈ 𝑀, and using the sublinearity of 𝐹, inequality (45) follows
𝜆∇𝑓 (0) + 𝜇∇𝑔 (0) = 0, 𝜇𝜏 𝜆𝜌 + = 0. 𝛼 (𝑥, 𝑥) 𝛽 (𝑥, 𝑥)
𝑚
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) ( ∑𝜇𝑗 ∇𝑔𝑗 (𝑥)) 𝑃) 𝑗=1
𝑚
< −∑
𝜇𝑗 𝜏𝑗
𝑗=1 𝛽𝑗
(𝑥, 𝑥)
(P)
𝑥 ∈ 𝑅.
𝑑2 (𝑥, 𝑥) ,
𝑖=1
𝑔 (𝑥) = 1 − 𝑒𝑥 ≦ 0,
(47)
2
𝑑 (𝑥, 𝑥) .
(51)
We observe that there exists no other 𝑥 ∈ 𝑋0 , such that 𝑓(𝑥) < 𝑓(𝑥). Hence, 𝑥 = 0 is an efficient solution for (P). Similarly, we can establish the following theorem.
By the sublinearity of 𝐹, we sum (46) and (47) to obtain 𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥)) 𝑃) 𝑚 𝜇𝑗 𝜏𝑗 𝜆 𝑖 𝜌𝑖 +∑ ) 𝑑2 (𝑥, 𝑥) , < − (∑ 𝛼 𝑥) 𝛽 𝑥) (𝑥, (𝑥, 𝑖=1 𝑖 𝑗=1 𝑗 𝑘
(48)
Theorem 19. Let 𝑥 ∈ 𝑅𝑛 be a feasible solution in problem (MP). Suppose that (i) there exists 𝜆 𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆 𝑖 = 1, 𝜇𝑗 ≧ 0 (at least one 𝜇𝑗 > 0, as 𝑗 ∈ 𝐽(𝑥)), 𝑖 ∈ 𝐾, 𝑗 ∈ 𝑀, such that 𝑘
𝑚
𝑖=1
𝑗=1
∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥) = 0;
from hypothesis (iii), which follows 𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑥)) 𝑃)
(49)
(52)
(ii) (𝑓, 𝑔) is quasipseudo-invex (𝐹, 𝑃)-type II at 𝑥; (iii) ∑𝑘𝑖=1 (𝜆 𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑥)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 /𝛽𝑗 (𝑥, 𝑥)) ≧ 0.
< 0.
Then 𝑥 is an efficient solution for (MP).
On the other hand, the hypothesis (i) implies 𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑥; 𝜂𝑇 (𝑥, 𝑥) (∑𝜆 𝑖 ∇𝑓𝑖 (𝑥) + ∑𝜇𝑗 ∇𝑔𝑗 (𝑥)) 𝑃)
4. Mond-Weir Duality (50)
= 0, which contradicts (49). Hence the conclusion of theorem is established.
In this section, a dual problem is considered for the class of multiobjective programming problem with the new invex functions. Consider the following Mond-Weir dual problem related to problem (MP):
Mathematical Problems in Engineering
7
Maximize 𝑓 (𝑢) = (𝑓1 (𝑢) , 𝑓2 (𝑢) , . . . , 𝑓𝑘 (𝑢)) subject to
𝑘
𝑚
𝑖=1
𝑗=1
From the feasibility of (𝑢, 𝜆, 𝜇) in Mond-Weir dual problem (MDI), it follows that 𝜆𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆𝑖 = 1, 𝜇𝑗 ≧ 0,
∑ 𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝜇𝑗 ∇𝑔𝑗 (𝑢) = 0, 𝜆𝑖 ≧ 0, 𝜇𝑗 ≧ 0,
(MDI)
𝑘
𝑗 ∈ 𝑀. Multiplying inequalities (58) by 𝜆𝑖 and 𝜇𝑗 , respectively, together with the sublinearity of 𝐹, we get 𝑘
∑𝜆𝑖 = 1, 𝑖 ∈ 𝐾,
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢)) 𝑃)
𝑖=1
𝑖=1
𝑗 ∈ 𝑀.
𝑘
𝜆𝑖 𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝛼 𝑢) (𝑥, 𝑖=1 𝑖
< −∑
∑𝑘𝑖=1
Let 𝑊1 = {(𝑢, 𝜆, 𝜇) ∈ 𝑋 × 𝑅𝑘 × 𝑅𝑚 : 𝜆𝑖 ∇𝑓𝑖 (𝑢) + 𝜇𝑗 ∇𝑔𝑗 (𝑢) = 0, 𝜆 ≥ 0, ∑𝑘𝑖=1 𝜆𝑖 = 1, 𝜇 ≧ 0} be the set of all feasible solutions in problem (MDI). ∑𝑚 𝑗=1
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ( ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) 𝑗=1
Theorem 20 (weak duality). Let 𝑥 and (𝑢, 𝜆, 𝜇) be feasible solutions for (MP) and (MDI), respectively. Moreover, assume that 𝑚 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 +∑ ≧ 0. 𝑖=1 𝛼𝑖 (𝑥, 𝑢) 𝑗=1 𝛽𝑗 (𝑥, 𝑢) 𝑘
∑
(53)
𝑚
≦ −∑
𝑗=1 𝛽𝑗 (𝑥, 𝑢)
𝑓 (𝑥) < 𝑓 (𝑢) .
(54)
Proof. Suppose contrary to the result that 𝑓 (𝑥) < 𝑓 (𝑢)
(55)
hold. By 𝑏(𝑥, 𝑢) > 0, the above inequality follows 𝑏 (𝑥, 𝑢) (𝑓 (𝑥) − 𝑓 (𝑢)) < 0;
(56)
by assumption (a), (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type I at 𝑢, which yields 𝐹 (𝑥, 𝑢; 𝛼𝑖 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑓𝑖 (𝑢) 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑢) < 0, 𝑖 ∈ 𝐾, 𝐹 (𝑥, 𝑢; 𝛽𝑗 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) + 𝜏𝑗 𝑑2 (𝑥, 𝑢)
𝑖=1
𝑗=1
(57)
By sublinearity of 𝐹 together with 𝛼𝑖 (𝑥, 𝑢) > 0 and 𝛽𝑗 (𝑥, 𝑢) > 0, the above inequalities yield 𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝛼𝑖 (𝑥, 𝑢)
𝑗 ∈ 𝑀.
𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
(61)
< 0. By the constraint condition of dual problem (MDI) and 𝐹(𝑥, 𝑢; 0) = 0, we have 𝑘
𝑚
𝑖=1
𝑗=1
(62)
= 0, which contradicts inequality (61). Thus, the conclusion of theorem holds. The proof of part (b) is similar to the proof of part (a). Theorem 21 (weak duality). Let 𝑥 and (𝑢, 𝜆, 𝜇) be feasible solutions for (MP) and (MDI), respectively. Moreover, assume that 𝑚 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 +∑ ≧ 0. 𝑖=1 𝛼𝑖 (𝑥, 𝑢) 𝑗=1 𝛽𝑗 (𝑥, 𝑢) 𝑘
𝑖 ∈ 𝐾, 𝑑2 (𝑥, 𝑢) ,
(60)
From assumption, ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 / 𝛽𝑗 (𝑥, 𝑢)) ≧ 0; inequality (60) implies
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝑗 ∈ 𝑀.
𝜏𝑗
𝑚
𝑘
then the following can not hold:
𝛽𝑗 (𝑥, 𝑢)
𝑘
𝑚 𝜇𝑗 𝜏𝑗 𝜆𝜌 ) 𝑑2 (𝑥, 𝑢) . < − (∑ 𝑖 𝑖 + ∑ 𝛼 𝛽 𝑢) 𝑢) (𝑥, (𝑥, 𝑖 𝑗 𝑖=1 𝑗=1
(b) (𝑓, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type I at 𝑢,
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) < −
𝑑2 (𝑥, 𝑢) .
Adding both sides of (59) with the sublinearity of 𝐹, we obtain
(a) (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type I at 𝑢,
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ∇𝑓𝑖 (𝑢) 𝑃) < −
𝜇𝑗 𝜏𝑗
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
If one of the following conditions is satisfied:
< 0,
(59)
𝑚
(58)
∑
If one of the following conditions is satisfied: (a) (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type II at 𝑢,
(63)
8
Mathematical Problems in Engineering (b) (𝑓, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type II at 𝑢,
Adding both sides of (70) with the sublinearity of 𝐹, we get
then the following can not hold: 𝑓 (𝑥) ≤ 𝑓 (𝑢) .
(64)
Proof. Suppose contrary to the result that 𝑓 (𝑥) ≤ 𝑓 (𝑢)
(65)
hold. By 𝑏(𝑥, 𝑢) > 0, the above inequality yields (66)
by assumption (a), (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type II at 𝑢, which yields 𝐹 (𝑥, 𝑢; 𝛼𝑖 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑓𝑖 (𝑢) 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑢) < 0, 𝑖 ∈ 𝐾, 𝐹 (𝑥, 𝑢; 𝛽𝑗 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) + 𝜏𝑗 𝑑2 (𝑥, 𝑢)
(67)
𝑖 ∈ 𝐾, 𝛽𝑗 (𝑥, 𝑢)
𝑑2 (𝑥, 𝑢) ,
(68)
𝑗 ∈ 𝑀.
𝑚
∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢) = 0. 𝑖=1
(69)
𝑗=1
For 𝜆𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆𝑖 = 1, 𝜇𝑗 ≧ 0, 𝑗 ∈ 𝑀. Multiplying inequalities (68) by 𝜆𝑖 and 𝜇𝑗 , respectively, together with the sublinearity of 𝐹, we have
𝑖=1
𝑘
𝜆𝑖 𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝑢) 𝛼 (𝑥, 𝑖 𝑖=1
𝑗=1
𝑚
≦ −∑
𝜇𝑗 𝜏𝑗
𝑗=1 𝛽𝑗 (𝑥, 𝑢)
𝑑2 (𝑥, 𝑢) .
(72)
𝑗∈𝐽(𝑢)
which contradicts the assumption 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 + ∑ ≧ 0. 𝛽𝑗 (𝑥, 𝑢) 𝑖=1 𝛼𝑖 (𝑥, 𝑢)
(73)
𝑗∈𝐽(𝑢)
Theorem 22 (strong duality). Assume that 𝑥 is a weakly efficient solution of (MP). Suppose that there exists 𝜆 ≥ 0 and 𝜇 ≧ 0, such that (𝑥, 𝜆, 𝜇) is feasible for (MDI). Furthermore, if the weak duality Theorem 20 holds for all feasible solutions of the problems (MP) and (MDI), then (𝑥, 𝜆, 𝜇) is a weakly efficient solution of (MDI). Proof. Suppose that (𝑥, 𝜆, 𝜇) is not a weakly efficient solution of (MDI); then there exists another feasible solution (𝑢, 𝜆, 𝜇) of (MDI) such that (74)
which is a contradiction to Theorem 20. Hence (𝑥, 𝜆, 𝜇) is a weakly efficient solution of (MDI). Theorem 23 (strong duality). Assume that 𝑥 is an efficient solution of (MP). Suppose that there exist 𝜆 ≥ 0 and 𝜇 ≧ 0, such that (𝑥, 𝜆, 𝜇) is feasible for (MDI). Furthermore, if the weak duality Theorem 21 holds for all feasible solutions of the problems (MP) and (MDI), then (𝑥, 𝜆, 𝜇) is an efficient solution of (MDI).
𝑓 (𝑥) ≤ 𝑓 (𝑢) ,
< −∑
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ( ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 + ∑ < 0, 𝛽𝑗 (𝑥, 𝑢) 𝑖=1 𝛼𝑖 (𝑥, 𝑢)
Proof. Suppose that (𝑥, 𝜆, 𝜇) is not an efficient solution of (MDI); then there exists another feasible solution (𝑢, 𝜆, 𝜇) of (MDI) such that
𝑘
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢)) 𝑃)
𝑚
(71)
𝑘
∑
𝑓 (𝑥) < 𝑓 (𝑢) ,
From the feasibility of (𝑢, 𝜆, 𝜇) in Mond-Weir dual problem (MDI), we have 𝑘
𝑚 𝜇𝑗 𝜏𝑗 𝜆𝜌 ) 𝑑2 (𝑥, 𝑢) . < − (∑ 𝑖 𝑖 + ∑ 𝛼 𝛽 𝑢) 𝑢) (𝑥, (𝑥, 𝑖 𝑗 𝑖=1 𝑗=1 𝑘
Thus, the conclusion of the theorem holds. The proof of part (b) is similar to the proof of part (a).
𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝛼𝑖 (𝑥, 𝑢) 𝜏𝑗
𝑗=1
𝑘
By sublinearity of 𝐹 together with 𝛼𝑖 (𝑥, 𝑢) > 0 and 𝛽𝑗 (𝑥, 𝑢) > 0, the above inequalities yield
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) < −
𝑖=1
∑
𝑗 ∈ 𝑀.
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ∇𝑓𝑖 (𝑢) 𝑃) < −
𝑚
Combining (69) and (71), we obtain
𝑏 (𝑥, 𝑢) (𝑓 (𝑥) − 𝑓 (𝑢)) ≤ 0;
< 0,
𝑘
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
(70)
(75)
which is a contradiction to Theorem 21. Hence (𝑥, 𝜆, 𝜇) is an efficient solution of (MDI). Theorem 24 (strict converse duality). Let 𝑥 and (𝑢, 𝜆, 𝜇) be feasible solutions for (MP) and (MDI), respectively. Suppose that 𝑓(𝑥) ≤ 𝑓(𝑢), and ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 / 𝛽𝑗 (𝑥, 𝑢)) ≧ 0. If one of the following conditions is satisfied:
Mathematical Problems in Engineering
9
(a) (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type II at 𝑢,
Adding both sides of (80) with the sublinearity of 𝐹, we get
(b) (𝑓, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type II at 𝑢, then 𝑥 = 𝑢. Proof. Suppose that 𝑥 ≠ 𝑢. By 𝑏(𝑥, 𝑢) > 0, the condition 𝑓(𝑥) ≤ 𝑓(𝑢) yields 𝑏 (𝑥, 𝑢) (𝑓 (𝑥) − 𝑓 (𝑢)) ≤ 0,
𝑚
𝑖=1
𝑗=1
(81) (76)
𝐹 (𝑥, 𝑢; 𝛼𝑖 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑓𝑖 (𝑢) 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑢) < 0, 𝑖 ∈ 𝐾, 𝑇
2
𝐹 (𝑥, 𝑢; 𝛽𝑗 (𝑥, 𝑢) 𝜂 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) + 𝜏𝑗 𝑑 (𝑥, 𝑢)
(77)
𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝛼𝑖 (𝑥, 𝑢)
𝛽𝑗 (𝑥, 𝑢)
𝑑2 (𝑥, 𝑢) ,
(78)
Because (𝑢, 𝜆, 𝜇) is feasible for (MDI), then 𝑘
𝑚
𝑖=1
(79)
For 𝜆𝑖 ≧ 0, ∑𝑘𝑖=1 𝜆𝑖 = 1, 𝜇𝑗 ≧ 0, 𝑗 ∈ 𝑀. Multiplying inequalities (78) by 𝜆𝑖 and 𝜇𝑗 , respectively, together with the sublinearity of 𝐹, we have
𝜆𝜌 < −∑ 𝑖 𝑖 𝑑2 (𝑥, 𝑢) , 𝑖=1 𝛼𝑖 (𝑥, 𝑢)
𝑗=1
(83)
which is a contradiction to (82). Then 𝑥 = 𝑢. The proof of part (b) is similar to the proof of part (a).
(a) (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type I at 𝑢,
then 𝑥 = 𝑢. Proof. Suppose that 𝑥 ≠ 𝑢. By 𝑏(𝑥, 𝑢) > 0, the condition 𝑓(𝑥) < 𝑓(𝑢) yields
(80)
(84)
𝐹 (𝑥, 𝑢; 𝛼𝑖 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑓𝑖 (𝑢) 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑢) < 0,
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ( ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝑖 ∈ 𝐾,
𝑗=1
𝑑2 (𝑥, 𝑢) .
𝑖=1
using assumption (a), (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type I at 𝑢, which yields
𝑘
𝑗=1 𝛽𝑗 (𝑥, 𝑢)
𝑚
𝑏 (𝑥, 𝑢) (𝑓 (𝑥) − 𝑓 (𝑢)) < 0;
𝑖=1
𝜇𝑗 𝜏𝑗
𝑘
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝑘
≦ −∑
(82)
Using inequality (79) together with the sublinearity of 𝐹, we obtain
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢)) 𝑃)
𝑚
𝑗=1
(b) (𝑓, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type I at 𝑢,
𝑗=1
𝑚
𝑖=1
Theorem 25 (strict converse duality). Let 𝑥 and (𝑢, 𝜆, 𝜇) be feasible solutions for (MP) and (MDI), respectively. Suppose that 𝑓(𝑥) < 𝑓(𝑢), and ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 / 𝛽𝑗 (𝑥, 𝑢)) ≧ 0. If one of the following conditions is satisfied:
𝑗 ∈ 𝑀.
∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢) = 0.
𝑚
= 0,
𝑖 ∈ 𝐾, 𝜏𝑗
𝑘
< 0.
By sublinearity of 𝐹 together with 𝛼𝑖 (𝑥, 𝑢) > 0 and 𝛽𝑗 (𝑥, 𝑢) > 0, the above inequalities yield
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) < −
Using the assumption ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 / 𝛽𝑗 (𝑥, 𝑢)) ≧ 0, inequality (81) follows that 𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝑗 ∈ 𝑀.
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ∇𝑓𝑖 (𝑢) 𝑃) < −
𝑚 𝜇𝑗 𝜏𝑗 𝜆𝜌 < − (∑ 𝑖 𝑖 + ∑ ) 𝑑2 (𝑥, 𝑢) . 𝛼 𝑢) 𝛽 𝑢) (𝑥, (𝑥, 𝑖 𝑗 𝑖=1 𝑗=1 𝑘
using assumption (a), (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type II at 𝑢, which yields
< 0,
𝑘
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝐹 (𝑥, 𝑢; 𝛽𝑗 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) + 𝜏𝑗 𝑑2 (𝑥, 𝑢) < 0,
𝑗 ∈ 𝑀.
(85)
10
Mathematical Problems in Engineering
By using the sublinearity of 𝐹 with 𝛼𝑖 (𝑥, 𝑢) > 0 and 𝛽𝑗 (𝑥, 𝑢) > 0 and (𝑢, 𝜆, 𝜇) ∈ 𝑊1 , inequalities (85) imply
Using the condition ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 / 𝛽𝑗 (𝑥, 𝑢)) ≧ 0, inequality (87) yields
𝑘
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢)) 𝑃) 𝑖=1
𝑘
𝜆𝑖 𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝛼 𝑢) (𝑥, 𝑖=1 𝑖 (86)
𝑚
𝐹 (𝑥, 𝑢; 𝜂 (𝑥, 𝑢) ( ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) ≦ −∑
𝜇𝑗 𝜏𝑗
𝑗=1 𝛽𝑗
(𝑥, 𝑢)
Adding both sides of (86) with the sublinearity of 𝐹, we have 𝑚
𝑖=1
𝑗=1
𝑗=1
𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝑑2 (𝑥, 𝑢) .
𝑘
𝑖=1
(88)
According to the constraint condition of (MDI) with 𝐹(𝑥, 𝑢; 0) = 0, we get
𝑗=1
𝑚
𝑚
< 0.
< −∑
𝑇
𝑘
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) (87) 𝑚 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 +∑ ) 𝑑2 (𝑥, 𝑢) . 𝑖=1 𝛼𝑖 (𝑥, 𝑢) 𝑗=1 𝛽𝑗 (𝑥, 𝑢) 𝑘
< − (∑
(89)
= 0, which is a contradiction to (88). Then 𝑥 = 𝑢. The proof of part (b) is similar to the proof of part (a).
5. Wolfe Type Duality In this section, we consider the Wolfe type dual for (MP) and establish various duality theorems. Let 𝑒 be the vector of 𝑅𝑘 whose components are all ones.
𝑚
𝑚
𝑚
𝑗=1
𝑗=1
𝑗=1
𝐺 (𝑢) = 𝑓 (𝑢) + 𝜇𝑇 𝑔 (𝑢) 𝑒 = (𝑓1 (𝑢) + ∑ 𝜇𝑗 𝑔𝑗 (𝑢) , 𝑓2 (𝑢) + ∑ 𝜇𝑗 𝑔𝑗 (𝑢) , . . . , 𝑓𝑘 (𝑢) + ∑𝜇𝑗 𝑔𝑗 (𝑢))
Maximize
𝑘
𝑚
𝑖=1
𝑗=1
∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝑡𝑗 ∇𝑔𝑗 (𝑢) = 0,
subject to
(MDII)
𝑡𝑗 = 2𝜇𝑗 , 𝑗 ∈ 𝑀, 𝑘
𝜆𝑖 ≧ 0,
∑𝜆𝑖 = 1, 𝑖 ∈ 𝐾,
𝜇𝑗 ≧ 0,
𝑗 ∈ 𝑀.
𝑖=1
Let 𝑊2 = {(𝑢, 𝜆, 𝜇, 𝑡) ∑𝑚 𝑗=1 𝑡𝑗 𝑔𝑗 (𝑢) = 0, 𝜆 ≥ 0,
∈ 𝑋 × 𝑅𝑘 × 𝑅𝑚 × 𝑅𝑚 : ∑𝑘𝑖=1 𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝑘𝑖=1 𝜆𝑖 = 1, 𝜇 ≧ 0, 𝑡 ≧ 0} be the set of all feasible solutions in problem (MDII).
then the following can not hold:
Theorem 26 (weak duality). Let 𝑥 ∈ 𝑋0 and (𝑢, 𝜆, 𝜇, 𝑡) ∈ 𝑊2 . Suppose that
Proof. If
𝑚 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 +∑ ≧ 0. 𝑖=1 𝛼𝑖 (𝑥, 𝑢) 𝑗=1 𝛽𝑗 (𝑥, 𝑢) 𝑘
∑
If one of the following conditions is satisfied: (a) (𝐺, 𝑔) is pseudoinvex (𝐹, 𝑃)-type I at 𝑢, (b) (𝐺, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type I at 𝑢,
(90)
𝑓 (𝑥) < 𝐺 (𝑢) .
(91)
𝑓 (𝑥) < 𝐺 (𝑢)
(92)
𝑓 (𝑥) < 𝑓 (𝑢) + 𝜇𝑇 𝑔 (𝑢) 𝑒.
(93)
holds, then we have
Since 𝑥 ∈ 𝑋0 and 𝜇 ≧ 0, the above inequality yields 𝑓 (𝑥) + 𝜇𝑇 𝑔 (𝑥) 𝑒 < 𝑓 (𝑢) + 𝜇𝑇 𝑔 (𝑢) 𝑒;
(94)
Mathematical Problems in Engineering
11
that is, 𝐺 (𝑥) < 𝐺 (𝑢) ;
(95)
𝑏 (𝑥, 𝑢) (𝐺 (𝑥) − 𝐺 (𝑢)) < 0;
(96)
by assumption (a), (𝐺, 𝑔) is pseudoinvex (𝐹, 𝑃)-type I at 𝑢, which implies 𝑚
𝐹 (𝑥, 𝑢; 𝛼𝑖 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) (∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢))
𝑖 ∈ 𝐾,
𝑚 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 +∑ ≧ 0. 𝑖=1 𝛼𝑖 (𝑥, 𝑢) 𝑗=1 𝛽𝑗 (𝑥, 𝑢)
𝑘
𝑚
𝑖=1
𝑗=1
(a) (𝐺, 𝑔) is pseudoinvex (𝐹, 𝑃)-type II at 𝑢, (b) (𝐺, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type II at 𝑢, then the following can not hold: 𝑓 (𝑥) ≤ 𝐺 (𝑢) .
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
𝑓 (𝑥) ≤ 𝐺 (𝑢)
(104)
𝑓 (𝑥) ≤ 𝑓 (𝑢) + 𝜇𝑇 𝑔 (𝑢) 𝑒.
(105)
𝑘
𝜆𝑖 𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝛼 𝑢) (𝑥, 𝑖=1 𝑖
(98)
𝑚
(103)
Proof. Suppose contrary to the result that
< −∑
hold. That is,
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ( ∑𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) 𝑗=1
Since 𝑥 ∈ 𝑋0 and 𝜇 ≧ 0, the above inequality yields
𝑑2 (𝑥, 𝑢) .
Adding both sides of (98) with the sublinearity of 𝐹, we
𝑓 (𝑥) + 𝜇𝑇 𝑔 (𝑥) 𝑒 ≤ 𝑓 (𝑢) + 𝜇𝑇 𝑔 (𝑢) 𝑒;
(106)
𝐺 (𝑥) ≤ 𝐺 (𝑢) .
(107)
that is,
get 𝑘
𝑚
𝑖=1
𝑗=1
By using 𝑏(𝑥, 𝑢) > 0, we get
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝑡𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) 𝑚 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 +∑ ) 𝑑2 (𝑥, 𝑢) , 𝛼 𝛽 𝑢) 𝑢) (𝑥, (𝑥, 𝑖 𝑗 𝑖=1 𝑗=1 𝑘
< − (∑
𝑏 (𝑥, 𝑢) (𝐺 (𝑥) − 𝐺 (𝑢)) ≤ 0. (99)
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝑡𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
By assumption (a), (𝐺, 𝑔) is pseudoinvex (𝐹, 𝑃)-type II at 𝑢, which yields 𝐹 (𝑥, 𝑢; 𝛼𝑖 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) (∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢))
From assumption, ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 /𝛽𝑗 (𝑥, 𝑢)) ≧ 0; inequality (99) implies 𝑘
(108)
𝑚
(𝑡𝑗 = 2𝜇𝑗 , 𝑗 ∈ 𝑀) .
< 0.
(102)
If one of the following conditions is satisfied:
By sublinearity of 𝐹 together with 𝛼𝑖 (𝑥, 𝑢) > 0, 𝛽𝑗 (𝑥, 𝑢) > 0, and the feasibility of (𝑢, 𝜆, 𝜇, 𝑡) in (MDII), the above inequalities yield
𝑗=1 𝛽𝑗 (𝑥, 𝑢)
(101)
𝑘
∑
2
𝑗 ∈ 𝑀.
≦ −∑
𝑗=1
which contradicts inequality (100). Thus, the conclusion of theorem holds. The proof of part (b) is similar to the proof of part (a).
𝐹 (𝑥, 𝑢; 𝛽𝑗 (𝑥, 𝑢) 𝜂 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) + 𝜏𝑗 𝑑 (𝑥, 𝑢)
𝜇𝑗 𝜏𝑗
𝑖=1
= 0,
(97)
𝑇
𝑚
𝑚
Theorem 27 (weak duality). Let 𝑥 ∈ 𝑋0 and (𝑢, 𝜆, 𝜇, 𝑡) ∈ 𝑊2 . Suppose that
𝑗=1
< 0,
𝑘
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝑡𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
by using 𝑏(𝑥, 𝑢) > 0, we have
⋅ 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑢) < 0,
By the constraint condition of (MDII) and 𝐹(𝑥, 𝑢; 0) = 0, we have
(100)
𝑗=1
⋅ 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑢) < 0,
𝑖 ∈ 𝐾,
𝐹 (𝑥, 𝑢; 𝛽𝑗 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) + 𝜏𝑗 𝑑2 (𝑥, 𝑢) < 0,
𝑗 ∈ 𝑀.
(109)
12
Mathematical Problems in Engineering
By sublinearity of 𝐹 together with 𝛼𝑖 (𝑥, 𝑢) > 0, 𝛽𝑗 (𝑥, 𝑢) > 0, and the feasibility of (𝑢, 𝜆, 𝜇, 𝑡) in (MDII), the above inequalities yield 𝑇
𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑢; 𝜂 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
For 𝑥 ∈ 𝑋0 and (𝑥, 𝜆, 𝜇, 𝑡) ∈ 𝑊2 , the above inequality yields 𝑓 (𝑥) < 𝑓 (𝑢) + 𝜇𝑇 𝑔 (𝑢) 𝑒;
(115)
𝑓 (𝑥) < 𝐺 (𝑢) ,
(116)
that is,
𝑘
𝜆𝑖 𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝑢) 𝛼 (𝑥, 𝑖=1 𝑖
< −∑
(110)
𝑚
which is a contradiction to Theorem 26. Hence (𝑥, 𝜆, 𝜇, 𝑡) is a weakly efficient solution of (MDII).
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) ( ∑𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) 𝑗=1
𝜇𝑗 𝜏𝑗
𝑚
≦ −∑
𝑗=1 𝛽𝑗
(𝑥, 𝑢)
𝑑2 (𝑥, 𝑢) .
Adding both sides of (110) with the sublinearity of 𝐹, we get 𝑘
𝑚
𝑖=1
𝑗=1
(a) (𝑓, 𝑔) is pseudoinvex (𝐹, 𝑃)-type I at 𝑢,
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝑡𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) 𝑚 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 +∑ ) 𝑑2 (𝑥, 𝑢) , 𝑖=1 𝛼𝑖 (𝑥, 𝑢) 𝑗=1 𝛽𝑗 (𝑥, 𝑢) 𝑘
< − (∑
(b) (𝑓, 𝑔) is pseudoquasi-invex (𝐹, 𝑃)-type I at 𝑢, (111)
(𝑡𝑗 = 2𝜇𝑗 , 𝑗 ∈ 𝑀) .
By using the constraint condition of (MDII) and 𝐹(𝑥, 𝑢; 0) = 0, inequality (111) implies 𝑘
𝑚
𝜇𝑗 𝜏𝑗
𝜆𝑖 𝜌𝑖 +∑ < 0, 𝛼 𝑖=1 𝑖 (𝑥, 𝑢) 𝑗=1 𝛽𝑗 (𝑥, 𝑢)
∑
(112)
which contradicts the assumption ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 /𝛽𝑗 (𝑥, 𝑢)) ≧ 0. Thus, the conclusion of theorem holds. The proof of part (b) is similar to the proof of part (a). Theorem 28 (strong duality). Assume that 𝑥 is a weakly efficient solution of (MP). Suppose that there exist 𝜆 ≥ 0, 𝜇 ≧ 0, and 𝑡 ≧ 0, such that (𝑥, 𝜆, 𝜇, 𝑡) is feasible for (MDII). Furthermore, if the weak duality Theorem 26 holds for all feasible solutions of the problems (MP) and (MDII), then (𝑥, 𝜆, 𝜇, 𝑡) is a weakly efficient solution of (MDII). Proof. Suppose that (𝑥, 𝜆, 𝜇, 𝑡) is not a weakly efficient solution of (MDII); then there exists another feasible solution (𝑥, 𝜆, 𝜇, 𝑡) of (MDII) such that 𝐺 (𝑥) < 𝐺 (𝑢) ;
Theorem 29 (strict converse duality). Let 𝑥 and (𝑢, 𝜆, 𝜇, 𝑡) be feasible solutions for (MP) and (MDII), respectively. Suppose that 𝑓(𝑥) < 𝐺(𝑢), and ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 /𝛽𝑗 (𝑥, 𝑢)) ≧ 0. If one of the following conditions is satisfied:
(113)
then 𝑥 = 𝑢. Proof. Suppose that 𝑥 ≠ 𝑢. From 𝑥 ∈ 𝑋0 and (𝑢, 𝜆, 𝜇, 𝑡) ∈ 𝑊2 , the condition 𝑓(𝑥) < 𝐺(𝑢) yields 𝑓 (𝑥) + 𝜇𝑇 𝑔 (𝑥) 𝑒 < 𝑓 (𝑢) + 𝜇𝑇 𝑔 (𝑢) 𝑒;
(117)
𝐺 (𝑥) < 𝐺 (𝑢) ,
(118)
that is,
by 𝑏(𝑥, 𝑢) > 0, which implies 𝑏 (𝑥, 𝑢) (𝐺 (𝑥) − 𝐺 (𝑢)) < 0;
using assumption (a), (𝐺, 𝑔) is pseudoinvex (𝐹, 𝑃)-type I at 𝑢, which yields 𝑚
𝐹 (𝑥, 𝑢; 𝛼𝑖 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) (∇𝑓𝑖 (𝑢) + ∑𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑗=1
⋅ 𝑃) + 𝜌𝑖 𝑑2 (𝑥, 𝑢) < 0,
𝑖 ∈ 𝐾;
𝐹 (𝑥, 𝑢; 𝛽𝑗 (𝑥, 𝑢) 𝜂𝑇 (𝑥, 𝑢) ∇𝑔𝑗 (𝑢) 𝑃) + 𝜏𝑗 𝑑2 (𝑥, 𝑢)
that is, 𝑓 (𝑥) + 𝜇𝑇 𝑔 (𝑥) 𝑒 < 𝑓 (𝑢) + 𝜇𝑇 𝑔 (𝑢) 𝑒.
(114)
(119)
< 0,
𝑗 ∈ 𝑀.
(120)
Mathematical Problems in Engineering
13
By sublinearity of 𝐹 together with 𝛼𝑖 (𝑥, 𝑢) > 0, 𝛽𝑗 (𝑥, 𝑢) > 0, and (𝑢, 𝜆, 𝜇, 𝑡) ∈ 𝑊2 , the above inequalities yield 𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) 𝑘
𝜆𝑖 𝜌𝑖 𝑑2 (𝑥, 𝑢) , 𝛼 𝑢) (𝑥, 𝑖=1 𝑖
< −∑
(121)
𝑚
𝑇
𝐹 (𝑥, 𝑢; 𝜂 (𝑥, 𝑢) ( ∑ 𝜇𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) 𝑗=1
𝜇𝑗 𝜏𝑗
𝑚
≦ −∑
𝑗=1 𝛽𝑗 (𝑥, 𝑢)
𝑑2 (𝑥, 𝑢) .
Adding both sides of (121) with the sublinearity of 𝐹, we have 𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑𝑡𝑗 ∇𝑔𝑗 (𝑢)) 𝑃) 𝑚 𝜇𝑗 𝜏𝑗 𝜆𝑖 𝜌𝑖 +∑ ) 𝑑2 (𝑥, 𝑢) , 𝛼 𝑢) 𝛽 𝑢) (𝑥, (𝑥, 𝑖 𝑗 𝑖=1 𝑗=1 𝑘
< − (∑
(122)
(𝑡𝑗 = 2𝜇𝑗 , 𝑗 ∈ 𝑀) .
Using the assumption ∑𝑘𝑖=1 (𝜆𝑖 𝜌𝑖 /𝛼𝑖 (𝑥, 𝑢)) + ∑𝑚 𝑗=1 (𝜇𝑗 𝜏𝑗 / 𝛽𝑗 (𝑥, 𝑢)) ≧ 0, inequality (122) implies 𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝑡𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
(123)
< 0. By using the constraint condition of (MDII) and 𝐹(𝑥, 𝑢; 0) = 0, we obtain 𝑘
𝑚
𝑖=1
𝑗=1
𝐹 (𝑥, 𝑢; 𝜂𝑇 (𝑥, 𝑢) (∑𝜆𝑖 ∇𝑓𝑖 (𝑢) + ∑ 𝑡𝑗 ∇𝑔𝑗 (𝑢)) 𝑃)
(124)
= 0, which is a contradiction to (123). Then 𝑥 = 𝑢. The proof of part (b) is similar to the proof of part (a).
6. Discussion and Conclusion In this paper, we study the multiobjective programming problems and two kinds of dual models. Then the sufficient optimality conditions, weak dual, strong dual, and strict converse dual, results are obtained and proved under a class of new generalized invex functions assumptions for the multiobjective programming.
Competing Interests The authors declare that they have no competing interests.
Acknowledgments This work is supported by National Natural Science Foundation of China under Grants no. 71103143 and no. 71473194; Science and Technology Foundation of Shaanxi Province of China under Grant no. 2013XJXX-40; Scientific Research Program Funded by Shaanxi Provincial Education Department (Program no. 15JK1456).
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