Univex Interval-Valued Mapping with Differentiability and Its

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 383692, 8 pages http://dx.doi.org/10.1155/2013/383692

Research Article Univex Interval-Valued Mapping with Differentiability and Its Application in Nonlinear Programming Lifeng Li,1,2 Sanyang Liu,1 and Jianke Zhang2 1 2

Department of Mathematics, School of Science, Xidian University, Xi’an 710126, China Department of Mathematics, School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China

Correspondence should be addressed to Sanyang Liu; [email protected] Received 25 November 2012; Accepted 26 May 2013 Academic Editor: Lotfollah Najjar Copyright © 2013 Lifeng Li et al. 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. Interval-valued univex functions are introduced for differentiable programming problems. Optimality and duality results are derived for a class of generalized convex optimization problems with interval-valued univex functions.

1. Introduction Imposing the uncertainty upon the optimization problems is an interesting research topic. The uncertainty may be interpreted as randomness, fuzziness, or interval-valued fuzziness. The randomness occurring in the optimization problems is categorized as the stochastic optimization problems, and the imprecision (fuzziness) occurring in the optimization problems is categorized as the fuzzy optimization problems. In order to perfectly match the real situations, intervalvalued optimization problems may provide an alternative choice for considering the uncertainty into the optimization problems. That is to say, the coefficients in the interval-valued optimization problems are assumed as closed intervals. Many approaches for interval-valued optimization problems have been explored in considerable details; see, for example, [1– 3]. Recently, Wu has extended the concept of convexity for real-valued functions to LU-convexity for interval-valued functions, then he has established the Karush-Tucker conditions [4–6] for an optimization problem with intervalvalued objective functions under the assumption of LUconvexity. Similar to the concept of nondominated solution in vector optimization problems, Wu has proposed a solution concept in optimization problems with interval-valued objective functions based on a partial ordering on the set of all closed intervals, then the interval-valued Wolfe duality theory [7] and Lagrangian duality theory [8] for intervalvalued optimization problems have been proposed. Recently,

Wu [9] has studied the duality theory for interval-valued linear programming problems. In 1981, Hanson [10] introduced the concept of invexity and established Karush-Tucker type sufficient optimality conditions for a nonlinear programming problem. In [11], Kaul et al. considered a differentiable multiobjective programming problem involving generalized type I functions. They investigated Karush-Tucker type necessary and sufficient conditions and obtained duality results under generalized type I functions. The class of B-vex functions has been introduced by Bector and Singh [12] as a generalization of convex functions, and duality results are established for vector valued B-invex programming in [13]. Bector et al. [14] introduced the concept of univex functions as a generalization of B-vex functions introduced by Bector et al. [15]. Combining the concepts of type I and univex functions, Rueda et al. [16] gave optimality conditions and duality results for several mathematical programming problems. Aghezzaf and Hachimi [17] introduced classes of generalized type I functions for a differentiable multiobjective programming problem and derived some Mond-Weir type duality results under the above generalized type I assumptions. Gulati et al. [18] introduced the concept of (𝐹, 𝛼, 𝜌, 𝑑)-𝑉-type I functions and also studied sufficiency optimality conditions and duality multiobjective programming problems. This paper aims at extending the Karush-Tucker optimality conditions to nonconvex optimization problem with interval-valued functions. First, we extend the concept of

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univexity for a real-valued function to an interval-valued function and present the concept of interval-valued univex functions. Then, the Karush-Tucker optimality conditions are proposed for an interval-valued function under the assumption of interval-valued univexity.

2. Preliminaries Let one denotes by I the class of all closed intervals in 𝑅. 𝐴 = [𝑎𝐿 , 𝑎𝑈] ∈ I denotes a closed interval, where 𝑎𝐿 and 𝑎𝑈 mean the lower and upper bounds of 𝐴, respectively. For every 𝑎 ∈ 𝑅, we denote 𝑎 = [𝑎, 𝑎]. Definition 1. Let 𝐴 = [𝑎𝐿 , 𝑎𝑈] and 𝐵 = [𝑏𝐿 , 𝑏𝑈] be in I; one has (i) 𝐴 + 𝐵 = {𝑎 + 𝑏 : 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵} = [𝑎𝐿 + 𝑏𝐿 , 𝑎𝑈 + 𝑏𝑈]; (ii) −𝐴 = {−𝑎 : 𝑎 ∈ 𝐴} = [−𝑎𝑈, −𝑎𝐿 ]; (iii) 𝐴 × 𝐵 = {𝑎𝑏 : 𝑎 ∈ 𝐴 and 𝑏 ∈ 𝐵} = [min𝑎𝑏 , max𝑎𝑏 ], where min𝑎𝑏 = min{𝑎𝐿 𝑏𝐿 , 𝑎𝐿 𝑏𝑈, 𝑎𝑈𝑏𝐿 , 𝑎𝑈𝑏𝑈} and max𝑎𝑏 = max{𝑎𝐿 𝑏𝐿 , 𝑎𝐿 𝑏𝑈, 𝑎𝑈𝑏𝐿 , 𝑎𝑈𝑏𝑈}. Then, we can see that 𝐴 − 𝐵 = 𝐴 + (−𝐵) = [𝑎𝐿 − 𝑏𝑈, 𝑎𝑈 − 𝑏𝐿 ] , 𝐿 𝑈 {[𝑘𝑎 , 𝑘𝑎 ] 𝑘𝐴 = {𝑘𝑎 : 𝑎 ∈ 𝐴} = { 𝑈 𝐿 [𝑘𝑎 , 𝑘𝑎 ] {

if 𝑘 ≥ 0,

(1)

if 𝑘 < 0,

where 𝑘 is a real number. By using Hausdorff metric, Neumaier [19] has proposed Hausdorff metric between the two closed intervals 𝐴 and 𝐵 as follows: 󵄨 󵄨 󵄨 󵄨 (2) 𝑑𝐻 (𝐴, 𝐵) = max {󵄨󵄨󵄨󵄨𝑎𝐿 − 𝑏𝐿 󵄨󵄨󵄨󵄨 , 󵄨󵄨󵄨󵄨𝑎𝑈 − 𝑏𝑈󵄨󵄨󵄨󵄨} . 𝐿

𝑈

𝐿

𝑈

Definition 2. Let 𝐴 = [𝑎 , 𝑎 ] and 𝐵 = [𝑏 , 𝑏 ] be two closed intervals in 𝑅. One writes 𝐴 ⪯ 𝐵 if and only if 𝑎𝐿 ≤ 𝑏𝐿 and 𝑎𝑈 ≤ 𝑏𝑈, 𝐴 ≺ 𝐵 if and only if 𝐴 ⪯ 𝐵 and 𝐴 ≠ 𝐵, that is, the following (a1), (a2), or (a3) is satisfied: (a1) 𝑎𝐿 < 𝑏𝐿 and 𝑎𝑈 ≤ 𝑏𝑈; (a2) 𝑎𝐿 ≤ 𝑏𝐿 and 𝑎𝑈 < 𝑏𝑈; (a3) 𝑎𝐿 < 𝑏𝐿 and 𝑎𝑈 < 𝑏𝑈. Definition 3 (see [20]). Let 𝐴 = [𝑎𝐿 , 𝑎𝑈] and 𝐵 = [𝑏𝐿 , 𝑏𝑈] be two closed intervals, the gH-difference of 𝐴 and 𝐵 is defined by 𝐿

𝑈

𝐿

𝑈

[𝑎 , 𝑎 ] ⊖𝑔 [𝑏 , 𝑏 ] = [min (𝑎𝐿 − 𝑏𝐿 , 𝑎𝑈 − 𝑏𝑈) , max (𝑎𝐿 − 𝑏𝐿 , 𝑎𝑈 − 𝑏𝑈)] . (3) For example, [1, 3]⊖𝑔 [0, 3] = [0, 1], [0, 3]⊖𝑔 [1, 3] = [−1, 0]. And 𝑎 − 𝑏 = [𝑎, 𝑎]⊖𝑔 [𝑏, 𝑏] = [𝑎 − 𝑏, 𝑎 − 𝑏] = 𝑎 − 𝑏. Proposition 4. (i) For every 𝐴, 𝐵 ∈ I, 𝐴⊖𝑔 𝐵 always exists and 𝐴⊖𝑔 𝐵 ∈ I. (ii) 𝐴⊖𝑔 𝐵 ⪯ 0 if and only if 𝐴 ⪯ 𝐵.

3. Interval-Valued Univex Functions Definition 5 (interval-valued function). The function 𝑓 : Ω → I is called an interval-valued function, where Ω ⊆ 𝑅𝑛 . Then, 𝑓(x) = 𝑓(𝑥1 , . . . , 𝑥𝑛 ) is a closed interval in 𝑅 for each x ∈ 𝑅𝑛 , and 𝑓(x) can be also written as 𝑓(x) = [𝑓𝐿 (x), 𝑓𝑈(x)], where 𝑓𝐿 (x) and 𝑓𝑈(x) are two real-valued functions defined on 𝑅𝑛 and satisfy 𝑓𝐿 (x) ≤ 𝑓𝑈(x) for every x ∈ Ω. Definition 6 (continuity of an interval-valued function). The function 𝑓 : Ω ⊆ 𝑅𝑛 → I is said to be continuous at 𝑥 ∈ Ω if both 𝑓𝐿 (x) and 𝑓𝑈(x) are continuous functions of x. The concept of gH-derivative of a function 𝑓 : (𝑎, 𝑏) → I is defined in [19]. Definition 7. Let 𝑥0 ∈ (𝑎, 𝑏) and ℎ be such that 𝑥0 + ℎ ∈ (𝑎, 𝑏), then the gH-derivative of a function 𝑓 : (𝑎, 𝑏) → I at 𝑥0 is defined as 𝑓󸀠 (𝑥0 ) = lim [𝑓 (𝑥0 + ℎ) ⊖𝑔 𝑓 (𝑥0 )] .

(4)

𝑥→0

If 𝑓󸀠 (𝑥0 ) ∈ I exists, then we say that 𝑓 is generalized Hukuhara differentiable (gH-differentiable, for short) at 𝑥0 . Moreover, [21] also proved the following theorem. Theorem 8. Let 𝑓 : (𝑎, 𝑏) → I be such that 𝑓(𝑥) = [𝑓𝐿 (𝑥), 𝑓𝑈(𝑥)]. The function 𝑓(𝑥) is gH-differentiable if and only if 𝑓𝐿 (𝑥) and 𝑓𝑈(𝑥) are differentiable real-valued functions. Furthermore, 󸀠

󸀠

𝑓󸀠 (𝑥) = [min {(𝑓𝐿 ) (𝑥) , (𝑓𝑈) (𝑥)} , 𝐿 󸀠

(5)

𝑈 󸀠

max {(𝑓 ) (𝑥) , (𝑓 ) (𝑥)}] . Definition 9 (gradient of an interval-valued function). Let 𝑓(x) be an interval-valued function defined on Ω, where Ω is an open subset of 𝑅𝑛 . Let 𝐷𝑥𝑖 (𝑖 = 1, 2, . . . , 𝑛) stand for the partial differentiation with respect to the 𝑖th variable 𝑥𝑖 . Assume that 𝑓𝐿 (x) and 𝑓𝑈(x) have continuous partial derivatives so that 𝐷𝑥𝑖 𝑓𝐿 (x) and 𝐷𝑥𝑖 𝑓𝑈(x) are continuous. For 𝑖 = 1, 2, . . . , 𝑛, define 𝐷𝑥𝑖 𝑓 (x) = [min (𝐷𝑥𝑖 𝑓𝐿 (x) , 𝐷𝑥𝑖 𝑓𝑈 (x)) ,

(6)

max (𝐷𝑥𝑖 𝑓𝐿 (x) , 𝐷𝑥𝑖 𝑓𝑈 (x))] . We will say that 𝑓(x) is differentiable at x, and we write 𝑡

∇𝑓 (x) = (𝐷𝑥1 𝑓 (x) , 𝐷𝑥2 𝑓 (x) , . . . , 𝐷𝑥𝑛 𝑓 (x)) .

(7)

We call ∇𝑓(x) the gradient of the interval-valued univex function at x.

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Example 10. Let 𝑓 : R2 → I defined by 𝑓(x) = [𝑥12 + 𝑥22 , 2𝑥12 + 2𝑥22 + 3]. So 𝑓𝐿 (x) = 𝑥12 + 𝑥22 and 𝑓𝑈(x) = 2𝑥12 + 2𝑥22 + 3. 𝐷𝑥1 𝑓𝐿 (x) = 2𝑥1 , 𝐷𝑥2 𝑓𝐿 (x) = 2𝑥2 , 𝐷𝑥1 𝑓𝑈(x) = 4𝑥1 , 𝐷𝑥2 𝑓𝑈(x) = 4𝑥2 . Thus, {[2𝑥1 , 4𝑥1 ] if 𝑥1 ≥ 0, 𝐷𝑥1 𝑓 (x) = { [4𝑥1 , 2𝑥1 ] if 𝑥1 < 0, { {[2𝑥2 , 4𝑥2 ] if 𝑥2 ≥ 0, 𝐷𝑥2 𝑓 (x) = { [4𝑥2 , 2𝑥2 ] if 𝑥2 < 0. {

(8)

𝑡

if 𝑥1 ≥ 0, 𝑥2 ≥ 0, if 𝑥1 ≥ 0, 𝑥2 < 0, if 𝑥1 < 0, 𝑥2 ≥ 0, if 𝑥1 < 0, 𝑥2 < 0. (9)

Further, 𝑡

(2𝑥1 , 2𝑥2 ) { { { { { 𝑡 { { (2𝑥1 , 4𝑥2 ) { { ∇𝐿 𝑓 (x) = { { 𝑡 { { (4𝑥1 , 2𝑥2 ) { { { { { 𝑡 {(4𝑥1 , 4𝑥2 )

if 𝑥1 ≥ 0, 𝑥2 ≥ 0,

𝑡

if 𝑥1 ≥ 0, 𝑥2 ≥ 0,

(4𝑥1 , 4𝑥2 ) { { { { { 𝑡 { { { {(4𝑥1 , 2𝑥2 ) 𝑈 ∇ 𝑓 (x) = { { 𝑡 { { (2𝑥1 , 4𝑥2 ) { { { { { 𝑡 {(2𝑥1 , 4𝑥2 )

if 𝑥1 < 0, 𝑥2 ≥ 0,

(12)

Example 15. Consider the real-valued function 𝜙1 given by 𝜙1 (𝑥) = 𝑥 + 1, 𝑥 ∈ 𝑅, then we can obtain Φ1 ([𝑎𝐿 , 𝑎𝑈]) = [𝑎𝐿 + 1, 𝑎𝑈 + 1]. If 𝜙2 (𝑥) = |𝑥|, 𝑥 ∈ 𝑅. Then if 𝑎𝐿 ≥ 0, [𝑎𝐿 , 𝑎𝑈] { { { { 𝑈 𝐿 { if 𝑎𝑈 ≤ 0, Φ2 ([𝑎𝐿 , 𝑎𝑈]) = {[−𝑎 , −𝑎 ] { { { {[0, max (−𝑎𝐿 , 𝑎𝑈)] if 𝑎𝐿 < 0, 𝑎𝑈 ≥ 0. { (13)

Example 17. Let 𝑓(𝑥) = [𝑥3 , 𝑥3 + 1], 𝑥 ∈ 𝑅, (10)

if 𝑥1 ≥ 0, 𝑥2 < 0,

2

{ 𝑦 if 𝑥 ≥ 𝑦, 𝑏 (𝑥, 𝑦) = { 𝑥 − 𝑦 if 𝑥 ≤ 𝑦, {0 𝜂 (𝑥, 𝑦) = {

if 𝑥1 < 0, 𝑥2 ≥ 0, if 𝑥1 < 0, 𝑥2 < 0.

Remark 11. If 𝑓𝐿 = 𝑓𝑈, then ∇𝑓(x) of interval-valued functions is the extension of ∇𝑓(x), where 𝑓 : Ω → 𝑅. The concept of convexity plays an important role in the optimization theory. In recent years, the concept of convexity has been generalized in several directions by using novel and innovative techniques. An important generalization of convex functions is the introduction of univex functions, which was introduced by Bector et al. [15]. Let 𝐾 be a nonempty open set in 𝑅𝑛 , and let 𝑓 : 𝐾 → 𝑅, 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , Φ : 𝑅 → 𝑅, and 𝑏 : 𝐾 × 𝐾 × [0, 1] → 𝑅+ , 𝑏 = 𝑏(x, y, 𝜆). If the function 𝑓 is differentiable, then 𝑏 does not depend on 𝜆; see [12] or [15]. Definition 12. A differentiable real-valued function 𝑓 is said to be univex at y ∈ 𝐾 with respect to 𝜂, Φ, 𝑏 if for all x ∈ 𝐾 𝑏 (x, y) Φ [𝑓 (x) − 𝑓 (y)] ≥ 𝜂𝑡 (x, y) ∇𝑓 (y) .

𝑏 (x, y) Φ [𝑓 (x) ⊖𝑔 𝑓 (y)] ⪰ 𝜂𝑡 (x, y) ∇𝑓 (y) .

Example 16. Let 𝑓(𝑥) = [𝑥2 , 2𝑥2 + 3], 𝑥 ∈ 𝑅, 𝑏 = 1, 𝜂(𝑥, 𝑦) = 𝑥 − 𝑦, Φ = Φ2 , then 𝑓(𝑥) is univex with respect to 𝑏, 𝜂, and Φ.

if 𝑥1 ≥ 0, 𝑥2 < 0,

if 𝑥1 < 0, 𝑥2 < 0.

Definition 13 (interval-valued univex function). A differentiable interval-valued function 𝑓 is said to be univex at y ∈ 𝐾 with respect to 𝜂, Φ, 𝑏 if for all x ∈ 𝐾

Remark 14. (i) An interval-valued univex function is the extension of a univex function by 𝑓𝐿 = 𝑓𝑈. (ii) Φ : I → I could be deduced from 𝜙 : 𝑅 → 𝑅 by Φ(𝐴) := {𝑦 : ∃𝑥 ∈ 𝐴, 𝜙(𝑥) = 𝑦, 𝑦 ∈ 𝑅}.

Thus, ([2𝑥1 , 4𝑥1 ] , [2𝑥2 , 4𝑥2 ]) { { { { { 𝑡 { { { {([2𝑥1 , 4𝑥1 ] , [4𝑥2 , 2𝑥2 ]) ∇𝑓 (x) = { { 𝑡 { { ([4𝑥1 , 2𝑥1 ] , [2𝑥2 , 4𝑥2 ]) { { { { { 𝑡 {([4𝑥1 , 2𝑥1 ] , [4𝑥2 , 4𝑥2 ])

Let 𝐾 be a nonempty open set in 𝑅𝑛 , and let 𝑓 : 𝐾 → I be an interval-valued function, 𝜂 : 𝐾 × 𝐾 → 𝑅𝑛 , Φ : I → I, and 𝑏 : 𝐾 × 𝐾 × [0, 1] → 𝑅+ , 𝑏 = 𝑏(x, y, 𝜆).

(11)

2

(14)

2

𝑥 + 𝑦 + 𝑥𝑦 if 𝑥 ≥ 𝑦, 𝑥−𝑦 if 𝑥 ≤ 𝑦.

Let Φ : I → I be defined by Φ([𝑎𝐿 , 𝑎𝑈]) = 3[𝑎𝐿 , 𝑎𝑈], then 𝑓(𝑥) is univex with respect to 𝑏, 𝜂 and Φ.

4. Optimality Criteria Let 𝑓(x), 𝑔1 (x), . . . , 𝑔𝑚 (x) be differentiable interval-valued functions defined on a nonempty open set 𝑋 ⊆ 𝑅𝑛 . Throughout this paper we consider the following primal problem (P): min

𝑓 (x)

s.t. 𝑔 (x) ⪯ 0,

𝑖 = 1, 2, . . . , 𝑚.

(P)

Let 𝑃 := {x ∈ 𝑋 : 𝑔(x) ⪯ 0, 𝑖 = 1, 2, . . . , 𝑚}. We say x∗ is an optimal solution of (P) if 𝑓(x) ⪰ 𝑓(x∗ ) for all P-feasible x. In this section, we obtain sufficient optimality conditions for a feasible solution x∗ to be efficient or properly efficient for (P) in the form of the following theorems.

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Theorem 18. Let x∗ be P-feasible. Suppose that

It is equivalent to

(i) there exist 𝜂, Φ0 , 𝑏0 , Φ𝑖 , 𝑏𝑖 , 𝑖 = 1, 2, . . . , 𝑚 such that 𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )] ⪰ 𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ ) , −𝑏𝑖 (x, x∗ ) Φ𝑖 [𝑔𝑖 (x∗ )] ⪰ 𝜂𝑡 (x, x∗ ) ∇𝑔𝑖 (x∗ )

(15) (16)

𝐿

𝑚

𝐿

{𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )} = −{𝜂𝑡 (x, x∗ ) ∑ 𝑦𝑖 ∇𝑔 (x∗ )} , 𝑖=1

𝑈

𝑚

𝑈

(28)

{𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )} = −{𝜂𝑡 (x, x∗ ) ∑ 𝑦𝑖 ∇𝑔 (x∗ )} . 𝑖=1

for all feasible x; From (16), we have

(ii) there exist y∗ = (𝑦1 , 𝑦2 , . . . , 𝑦𝑚 )𝑡 ∈ 𝑅𝑚 such that 𝑚

∗

∗

∇𝑓 (x ) + ∑ 𝑦𝑖 ∇𝑔 (x ) = 0,

(17)

y∗ ≥ 0.

(18)

𝐿

𝐿

𝑈

𝑈

𝑈

𝐿

{−𝑏𝑖 (x, x∗ ) Φ𝑖 [𝑔𝑖 (x∗ )]} ≥ {𝜂𝑡 (x, x∗ ) ∇𝑔𝑖 (x∗ )} , {−𝑏𝑖 (x, x∗ ) Φ𝑖 [𝑔𝑖 (x∗ )]} ≥ {𝜂𝑡 (x, x∗ ) ∇𝑔𝑖 (x∗ )} .

𝑖=1

From Definition 1, we have

Further, suppose that

−{𝑏𝑖 (x, x∗ ) Φ𝑖 [𝑔𝑖 (x∗ )]} ≥ {𝜂𝑡 (x, x∗ ) ∇𝑔𝑖 (x∗ )} ,

Φ0 (𝐴) ⪰ 0 󳨐⇒ 𝐴 ⪰ 0,

(19)

𝐴 ⪯ 0 󳨐⇒ Φ𝑖 (𝐴) ⪰ 0,

(20)

𝑏0 (x, x∗ ) > 0,

𝑏𝑖 (x, x∗ ) > 0

∗

∗

𝐿

𝑡

∗

𝑈

∗

Thus, {𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )]}

∗

for all feasible x. Then, x is an optimal solution of (P).

≥ {𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )}

Proof. Let x be P-feasible. Then, (22)

𝑡

∗

𝐿

𝐿

𝑚

𝐿 ∗

= −{𝜂 (x, x ) ∑ 𝑦𝑖 ∇𝑔 (x )} 𝑖=1

From (20), we conclude that 𝑚

Φ𝑖 [𝑔𝑖 (x)] ⪰ 0.

≥ ∑ {𝑏𝑖 (x, x∗ ) Φ𝑖 [𝑔𝑖 (x∗ )]}

(23)

𝑈

𝑖=1

Thus,

(31)

𝑚

≥ ∑ {𝑏𝑖 (x, x∗ ) Φ𝑖 [𝑔𝑖 (x∗ )]}

Φ𝐿𝑖 [𝑔𝑖 (x)] ≥ 0, Φ𝑈 𝑖 [𝑔𝑖

(30)

−{𝑏𝑖 (x, x ) Φ𝑖 [𝑔𝑖 (x )]} ≥ {𝜂 (x, x ) ∇𝑔𝑖 (x )} .

(21)

𝑔𝑖 (x) ⪯ 0.

(29)

𝑖=1

(24)

(x)] ≥ 0.

𝐿

≥ 0,

By (15) and Definition 2, we have 𝐿

{𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )]}

𝐿

{𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )]} ≥ {𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )} , 𝑈

𝑈

≥ {𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )]}

𝑈

{𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )]} ≥ {𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )} . (25)

𝐿

≥ 0. So,

From (17), 𝑡

∗

∗

𝑡

∗

𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )] ⪰ 0.

𝑚

∗

𝜂 (x, x ) ∇𝑓 (x ) + 𝜂 (x, x ) ∑ 𝑦𝑖 ∇𝑔 (x ) = 0.

(26)

𝑖=1

From (21), it follows that

It follows from Definition 2 that 𝑡

∗

∗

𝐿

𝑡

∗

𝑚

𝑖=1

∗

∗

𝑈

𝑡

∗

𝑚

Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )] ⪰ 0.

(33)

𝑓 (x) ⊖𝑔 𝑓 (x∗ ) ⪰ 0.

(34)

By (19),

𝐿 ∗

{𝜂 (x, x ) ∇𝑓 (x )} + {𝜂 (x, x ) ∑ 𝑦𝑖 ∇𝑔 (x )} = 0, 𝑡

(32)

From Proposition 4, it follows that

𝑈 ∗

{𝜂 (x, x ) ∇𝑓 (x )} + {𝜂 (x, x ) ∑ 𝑦𝑖 ∇𝑔 (x )} = 0. 𝑖=1

(27)

𝑓 (x) ⪰ 𝑓 (x∗ ) . Therefore, x∗ is an optimal solution of (P).

(35)

Journal of Applied Mathematics

5

Theorem 19. Let x∗ be P-feasible. Suppose that

From (38) and Definition 2, it follows that

(i) there exist 𝜂, Φ0 , 𝑏0 , Φ𝑖 , 𝑏𝑖 , 𝑖 = 1, 2, . . . , 𝑚 such that 𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ ) ⪰ 0 󳨐⇒ 𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )] ⪰ 0, (36)

𝐿

𝑚

𝐿

{𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )} + {𝜂𝑡 (x, x∗ ) ∑ 𝑦𝑖 ∇𝑔 (x∗ )} = 0, 𝑖=1

𝑡

∗

𝑈

∗

𝑡

∗

𝑚

𝑈 ∗

{𝜂 (x, x ) ∇𝑓 (x )} + {𝜂 (x, x ) ∑ 𝑦𝑖 ∇𝑔 (x )} = 0.

−𝑏𝑖 (x, x∗ ) Φ𝑖 [𝑔𝑖 (x∗ )] ⪯ 0 󳨐⇒ 𝜂𝑡 (x, x∗ ) ∇𝑔𝑖 (x∗ ) ⪯ 0

𝑖=1

(37)

(48) It is equivalent to

for all feasible x; 𝑡

∗

𝑚

(ii) there exist y = (𝑦1 , 𝑦2 , . . . , 𝑦𝑚 ) ∈ 𝑅 such that 𝑚

∇𝑓 (x∗ ) + ∑ 𝑦𝑖 ∇𝑔 (x∗ ) = 0,

𝑚

𝑈

y∗ ≥ 0.

𝐿

𝑖=1

(38)

𝑖=1

(39)

𝑈

(49)

{𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )} = −{𝜂𝑡 (x, x∗ ) ∑ 𝑦𝑖 ∇𝑔 (x∗ )} . 𝑖=1

Therefore,

Further, suppose that Φ0 (𝐴) ⪰ 0 󳨐⇒ 𝐴 ⪰ 0,

(40)

𝐴 ⪯ 0 󳨐⇒ Φ𝑖 (𝐴) ⪰ 0,

(41)

∗

∗

𝑏0 (x, x ) > 0,

𝑏𝑖 (x, x ) > 0

(42)

𝐿

{𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )} ≥ 0, 𝑡

Proof. Let x be P-feasible. Then, 𝑔𝑖 (x∗ ) ⪯ 0, from (41), we obtain that ∗

Φ𝑖 [𝑔𝑖 (x )] ⪰ 0.

(43)

∗

𝑈

𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ ) ⪰ 0.

(51)

𝑏0 (x, x∗ ) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (x∗ )] ⪰ 0.

(52)

By (36),

Then, from (40) and (42), we have 𝑓 (x) ⊖𝑔 𝑓 (x∗ ) ⪰ 0.

−𝑏𝑖 (x, x∗ ) Φ𝑖 [𝑔𝑖 (x∗ )] ⪯ 0.

(44)

𝑓 (x) ⪰ 𝑓 (x∗ ) .

𝜂 (x, x ) ∇𝑔𝑖 (x ) ⪯ 0.

(45)

(53)

From Proposition 4, it follows that

By (37), ∗

(50)

From Definition 2, we obtain that

So,

∗

∗

{𝜂 (x, x ) ∇𝑓 (x )} ≥ 0.

for all feasible x. Then, x∗ is an optimal solution of (P).

𝑡

𝑚

𝐿

{𝜂𝑡 (x, x∗ ) ∇𝑓 (x∗ )} = −{𝜂𝑡 (x, x∗ ) ∑ 𝑦𝑖 ∇𝑔 (x∗ )} ,

(54)

Therefore, x∗ is an optimal solution of (P).

Thus,

5. Duality

𝑚

𝑡

∗

∗

− ∑ 𝑦𝑖 𝜂 (x, x ) ∇𝑔𝑖 (x ) ⪰ 0.

(46)

𝑖=1

Consider the following: max

Then, we have 𝑚

𝑚

𝑈 𝑡

∗

𝑓 (u)

s.t.

∗

− {∑ 𝑦𝑖 𝜂 (x, x ) ∇𝑔𝑖 (x )}

∇𝑓 (u) + ∑ 𝑦𝑖 ∇𝑔𝑖 (u) = 0, 𝑖=1

𝑖=1

𝑦𝑖 ∇𝑔𝑖 (u) ⪰ 0

𝐿

𝑚

𝑡

∗

∗

= {− ∑ 𝑦𝑖 𝜂 (x, x ) ∇𝑔𝑖 (x )} ≥ 0, 𝑖=1

𝑚

𝑡

∗

∗

− {∑ 𝑦𝑖 𝜂 (x, x ) ∇𝑔𝑖 (x )} 𝑖=1

𝑚

𝑈

= {− ∑ 𝑦𝑖 𝜂𝑡 (x, x∗ ) ∇𝑔𝑖 (x∗ )} ≥ 0. 𝑖=1

𝑦𝑖 ≥ 0. (47)

𝐿

(D)

Theorem 20 (weak duality). Let x be P-feasible, and let (u, y) be D-feasible. Assume that there exist 𝜂, Φ0 , 𝑏0 , Φ𝑖 , 𝑏𝑖 , 𝑖 = 1, 2, . . . , 𝑚 such that 𝑏0 (x, u) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (u)] ⪰ 𝜂𝑡 (x, u) ∇𝑓 (u) , −𝑏𝑖 (x, u) Φ𝑖 [𝑔𝑖 (u)] ⪰ 𝜂𝑡 (x, u) ∇𝑔𝑖 (u)

(55)

6

Journal of Applied Mathematics

at u;

Since (u, y) is D-feasible we can obtain that, 𝑚

Φ0 (𝐴) ⪰ 0 󳨐⇒ 𝐴 ⪰ 0, 𝑏0 (x, u) > 0,

∇𝑓 (u) + ∑𝑦𝑖 ∇𝑔𝑖 (u) = 0.

(56)

𝑏𝑖 (x, u) ≥ 0

So,

and ∑𝑚 𝑖=1 𝑏𝑖 (x, u)𝑦𝑖 Φ𝑖 (𝑔𝑖 (u)) ⪰ 0. Then, 𝑓(x) ⪰ 𝑓(u).

𝑚

𝜂𝑡 (x, u) ∇𝑓 (u) + 𝜂𝑡 (x, u) ∑ 𝑦𝑖 ∇𝑔𝑖 (u) = 0.

Proof. It is similar to the proof of Theorem 18.

𝜂𝑡 (x, u) ∇𝑓 (u) ⪰ 0 󳨐⇒ 𝑏0 (x, u) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (u)] ⪰ 0, (57) 𝑚

𝑖=1

𝑖=1

𝑡

−𝑏1 (x, u) Φ1 [∑ 𝑦𝑖 𝑔𝑖 (u)] ⪯ 0 󳨐⇒ ∑ 𝑦𝑖 𝜂 (x, u) ∇𝑔𝑖 (u) ⪯ 0 (58) at u;

By Definition 2, it follows that 𝑚

𝐿

𝐿

{𝜂𝑡 (x, u) ∇𝑓 (u)} + {𝜂𝑡 (x, u) ∑ 𝑦𝑖 ∇𝑔𝑖 (u)} = 0, 𝑖=1 𝑚

𝑈

𝑈

Φ0 (𝐴) ⪰ 0 󳨐⇒ 𝐴 ⪰ 0,

(59)

𝐴 ⪰ 0 󳨐⇒ Φ1 (𝐴) ⪰ 0,

(60)

𝑏1 (x, u) ≥ 0.

(61)

Then, 𝑓(x) ⪰ 𝑓(u).

𝑖=1

Therefore, 𝑚

𝐿

𝐿

{𝜂𝑡 (x, u) ∇𝑓 (u)} = −{𝜂𝑡 (x, u) ∑ 𝑦𝑖 ∇𝑔𝑖 (u)} ≥ 0, 𝑚

𝑈

𝑈

(68)

{𝜂𝑡 (x, u) ∇𝑓 (u)} = −{𝜂𝑡 (x, u) ∑ 𝑦𝑖 ∇𝑔𝑖 (u)} ≥ 0. 𝑖=1

Then, 𝜂𝑡 (x, u) ∇𝑓 (u) ⪰ 0.

(69)

𝑏0 (x, u) Φ0 [𝑓 (x) ⊖𝑔 𝑓 (u)] ⪰ 0.

(70)

By (57),

Since (u, y) is D-feasible, then 𝑦𝑖𝑡 ∇𝑔𝑖 (u)

Proof. and (61),

⪰ 0, from (60)

From (59) and (61),

𝑚

−𝑏1 (x, u) Φ1 [∑ 𝑦𝑖 𝑔𝑖 (u)] ⪯ 0.

(62)

𝑖=1

𝑚

𝑡

∑𝑦𝑖 𝜂 (x, u) ∇𝑔𝑖 (u) ⪯ 0.

𝑚

− ∑ 𝑦𝑖 𝜂𝑡 (x, u) ∇𝑔𝑖 (u) ⪰ 0, 𝑖=1

𝑈

𝑚

− {∑ 𝑦𝑖 𝜂𝑡 (x, u) ∇𝑔𝑖 (u)} 𝑖=1

𝐿 𝑡

= {− ∑ 𝑦𝑖 𝜂 (x, u) ∇𝑔𝑖 (u)} ≥ 0, 𝑖=1

(72)

(64)

Proof. Since a constraint qualification is satisfied at x∗ , there exists y∗ ∈ 𝑅𝑚 such that the following Kuhn-Tucker conditions are satisfied: 𝑚

∇𝑓 (x∗ ) + ∑ 𝑦𝑖 ∇𝑔𝑖 (x∗ ) = 0,

𝐿 𝑡

− {∑ 𝑦𝑖 𝜂 (x, u) ∇𝑔𝑖 (u)}

𝑚

𝑖=1

∑ 𝑦𝑖 ∇𝑔𝑖 (x∗ ) = 0,

𝑖=1

𝑈

= {− ∑ 𝑦𝑖 𝜂𝑡 (x, u) ∇𝑔𝑖 (u)} ≥ 0. 𝑖=1

𝑓 (x) ⪰ 𝑓 (u) .

Theorem 22 (strong duality). If x∗ is P-optimal and a constraint qualification is satisfied at x∗ , then there exists y∗ = (𝑦1 , 𝑦2 , . . . , 𝑦𝑚 )𝑡 ∈ 𝑅𝑚 such that (x∗ , y∗ ) is D-feasible and the values of the objective functions for (P) and (D) are equal at x∗ and (x∗ , y∗ ), respectively. Furthermore, if for all P-feasible x and D-feasible (u, y), the hypotheses of Theorem 19 are satisfied, then (x∗ , y∗ ) is D-optimal.

Thus,

𝑚

(71)

(63)

𝑖=1

𝑚

𝑓 (x) ⊖𝑔 𝑓 (u) ⪰ 0. thus,

Then, we have

𝑚

(67)

{𝜂𝑡 (x, u) ∇𝑓 (u)} + {𝜂𝑡 (x, u) ∑ 𝑦𝑖 ∇𝑔𝑖 (u)} = 0.

𝑖=1

𝑏0 (x, u) > 0,

(66)

𝑖=1

Theorem 21 (weak duality). Let x be P-feasible, and let (u, y) be D-feasible. Assume that there exist 𝜂, Φ0 , 𝑏0 , Φ1 , 𝑏1 such that

𝑚

(65)

𝑖=1

𝑖=1

𝑦𝑖 ≥ 0.

Therefore, (x∗ , y∗ ) is D-feasible.

(73)

Journal of Applied Mathematics

7

Suppose that (x∗ , y∗ ) is not D-optimal. Then, there exists a D-feasible (u, y) such that 𝑓(u) ≻ 𝑓(x∗ ). This contradicts Theorem 20. Therefore, (x∗ , y∗ ) is D-optimal.

6. Numerical Example Consider the following example: minimize 𝑓 (x) = [𝑥1 − sin (𝑥2 ) + 1, 𝑥1 − sin (𝑥2 ) + 3] subject to

𝑔1 (x) = [sin (𝑥1 ) − 4 sin (𝑥2 ) − 2, sin (𝑥1 ) − 4 sin (𝑥2 )] ⪯ 0, 𝑔2 (x) = [2 sin (𝑥1 ) + 7 sin (𝑥2 ) + 𝑥1 − 7, 2 sin (𝑥1 ) + 7 sin (𝑥2 ) + 𝑥1 − 6] ⪯ 0, 𝑔3 (x) = [2𝑥1 + 2𝑥2 − 5, 2𝑥1 + 2𝑥2 − 3] ⪯ 0, 𝑔4 (x) = [4𝑥12 + 4𝑥22 − 12, 4𝑥12 + 4𝑥22 − 9] ⪯ 0, 𝑔5 (x) = [− sin (𝑥1 ) − 1, − sin (𝑥1 )] ⪯ 0, 𝑔6 (x) = [− sin (𝑥2 ) − 1, − sin (𝑥2 )] ⪯ 0. (74)

Note that the interval-valued objective function is univex with respect to 𝑏 = 1, 𝜂(x, u) = x − u, Φ([𝑎𝐿 , 𝑎𝑈]) = [𝑎𝐿 , 𝑎𝑈], and every 𝑔𝑖 (𝑖 = 1, 2, . . . , 6) is univex with respect to 𝑏 = 1, Φ([𝑎𝐿 , 𝑎𝑈]) = [𝑎𝐿 , 𝑎𝑈] 𝜂 (x, u) = (

sin 𝑥1 − sin 𝑢1 sin 𝑥2 − sin 𝑢2 𝑇 , ) , cos 𝑢1 cos 𝑢2

(75)

where x = (𝑥1 , 𝑥2 )𝑇 and u = (𝑢1 , 𝑢2 )𝑇 . It is easy to see that the problem satisfies the assumptions of Theorem 18. Then, 𝑇

𝑇

(1, − cos 𝑥2 ) + 𝜇1 (cos 𝑥1 , −4 cos 𝑥2 ) 𝑇

+ 𝜇2 (2 cos 𝑥1 + 1, 7 cos 𝑥2 ) + 𝜇3 (2, 2)𝑇 𝑇

+ 𝜇4 (8𝑥1 , 8𝑥2 ) + 𝜇5 (− cos 𝑥1 , 0)

𝑇

(76)

𝑇

+ 𝜇6 (0, − cos 𝑥2 ) = (0, 0)𝑇 . After some algebraic calculations, we obtain that x∗ = (0, sin−1 (6/7))𝑇 and u∗ = (0, 1/7, 0, 0, 10/7, 0)𝑇 . Therefore, x∗ is a solution.

Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant no. 60974082), and the Science Plan Foundation of the Education Bureau of Shaanxi Province (nos. 11JK1051, 2013JK1098, 2013JK1130, and 2013JK1182).

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