Univex Interval-Valued Mapping with Differentiability and Its Application in Nonlinear Programming

Abstract

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, interval-valued 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 interval-valued objective functions under the assumption of LU-convexity. 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 interval-valued 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 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=[𝑎^𝐿,𝑎^𝑈]∈ℐ denotes a closed interval, where 𝑎^𝐿 and 𝑎^𝑈 mean the lower and upper bounds of 𝐴, respectively. For every 𝑎∈𝑅, we denote 𝑎=[𝑎,𝑎].

Definition 1. Let 𝐴=[𝑎^𝐿,𝑎^𝑈] and 𝐵=[𝑏^𝐿,𝑏^𝑈] be in ℐ; one has (i)𝐴+𝐵={𝑎+𝑏∶𝑎∈𝐴 and 𝑏∈𝐵}=[𝑎^𝐿+𝑏^𝐿,𝑎^𝑈+𝑏^𝑈]; (ii) −𝐴={−𝑎∶𝑎∈𝐴}=[−𝑎^𝑈,−𝑎^𝐿]; (iii)𝐴×𝐵={𝑎𝑏∶𝑎∈𝐴and𝑏∈𝐵}=[min_𝑎𝑏,max_𝑎𝑏], where min_𝑎𝑏=min{𝑎^𝐿𝑏^𝐿,𝑎^𝐿𝑏^𝑈,𝑎^𝑈𝑏^𝐿,𝑎^𝑈𝑏^𝑈} and max_𝑎𝑏=max{𝑎^𝐿𝑏^𝐿,𝑎^𝐿𝑏^𝑈,𝑎^𝑈𝑏^𝐿,𝑎^𝑈𝑏^𝑈}.
Then, we can see that 𝐴−𝐵=𝐴+(−𝐵)=𝑎^𝐿−𝑏^𝑈,𝑎^𝑈−𝑏^𝐿,𝑘𝐴={𝑘𝑎∶𝑎∈𝐴}=𝑘𝑎^𝐿,𝑘𝑎^𝑈if𝑘≥0,𝑘𝑎^𝑈,𝑘𝑎^𝐿if𝑘<0,(1) 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: 𝑑_𝐻(𝐴,𝐵)=max𝑎^𝐿−𝑏^𝐿,𝑎^𝑈−𝑏^𝑈.(2)

Definition 2. Let 𝐴=[𝑎^𝐿,𝑎^𝑈] and 𝐵=[𝑏^𝐿,𝑏^𝑈] be two closed intervals in 𝑅. One writes 𝐴⪯𝐵 if and only if 𝑎^𝐿≤𝑏^𝐿 and 𝑎^𝑈≤𝑏^𝑈, 𝐴≺𝐵 if and only if 𝐴⪯𝐵 and 𝐴≠𝐵, that is, the following (a1), (a2), or (a3) is satisfied:(a1)𝑎^𝐿<𝑏^𝐿 and 𝑎^𝑈≤𝑏^𝑈;(a2)𝑎^𝐿≤𝑏^𝐿 and 𝑎^𝑈<𝑏^𝑈;(a3)𝑎^𝐿<𝑏^𝐿 and 𝑎^𝑈<𝑏^𝑈.

Definition 3 (see [20]). Let 𝐴=[𝑎^𝐿,𝑎^𝑈] and 𝐵=[𝑏^𝐿,𝑏^𝑈] be two closed intervals, the gH-difference of 𝐴 and 𝐵 is defined by 𝑎^𝐿,𝑎^𝑈⊖_𝑔𝑏^𝐿,𝑏^𝑈=min𝑎^𝐿−𝑏^𝐿,𝑎^𝑈−𝑏^𝑈,max𝑎^𝐿−𝑏^𝐿,𝑎^𝑈−𝑏^𝑈.(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⊖_𝑔𝐵 always exists and 𝐴⊖_𝑔𝐵∈ℐ.(ii)𝐴⊖_𝑔𝐵⪯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⊆𝑅^𝑛. Then, 𝑓(𝐱)=𝑓(𝑥₁,…,𝑥_𝑛) is a closed interval in 𝑅 for each 𝐱∈𝑅^𝑛, and 𝑓(𝐱) can be also written as 𝑓(𝐱)=[𝑓^𝐿(𝐱),𝑓^𝑈(𝐱)], where 𝑓^𝐿(𝐱) and 𝑓^𝑈(𝐱) are two real-valued functions defined on 𝑅^𝑛 and satisfy 𝑓^𝐿(𝐱)≤𝑓^𝑈(𝐱) for every 𝐱∈Ω.

Definition 6 (continuity of an interval-valued function). The function 𝑓∶Ω⊆𝑅^𝑛→ℐ is said to be continuous at 𝑥∈Ω if both 𝑓^𝐿(𝐱) and 𝑓^𝑈(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_𝑥→0𝑓𝑥₀+ℎ⊖_𝑔𝑓𝑥₀.(4) 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)=[𝑓^𝐿(𝑥),𝑓^𝑈(𝑥)]. The function 𝑓(𝑥) is gH-differentiable if and only if 𝑓^𝐿(𝑥) and 𝑓^𝑈(𝑥) are differentiable real-valued functions. Furthermore, 𝑓^′(𝑥)=min𝑓^𝐿′(𝑥),𝑓^𝑈′(𝑥,𝑓)max^𝐿′(𝑥),𝑓^𝑈′(𝑥).(5)

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 𝑅^𝑛. Let 𝐷_𝑥𝑖(𝑖=1,2,…,𝑛) stand for the partial differentiation with respect to the 𝑖th variable 𝑥_𝑖. Assume that 𝑓^𝐿(𝐱) and 𝑓^𝑈(𝐱) have continuous partial derivatives so that 𝐷_𝑥𝑖𝑓^𝐿(𝐱) and 𝐷_𝑥𝑖𝑓^𝑈(𝐱) are continuous. For 𝑖=1,2,…,𝑛, define 𝐷_𝑥𝑖𝑓(𝐱)=min𝐷_𝑥𝑖𝑓^𝐿(𝐱),𝐷_𝑥𝑖𝑓^𝑈(𝐱,𝐷)max_𝑥𝑖𝑓^𝐿(𝐱),𝐷_𝑥𝑖𝑓^𝑈(𝐱).(6) We will say that 𝑓(𝐱) is differentiable at 𝐱, and we write ∇𝑓(𝐱)=𝐷_𝑥1𝑓(𝐱),𝐷_𝑥2𝑓(𝐱),…,𝐷_𝑥𝑛𝑓(𝐱)^𝑡.(7) We call ∇𝑓(𝐱) the gradient of the interval-valued univex function at 𝐱.

Example 10. Let 𝑓∶𝐑²→ℐ defined by 𝑓(𝐱)=[𝑥²₁+𝑥²₂,2𝑥²₁+2𝑥²₂+3]. So 𝑓^𝐿(𝐱)=𝑥²₁+𝑥²₂ and 𝑓^𝑈(𝐱)=2𝑥²₁+2𝑥²₂+3. 𝐷_𝑥1𝑓^𝐿(𝐱)=2𝑥₁,𝐷_𝑥2𝑓^𝐿(𝐱)=2𝑥₂,𝐷_𝑥1𝑓^𝑈(𝐱)=4𝑥₁,𝐷_𝑥2𝑓^𝑈(𝐱)=4𝑥₂. Thus, 𝐷_𝑥1𝑓(𝐱)=2𝑥₁,4𝑥₁if𝑥₁≥0,4𝑥₁,2𝑥₁if𝑥₁𝐷<0,_𝑥2𝑓(𝐱)=2𝑥₂,4𝑥₂if𝑥₂≥0,4𝑥₂,2𝑥₂if𝑥₂<0.(8)
Thus, ∇𝑓(𝐱)=2𝑥₁,4𝑥₁,2𝑥₂,4𝑥₂^𝑡if𝑥₁≥0,𝑥₂≥0,2𝑥₁,4𝑥₁,4𝑥₂,2𝑥₂^𝑡if𝑥₁≥0,𝑥₂<0,4𝑥₁,2𝑥₁,2𝑥₂,4𝑥₂^𝑡if𝑥₁<0,𝑥₂≥0,4𝑥₁,2𝑥₁,4𝑥₂,4𝑥₂^𝑡if𝑥₁<0,𝑥₂<0.(9) Further, ∇^𝐿𝑓(𝐱)=2𝑥₁,2𝑥₂^𝑡if𝑥₁≥0,𝑥₂≥0,2𝑥₁,4𝑥₂^𝑡if𝑥₁≥0,𝑥₂<0,4𝑥₁,2𝑥₂^𝑡if𝑥₁<0,𝑥₂≥0,4𝑥₁,4𝑥₂^𝑡if𝑥₁<0,𝑥₂∇<0._𝑈𝑓(𝐱)=4𝑥₁,4𝑥₂^𝑡if𝑥₁≥0,𝑥₂≥0,4𝑥₁,2𝑥₂^𝑡if𝑥₁≥0,𝑥₂<0,2𝑥₁,4𝑥₂^𝑡if𝑥₁<0,𝑥₂≥0,2𝑥₁,4𝑥₂^𝑡if𝑥₁<0,𝑥₂<0.(10)

Remark 11. If 𝑓^𝐿=𝑓^𝑈, 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 𝑅^𝑛, and let 𝑓∶𝐾→𝑅, 𝜂∶𝐾×𝐾→𝑅^𝑛, 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∈𝐾𝑏(𝐱,𝐲)Φ𝑓(𝐱)−𝑓(𝐲)≥𝜂^𝑡(𝐱,𝐲)∇𝑓(𝐲).(11)
Let 𝐾 be a nonempty open set in 𝑅^𝑛, and let 𝑓∶𝐾→ℐ be an interval-valued function, 𝜂∶𝐾×𝐾→𝑅^𝑛,Φ∶ℐ→ℐ, and 𝑏∶𝐾×𝐾×[0,1]→𝑅⁺,𝑏=𝑏(𝐱,𝐲,𝜆).

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∈𝐾𝑏(𝐱,𝐲)Φ𝑓(𝐱)⊖_𝑔𝑓(𝐲)≽𝜂^𝑡(𝐱,𝐲)∇𝑓(𝐲).(12)

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

Example 15. Consider the real-valued function 𝜙₁ given by 𝜙₁(𝑥)=𝑥+1,𝑥∈𝑅, then we can obtain Φ₁([𝑎^𝐿,𝑎^𝑈])=[𝑎^𝐿+1,𝑎^𝑈+1]. If 𝜙₂(𝑥)=|𝑥|,𝑥∈𝑅. Then Φ₂𝑎^𝐿,𝑎^𝑈=𝑎^𝐿,𝑎^𝑈if𝑎^𝐿≥0,−𝑎^𝑈,−𝑎^𝐿if𝑎^𝑈≤0,0,max−𝑎^𝐿,𝑎^𝑈if𝑎^𝐿<0,𝑎^𝑈≥0.(13)

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.

Example 17. Let 𝑓(𝑥)=[𝑥³,𝑥³+1],𝑥∈𝑅, 𝑏=𝑦𝑥,𝑦²𝜂=𝑥𝑥−𝑦if𝑥≥𝑦,0if𝑥≤𝑦,𝑥,𝑦²+𝑦²+𝑥𝑦if𝑥≥𝑦,𝑥−𝑦if𝑥≤𝑦.(14) Let Φ∶ℐ→ℐ be defined by Φ([𝑎^𝐿,𝑎^𝑈])=3[𝑎^𝐿,𝑎^𝑈], then 𝑓(𝑥) is univex with respect to 𝑏,𝜂 and Φ.

4. Optimality Criteria

Let 𝑓(𝐱),𝑔₁(𝐱),…,𝑔_𝑚(𝐱) be differentiable interval-valued functions defined on a nonempty open set 𝑋⊆𝑅^𝑛. Throughout this paper we consider the following primal problem (P): (𝐱)(𝐱)(P)min𝑓s.t.𝑔⪯0,𝑖=1,2,…,𝑚.

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.

Theorem 18. Let 𝐱^∗ be P-feasible. Suppose that(i) there exist 𝜂, Φ₀, 𝑏₀, Φ_𝑖, 𝑏_𝑖, 𝑖=1,2,…,𝑚 such that 𝑏₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗≽𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗,(15)−𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗≽𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗(16) for all feasible 𝐱;(ii) there exist 𝐲^∗=(𝑦₁,𝑦₂,…,𝑦_𝑚)^𝑡∈𝑅^𝑚 such that ∇𝑓𝐱^∗+^𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝐲=0,(17)_∗≥0.(18)Further, suppose that Φ₀(𝐴)≽0⇒𝐴≽0,(19)𝐴⪯0⇒Φ_𝑖(𝐴)𝑏≽0,(20)₀𝐱,𝐱^∗>0,𝑏_𝑖𝐱,𝐱^∗>0(21) for all feasible 𝐱. Then, 𝐱^∗ is an optimal solution of (P).

Proof. Let 𝐱 be P-feasible. Then, 𝑔_𝑖(𝐱)⪯0.(22) From (20), we conclude that Φ_𝑖𝑔_𝑖(𝐱)≽0.(23) Thus, Φ^𝐿_𝑖𝑔_𝑖(𝐱Φ)≥0,_𝑈𝑖𝑔_𝑖(𝐱)≥0.(24) By (15) and Definition 2, we have 𝑏₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗𝐿≥𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝐿,𝑏₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗𝑈≥𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝑈.(25) From (17), 𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗+𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗=0.(26) It follows from Definition 2 that 𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝐿+𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝐿𝜂=0,_𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝑈+𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝑈=0.(27) It is equivalent to 𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝐿=−𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝐿,𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝑈=−𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝑈.(28) From (16), we have −𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗𝐿≥𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗𝐿,−𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗𝑈≥𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗𝑈.(29) From Definition 1, we have −𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗𝑈≥𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗𝐿,−𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗𝐿≥𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗𝑈.(30) Thus, 𝑏₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗𝐿≥𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝐿=−𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝐿≥^𝑚_𝑖=1𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗𝑈≥^𝑚_𝑖=1𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗𝐿𝑏≥0,₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗𝑈≥𝑏₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗𝐿≥0.(31) So, 𝑏₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗≽0.(32) From (21), it follows that Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗≽0.(33) By (19), 𝑓(𝐱)⊖_𝑔𝑓𝐱^∗≽0.(34) From Proposition 4, it follows that 𝑓(𝐱)≽𝑓𝐱^∗.(35) Therefore, 𝐱^∗ is an optimal solution of (P).

Theorem 19. Let 𝐱^∗ be P-feasible. Suppose that(i)there exist 𝜂, Φ₀, 𝑏₀, Φ_𝑖, 𝑏_𝑖, 𝑖=1,2,…,𝑚 such that 𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗≽0⇒𝑏₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗≽0,(36)−𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗⪯0⇒𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗⪯0(37) for all feasible 𝐱;(ii)there exist 𝐲^∗=(𝑦₁,𝑦₂,…,𝑦_𝑚)^𝑡∈𝑅^𝑚 such that ∇𝑓𝐱^∗+^𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝐲=0,(38)_∗≥0.(39) Further, suppose that Φ₀(𝐴)≽0⇒𝐴≽0,(40)𝐴⪯0⇒Φ_𝑖(𝐴)𝑏≽0,(41)₀𝐱,𝐱^∗>0,𝑏_𝑖𝐱,𝐱^∗>0(42) for all feasible 𝐱. Then, 𝐱^∗ is an optimal solution of (P).

Proof. Let 𝐱 be P-feasible. Then, 𝑔_𝑖(𝐱^∗)⪯0, from (41), we obtain that Φ_𝑖𝑔_𝑖𝐱^∗≽0.(43) So, −𝑏_𝑖𝐱,𝐱^∗Φ_𝑖𝑔_𝑖𝐱^∗⪯0.(44) By (37), 𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗⪯0.(45) Thus, −^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗≽0.(46) Then, we have −^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗𝑈=−^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗𝐿−≥0,_𝑚𝑖=1𝑦_𝑖𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗𝐿=−^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡𝐱,𝐱^∗∇𝑔_𝑖𝐱^∗𝑈≥0.(47) From (38) and Definition 2, it follows that 𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝐿+𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝐿𝜂=0,_𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝑈+𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝑈=0.(48) It is equivalent to 𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝐿=−𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝐿,𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝑈=−𝜂^𝑡𝐱,𝐱^∗𝑚_𝑖=1𝑦_𝑖∇𝑔𝐱^∗𝑈.(49) Therefore, 𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝐿𝜂≥0,_𝑡𝐱,𝐱^∗∇𝑓𝐱^∗𝑈≥0.(50) From Definition 2, we obtain that 𝜂^𝑡𝐱,𝐱^∗∇𝑓𝐱^∗≽0.(51) By (36), 𝑏₀𝐱,𝐱^∗Φ₀𝑓(𝐱)⊖_𝑔𝑓𝐱^∗≽0.(52) Then, from (40) and (42), we have 𝑓(𝐱)⊖_𝑔𝑓𝐱^∗≽0.(53) From Proposition 4, it follows that 𝑓(𝐱)≽𝑓𝐱^∗.(54) Therefore, 𝐱^∗ is an optimal solution of (P).

5. Duality

Consider the following: (𝐮)(𝐮)+max𝑓s.t.∇𝑓^𝑚_𝑖=1𝑦_𝑖∇𝑔_𝑖(𝐮)𝑦=0,_𝑖∇𝑔_𝑖(𝐮)𝑦≽0_𝑖(D)≥0.

Theorem 20 (Weak duality). Let 𝐱 be P-feasible, and let (𝐮,𝐲) be D-feasible. Assume that there exist 𝜂, Φ₀, 𝑏₀, Φ_𝑖, 𝑏_𝑖, 𝑖=1,2,…,𝑚 such that 𝑏₀(𝐱,𝐮)Φ₀𝑓(𝐱)⊖_𝑔𝑓(𝐮)≽𝜂^𝑡(𝐱,𝐮)∇𝑓(𝐮),−𝑏_𝑖(𝐱,𝐮)Φ_𝑖𝑔_𝑖(𝐮)≽𝜂^𝑡(𝐱,𝐮)∇𝑔_𝑖(𝐮)(55) at 𝐮; Φ₀(𝐴)𝑏≽0⇒𝐴≽0,₀(𝐱,𝐮)>0,𝑏_𝑖(𝐱,𝐮)≥0(56) and ∑^𝑚_𝑖=1𝑏_𝑖(𝐱,𝐮)𝑦_𝑖Φ_𝑖(𝑔_𝑖(𝐮))≽0. Then, 𝑓(𝐱)≽𝑓(𝐮).

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

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 𝜂^𝑡(𝐱,𝐮)∇𝑓(𝐮)≽0⇒𝑏₀(𝐱,𝐮)Φ₀𝑓(𝐱)⊖_𝑔𝑓(𝐮)≽0,(57)−𝑏₁(𝐱,𝐮)Φ₁^𝑚_𝑖=1𝑦_𝑖𝑔_𝑖(𝐮)⪯0⇒^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡(𝐱,𝐮)∇𝑔_𝑖(𝐮)⪯0(58) at 𝐮; Φ₀(𝐴)≽0⇒𝐴≽0,(59)𝐴≽0⇒Φ₁(𝐴)𝑏≽0,(60)₀(𝐱,𝐮)>0,𝑏₁(𝐱,𝐮)≥0.(61) Then, 𝑓(𝐱)≽𝑓(𝐮).

Proof. Since (𝐮,𝐲) is D-feasible, then 𝑦^𝑡_𝑖∇𝑔_𝑖(𝐮)≽0, from (60) and (61), −𝑏₁(𝐱,𝐮)Φ₁^𝑚_𝑖=1𝑦_𝑖𝑔_𝑖(𝐮)⪯0.(62) Then, we have ^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡(𝐱,𝐮)∇𝑔_𝑖(𝐮)⪯0.(63) Thus, −^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡(𝐱,𝐮)∇𝑔_𝑖(𝐮)−≽0,_𝑚𝑖=1𝑦_𝑖𝜂^𝑡(𝐱,𝐮)∇𝑔_𝑖(𝐮)^𝑈=−^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡(𝐱,𝐮)∇𝑔_𝑖(𝐮)^𝐿−≥0,_𝑚𝑖=1𝑦_𝑖𝜂^𝑡(𝐱,𝐮)∇𝑔_𝑖(𝐮)^𝐿=−^𝑚_𝑖=1𝑦_𝑖𝜂^𝑡(𝐱,𝐮)∇𝑔_𝑖(𝐮)^𝑈≥0.(64) Since (𝐮,𝐲) is D-feasible we can obtain that, ∇𝑓(𝐮)+^𝑚_𝑖=1𝑦_𝑖∇𝑔_𝑖(𝐮)=0.(65) So, 𝜂^𝑡(𝐱,𝐮)∇𝑓(𝐮)+𝜂^𝑡(𝐱,𝐮)^𝑚_𝑖=1𝑦_𝑖∇𝑔_𝑖(𝐮)=0.(66)
By Definition 2, it follows that 𝜂^𝑡(𝐱,𝐮)∇𝑓(𝐮)^𝐿+𝜂^𝑡(𝐱,𝐮)^𝑚_𝑖=1𝑦_𝑖∇𝑔_𝑖(𝐮)^𝐿𝜂=0,_𝑡(𝐱,𝐮)∇𝑓(𝐮)^𝑈+𝜂^𝑡(𝐱,𝐮)^𝑚_𝑖=1𝑦_𝑖∇𝑔_𝑖(𝐮)^𝑈=0.(67) Therefore, 𝜂^𝑡(𝐱,𝐮)∇𝑓(𝐮)^𝐿=−𝜂^𝑡(𝐱,𝐮)^𝑚_𝑖=1𝑦_𝑖∇𝑔_𝑖(𝐮)^𝐿𝜂≥0,_𝑡(𝐱,𝐮)∇𝑓(𝐮)^𝑈=−𝜂^𝑡(𝐱,𝐮)^𝑚_𝑖=1𝑦_𝑖∇𝑔_𝑖(𝐮)^𝑈≥0.(68) Then, 𝜂^𝑡(𝐱,𝐮)∇𝑓(𝐮)≽0.(69) By (57), 𝑏₀(𝐱,𝐮)Φ₀𝑓(𝐱)⊖_𝑔𝑓(𝐮)≽0.(70) From (59) and (61), 𝑓(𝐱)⊖_𝑔𝑓(𝐮)≽0.(71) thus, 𝑓(𝐱)≽𝑓(𝐮).(72)

Theorem 22 (strong duality). If 𝐱^∗ is P-optimal and a constraint qualification is satisfied at 𝐱^∗, then there exists 𝐲^∗=(𝑦₁,𝑦₂,…,𝑦_𝑚)^𝑡∈𝑅^𝑚 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.

Proof. Since a constraint qualification is satisfied at 𝐱^∗, there exists 𝐲^∗∈𝑅^𝑚 such that the following Kuhn-Tucker conditions are satisfied: ∇𝑓𝐱^∗+^𝑚_𝑖=1𝑦_𝑖∇𝑔_𝑖𝐱^∗=0,_𝑚𝑖=1𝑦_𝑖∇𝑔_𝑖𝐱^∗𝑦=0,_𝑖≥0.(73)
Therefore, (𝐱^∗,𝐲^∗) is D-feasible.
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: minimize𝑓(𝐱)=𝑥₁−sin𝑥₂+1,𝑥₁−sin𝑥₂+3subjectto𝑔₁(𝐱)=sin𝑥₁−4sin𝑥₂𝑥−2,sin₁−4sin𝑥₂𝑔⪯0,₂(𝐱)=2sin𝑥₁+7sin𝑥₂+𝑥₁𝑥−7,2sin₁+7sin𝑥₂+𝑥₁𝑔−6⪯0,₃(𝐱)=2𝑥₁+2𝑥₂−5,2𝑥₁+2𝑥₂𝑔−3⪯0,₄(𝐱)=4𝑥²₁+4𝑥²₂−12,4𝑥²₁+4𝑥²₂𝑔−9⪯0,₅(𝐱)=−sin𝑥₁−1,−sin𝑥₁𝑔⪯0,₆(𝐱)=−sin𝑥₂−1,−sin𝑥₂⪯0.(74)

Note that the interval-valued objective function is univex with respect to 𝑏=1, 𝜂(𝐱,𝐮)=𝐱−𝐮, Φ([𝑎^𝐿,𝑎^𝑈])=[𝑎^𝐿,𝑎^𝑈], and every 𝑔_𝑖 (𝑖=1,2,…,6) is univex with respect to 𝑏=1, Φ([𝑎^𝐿,𝑎^𝑈])=[𝑎^𝐿,𝑎^𝑈]𝜂(𝐱,𝐮)=sin𝑥₁−sin𝑢₁cos𝑢₁,sin𝑥₂−sin𝑢₂cos𝑢₂^𝑇,(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, 1,−cos𝑥₂^𝑇+𝜇₁cos𝑥₁,−4cos𝑥₂^𝑇+𝜇₂2cos𝑥₁+1,7cos𝑥₂^𝑇+𝜇₃(2,2)^𝑇+𝜇₄8𝑥₁,8𝑥₂^𝑇+𝜇₅−cos𝑥₁,0^𝑇+𝜇₆0,−cos𝑥₂^𝑇=(0,0)^𝑇.(76)

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).