New Optimality Conditions for a Nondifferentiable Fractional Semipreinvex Programming Problem

Abstract

We study a nondifferentiable fractional programming problem as follows: (𝑃)min_𝑥∈𝐾𝑓(𝑥)/𝑔(𝑥) subject to 𝑥∈𝐾⊆𝑋,ℎ_𝑖(𝑥)≤0,𝑖=1,2,…,𝑚, where 𝐾 is a semiconnected subset in a locally convex topological vector space 𝑋, 𝑓∶𝐾→ℝ, 𝑔∶𝐾→ℝ₊ and ℎ_𝑖∶𝐾→ℝ, 𝑖=1,2,…,𝑚. If 𝑓, −𝑔, and ℎ_𝑖, 𝑖=1,2,…,𝑚, are arc-directionally differentiable, semipreinvex maps with respect to a continuous map 𝛾∶[0,1]→𝐾⊆𝑋 satisfying 𝛾(0)=0 and 𝛾^′(0⁺)∈𝐾, then the necessary and sufficient conditions for optimality of (𝑃) are established.

1. Introduction

In recent years, there has been an increasing interest in studying the develpoment of optimality conditions for nondifferentiable multiobjective programming problems. Many authors established and employed some different Kuhn and Tucker type necessary conditions or other type necessary conditions to research optimal solutions; see [1–27] and references therein. In [7], Lai and Ho used the Pareto optimality condition to investigate multiobjective programming problems for semipreinvex functions. Lai [6] had obtained the necessary and sufficient conditions for optimality programming problems with semipreinvex assumptions. Some Pareto optimality conditions are established by Lai and Lin in [8]. Lai and Szilágyi [9] studied the programming with convex set functions and proved that the alternative theorem is valid for convex set functions defined on convex subfamily 𝑆 of measurable subsets in 𝑋 and showed that if the system 𝑓(Ω)𝑔(Ω)≪𝜃,<𝜃 (1) has on solution, where 𝜃 stands for zero vector in a topological vector space, then there exists a nonzero continuous linear function (𝑦^∗,𝑧^∗)∈𝐶^∗×𝐷^∗ such that 𝑓(Ω),𝑦^∗+𝑔(Ω),𝑧^∗≥0∀Ω∈𝑆. (2) In this paper, we study the following optimization problem: min_𝑥∈𝐾𝑓(𝑥)𝑔(𝑥)subjectto𝑥∈𝐾⊆𝑋,ℎ_𝑖(𝑥)≤0,𝑖=1,2,…,𝑚, (𝑃) where 𝐾 is a semiconnected subset in a locally convex topological vector space 𝑋, 𝑓∶𝐾→ℝ, 𝑔∶𝐾→ℝ₊ and ℎ_𝑖∶𝐾→(−∞,0], 𝑖=1,2,…,𝑚, are functions satisfying some suitable conditions. The purpose of this study is dealt with such constrained fractional semipreinvex programming problem. Finally, we established the Fritz John type necessary and sufficient conditions for the optimality of a fractional semipreinvex programming problem.

2. Preliminaries

Throughout this paper, we let 𝑋 be a locally convex topological vector space over the real field ℝ. Denote 𝐿¹(𝑋) by the space of all linear operators from 𝑋 into ℝ.

Let 𝑊 be a nonempty convex subset of 𝑋. Let 𝑓∶𝑊→ℝ be differentiable at 𝑥₀∈𝐾. Then there is a linear operator 𝐴=𝑓^′(𝑥₀)∈𝐿¹(𝑋), such that lim_𝛼→0𝑓(1−𝛼)𝑥₀+𝛼𝑥−𝑓𝑥₀𝛼=𝑓^′𝑥₀𝑥−𝑥₀. (3) Recall that a function 𝑓∶𝑊→ℝ is called convex on 𝑊, if 𝑓(1−𝛼)𝑥₀+𝛼𝑥≤(1−𝛼)𝑓𝑥₀+𝛼𝑓(𝑥) (4) or 𝑓(1−𝛼)𝑥₀+𝛼𝑥−𝑓𝑥₀𝛼≤𝑓(𝑥)−𝑓𝑥₀. (5) If 𝑓∶𝑊→ℝ is convex and differentiable at 𝑥₀∈𝐾, then by (3) and (5), we have 𝑓^′𝑥₀𝑥−𝑥₀≤𝑓(𝑥)−𝑓𝑥₀. (6) In 1981, Hanson [13, 14] introduced a generalized convexity on 𝑋, so-called invexity; that is, 𝑥−𝑥₀ is replaced by a vector 𝜏(𝑥₀,𝑥)∈𝑋 in (6), or 𝑓^′𝑥₀𝜏𝑥₀,𝑥≤𝑓(𝑥)−𝑓𝑥₀. (7) So an invex function is indeed a generalization of a convex differentiable function.

Definition 1 (see [6]). (1) A set 𝐾⊆𝑋 is said to be semiconnected with respect to a given 𝜏∶𝑋×𝑋→ℝ if 𝑥,𝑦∈𝐾,0≤𝛼≤1⇒𝑦+𝛼𝜏𝑥,𝑦,𝛼∈𝐾. (8)
(2) A map 𝑓∶𝑋→ℝ is said to be semipreinvex on a semiconnected subset 𝐾⊂𝑋 if each (𝑥,𝑦,𝛼)∈𝐾×𝐾×[0,1] corresponds a vector 𝜏(𝑥,𝑦,𝛼)∈𝑋 such that 𝑓𝑥+𝛼𝜏𝑥,𝑦,𝛼≤(1−𝛼)𝑓(𝑥)+𝛼𝑓𝑦,lim_𝛼↓0𝛼𝜏𝑥,𝑦,𝛼=𝜃, (9) where 𝜃 stands for the zero vector of 𝑋.

The following is an example of a bounded semiconnected set in ℝ, which is semiconnected with respect to a nontrivial 𝜏.

Example 2. Let 𝐴∶=[4,8], 𝐵∶=[−8,−4] and 𝐾∶=𝐴∪𝐵 be bounded sets. Let 𝜏∶𝐾×𝐾×[0,1]→ℝ be defined by 𝜏𝑥,𝑦,𝛼=𝑥−𝑦11−𝛼,for𝑥,𝑦,𝛼∈𝐴×𝐴×0,2,𝜏𝑥,𝑦,𝛼=𝑥−𝑦11−𝛼,for𝑥,𝑦,𝛼∈𝐵×𝐵×0,2,𝜏𝑥,𝑦,𝛼=−8−𝑦11−𝛼,for𝑥,𝑦,𝛼∈𝐴×𝐵×0,2,𝜏𝑥,𝑦,𝛼=4−𝑦11−𝛼,for𝑥,𝑦,𝛼∈𝐵×𝐴×0,2,𝜏𝑥,𝑦,𝛼=𝑥−𝑦𝛼1,for𝑥,𝑦,𝛼∈𝐴×𝐴×2,𝜏=,1𝑥,𝑦,𝛼𝑥−𝑦𝛼1,for𝑥,𝑦,𝛼∈𝐵×𝐵×2,𝜏=,1𝑥,𝑦,𝛼−8−𝑦𝛼1,for𝑥,𝑦,𝛼∈𝐴×𝐵×2,𝜏=,1𝑥,𝑦,𝛼4−𝑦𝛼1,for𝑥,𝑦,𝛼∈𝐵×𝐴×2,1. (10) Then 𝐾 is a bound semiconnected set with respect to 𝜏.

Theorem 3 (see [6, Theorem   2.2]). Let 𝐾⊂𝑋 be a semiconnected subset and 𝑓∶𝐾→ℝ a semipreinvex map. Then any local minimum of 𝑓 is also a global minimum of 𝑓 over 𝐾.

From the assumption in problem (9), there exists a positive number 𝜆 such that 𝑓𝑦𝑔𝑦𝑓𝑦𝑦≥𝜆∀𝑦∈𝑋,−𝜆𝑔≥0. (11) Consequently, we can reduce the problem (9) to an equivalent nonfractional parametric problem: 𝜐(𝜆)∶=min_𝑦∈𝑋𝑓𝑦−𝜆𝑔𝑦≥0, 𝑃_𝜆 where 𝜆∈[0,∞) is a parameter.

We will prove that the problem (𝑃) is equivalent to the problem (𝑃_𝜆^∗) for the optimal value 𝜆^∗. The following result is our main technique to derive the necessary and sufficient optimality conditions for problem (𝑃).

Theorem 4. Problem (𝑃) has an optimal solution 𝑦₀ with optimal value 𝜆^∗ if and only if 𝑣(𝜆^∗)=0 and 𝑦₀ is an optimal solution of (𝑃_𝜆^∗).

Proof. If 𝑦₀ is an optimal solution of (𝑃) with optimal value 𝜆^∗, that is, 𝜆^∗𝑓𝑦∶=⁰𝑔𝑦₀=min_𝑧∈𝑋𝑓(𝑧)𝑔(𝑧)≤𝑓(𝑧)𝑔(𝑧)∀𝑧∈𝑋. (12) It follows from (12) that 𝑓(𝑧)−𝜆^∗𝑔(𝑧)𝑓𝑦≥0∀𝑧∈𝑋,₀−𝜆^∗𝑔𝑦₀=0. (13) Thus, we have 0≤min_𝑧∈𝑋𝑓(𝑧)−𝜆^∗𝑔(𝑧)≤𝑓𝑦₀−𝜆^∗𝑔𝑦₀=0. (14) Then, by (14), we get 𝜈𝜆^∗=min_𝑧∈𝑋𝑓(𝑧)−𝜆^∗𝑔(𝑧)=𝑓𝑦₀−𝜆^∗𝑔𝑦₀=0. (15) Therefore, 𝑦₀ is an optimal solution of (𝑃_𝜆^∗) and 𝜈(𝜆^∗)=0.
Conversely, if 𝑦₀ is an optimal solution of (𝑃_𝜆^∗) with optimal value 𝜈(𝜆^∗)=0, then 𝑓𝑦₀−𝜆^∗𝑔𝑦₀=min_𝑧∈𝑋𝑓(𝑧)−𝜆^∗𝑔(𝑧)=0. (16) So 𝑓(𝑧)−𝜆^∗𝑔(𝑧)≥0=𝑓𝑦₀−𝜆^∗𝑔𝑦₀∀𝑧∈𝑋. (17) It follows from (17) that 𝑓(𝑧)𝑔(𝑧)≥𝜆^∗𝑓𝑦∀𝑧∈𝑋,₀𝑔𝑦₀=𝜆^∗, (18) and hence min_𝑧∈𝑋𝑓(𝑧)𝑔(𝑧)≥𝜆^∗,min_𝑧∈𝑋𝑓(𝑧)𝑔(𝑧)≤𝑓𝑦₀𝑔𝑦₀=𝜆^∗. (19) Therefore, min_𝑧∈𝑋𝑓(𝑧)𝑔(𝑧)=𝜆^∗=𝑓𝑦₀𝑔𝑦₀ (20) and we know 𝑦₀ is an optimal solution of (𝑃) with optimal value 𝜆^∗.

3. The Existence of the Necessary and Sufficient Conditions for Semipreinvex Functions

Definition 5 (see [6]). A mapping 𝑓∶𝐾⊂𝑋→ℝ is said to be arcwise directionally (in short, arc-directionally) differentiable at 𝑥₀∈𝐾 with respect to a continuous arc 𝛽∶[0,1]→𝐾⊂𝑋 if 𝑥₀+𝛽(𝑡)∈𝐾 for 𝑡∈[0,1] with 𝛽(0)=𝜃,𝛽^′0⁺=𝑢(in𝑋), (21) that is, the continuous function 𝛽 is differentiable from right at 0, and the limit lim_𝑡↓0𝑓𝑥₀+𝛽(𝑡)−𝑓𝑥₀𝑡≅𝑓^′𝑥₀;𝑢exists. (22)

Note that the arc directional derivative 𝑓^′(𝑥₀;⋅) is a mapping from 𝑋 into ℝ. Moreover, how can we make 𝐾 to be a semiconnected set? Indeed, we can construct a function 𝜏 concerned with 𝛽 defined as follows.

For any 𝑥,𝑦∈𝐾 and 𝑡∈[0,1], we choose a vector 𝜏𝛽(𝑡)𝑥,𝑦,𝑡∶=𝑡=𝛽(𝑡)−𝛽(0),𝑡−0 (23) then lim_𝑡↓0𝜏𝑥,𝑦,𝑡=𝛽^′0⁺𝑑=𝑢,𝑑𝑡𝑡𝜏𝑥,𝑦,𝑡_𝑡=0+=𝛽^′0⁺=𝑢. (24) Let 𝑓∶𝑋→ℝ, −𝑔∶𝑋→ℝ₋ and ℎ_𝑖∶𝑋→ℝ₋, 𝑖=1,2,…,𝑚, be semipreinvex maps on a semiconnected subset 𝐾 in 𝑋. Consider a constrained programming problem as (𝑃).

The following Fritz John type theorem is essential in this section for programming problem (𝑃).

Theorem 6 (Necessary Optimality Condition). Suppose that 𝑓, −𝑔 and ℎ_𝑖,  𝑖=1,2,…,𝑚 are arc-directionally differentiable at 𝑥₀∈𝐾 and semipreinvex on 𝐾 with respect to a continuous arc 𝛽 defined as in Definition 5. If 𝑥₀ minimizes locally for the semipreinvex programming problem (𝑃), then there exist 𝜆^∗∈(0,∞) and {𝛾_𝑖}^𝑚_𝑖=1⊆[0,∞) such that 𝑓^′𝑥₀;𝑢−𝜆^∗𝑔^′𝑥₀;𝑢+^𝑚_𝑖=1𝛾_𝑖ℎ^′_𝑖𝑥₀;𝑢≥0, (25) where 𝑢=𝛽^′(0⁺) and ^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖𝑥₀=0. (26)

Proof. By Theorem 4, the minimum solution to (𝑃) is also a minimum to (𝑃_𝜆^∗). Then 𝑥₀ is the local minimal solution to (𝑃_𝜆^∗). By Theorem 3, we have 𝑥₀ is the global minimal solution to (𝑃_𝜆). It follows that the system 𝑓(𝑥)−𝜆^∗𝑔(𝑥)−𝑓𝑥₀−𝜆^∗𝑔𝑥₀ℎ<0,_𝑖(𝑥)≤0,𝑖=1,2,…,𝑚 (27) has no solution in 𝐾, then we have 𝑓(𝑥)−𝜆^∗𝑔(𝑥)−𝑓𝑥₀−𝜆^∗𝑔𝑥₀+^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖(𝑥)<0 (28) has no solution in 𝐾 for any {𝛾_𝑖}^𝑚_𝑖=1⊆[0,∞). Thus for any 𝑥∈𝐾, 𝑓(𝑥)−𝜆^∗𝑔(𝑥)−𝑓𝑥₀−𝜆^∗𝑔𝑥₀+^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖(𝑥)≥0 (29) for some {𝛾_𝑖}^𝑚_𝑖=1⊆[0,∞). Putting 𝑥=𝑥₀ in (29), we get ^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖𝑥₀≥0. (30) Since 𝛾_𝑖≥0 and ℎ_𝑖(𝑥₀)≤0, it follows that ^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖𝑥₀=0. (31) So (26) is proved.
As 𝐾 is a semiconnected set, for any 𝑥∈𝐾 and 𝑡∈[0,1], we have 𝑥₀+𝑡𝜏𝑥₀,𝑥,𝑡∈𝐾. (32) For 𝑡≠0, the point 𝑥=𝑥₀+𝑡𝜏(𝑥₀,𝑥,𝑡)≠𝑥₀ does not solve the system (27). So substituting 𝑥 in (29) and using the result (26), we obtain 𝑓𝑥₀+𝑡𝜏𝑥₀,𝑥,𝑡−𝑓𝑥₀−𝜆^∗𝑔𝑥₀+𝑡𝜏𝑥₀,𝑥,𝑡−𝑔𝑥₀+^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖𝑥₀+𝑡𝜏𝑥₀,𝑥,𝑡−ℎ_𝑖𝑥₀≥0. (33) Since 𝑓 and 𝑔 are arc-directionally differentiable with respect to 𝛽, choose a vector 𝜏(𝑥₀,𝑥,𝑡) as (23), so that (24) hold. It follows that if we divide (33) by 𝑡≠0 and take the limit as 𝑡↓0, then we have 𝑓^′𝑥₀;𝑢−𝜆^∗𝑔^′𝑥₀;𝑢+^𝑚_𝑖=1𝛾_𝑖ℎ^′_𝑖𝑥₀;𝑢≥0, (34) which proves (25) and the proof of theorem is completed.

Theorem 7 (Sufficient Optimality Condition). Let 𝑓, −𝑔 and ℎ_𝑖, 𝑖=1,2,…,𝑚 be arc-directionally differentiable at 𝑥₀∈𝐾 and semipreinvex on 𝐾 with respect to a continuous arc 𝛽 defined as in Definition 5. If there exist 𝜆∈(0,∞) and {𝛾_𝑖}^𝑚_𝑖=1⊆[0,∞) satisfying 𝑓^′𝑥₀;𝑢−𝜆𝑔^′𝑥₀;𝑢+^𝑚_𝑖=1𝛾_𝑖ℎ^′_𝑖𝑥₀;𝑢≥0, (35) with 𝑢=𝛽^′(0⁺) and ^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖𝑥₀=0, (36) then 𝑥₀ is an optimal solution for problem (𝑃).

Proof. Suppose to the contrary that 𝑥₀ is not optimal for problem (𝑃) and 𝜆=𝑓(𝑥₀)/𝑔(𝑥₀). Then 𝑓(𝑥₀)−𝜆𝑔(𝑥₀)=0. Therefore, 0≤min_𝑥∈𝑋𝑓(𝑥)−𝜆𝑔(𝑥)≤𝑓𝑥₀−𝜆𝑔𝑥₀=0, (37) thus 𝜈(𝜆)=min_𝑥∈𝑋(𝑓(𝑥)−𝜆𝑔(𝑥))=0.
By Theorem 4, 𝑥₀ was not optimal for problem 𝑃_𝜆. Then there is an 𝑥∈𝑋 such that 𝑓(𝑥)−𝜆𝑔(𝑥)<𝑓𝑥₀−𝜆𝑔𝑥₀,ℎ_𝑖(𝑥)≤0 (38) for 𝑖=1,2,…,𝑚. Moreover, we have 𝑓(𝑥)−𝜆𝑔(𝑥)−𝑓𝑥₀−𝜆𝑔𝑥₀<0,_𝑚𝑖=1𝛾_𝑖ℎ_𝑖(𝑥)−ℎ_𝑖𝑥₀≤0since^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖𝑥₀=0 (39)(40) for any {𝛾_𝑖}^𝑚_𝑖=1⊆[0,∞). Thus𝑓(𝑥)−𝜆𝑔(𝑥)−𝑓𝑥₀−𝜆𝑔𝑥₀+^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖(𝑥)−ℎ_𝑖𝑥₀<0. (41) Since the semi-preinvex maps 𝑓,  −𝑔 and ℎ_𝑖,  𝑖=1,2,…,𝑚 are arc-directionally differentiable, it follows that for (𝑥,𝑥₀,𝑡)∈𝐾×𝐾×[0,1] there corresponds a vector 𝜏(𝑥,𝑥₀,𝑡)∈𝑋 such that 𝑓𝑥₀+𝑡𝜏𝑥,𝑥₀,𝑡≤(1−𝑡)𝑓𝑥₀(𝑥),𝑥+𝑡𝑓−𝑔₀+𝑡𝜏𝑥,𝑥₀,𝑡≤(1−𝑡)−𝑔𝑥₀𝑥),ℎ+𝑡−𝑔(_𝑖𝑥₀+𝑡𝜏𝑥,𝑥₀,𝑡≤(1−𝑡)ℎ_𝑖𝑥₀+𝑡ℎ_𝑖(𝑥), (42) and so 𝑓𝑥₀+𝑡𝜏𝑥,𝑥₀,𝑡−𝑓𝑥₀𝑡≤𝑓(𝑥)−𝑓𝑥₀,−𝑔𝑥₀+𝑡𝜏𝑥,𝑥₀,𝑡+𝑔𝑥₀𝑡≤−𝑔(𝑥)+𝑔𝑥₀,ℎ_𝑖𝑥₀+𝑡𝜏𝑥,𝑥₀,𝑡−ℎ_𝑖𝑥₀𝑡≤ℎ_𝑖(𝑥)−ℎ_𝑖𝑥₀. (43) Letting 𝑡↓0, we have lim_𝑡↓0𝜏(𝑥,𝑥₀,𝑡)=𝛽^′(0⁺)=𝑢 and the last inequalities imply 𝑓^′𝑥₀,𝑢≤𝑓(𝑥)−𝑓𝑥₀,−𝑔^′𝑥₀,𝑢≤−𝑔(𝑥)−𝑔𝑥₀,ℎ^′_𝑖𝑥₀,𝑢≤ℎ_𝑖(𝑥)−ℎ_𝑖𝑥₀. (44) Consequently, from (41) and (44), we obtain 𝑓^′𝑥₀;𝑢−𝜆𝑔^′𝑥₀;𝑢+^𝑚_𝑖=1𝛾_𝑖ℎ^′_𝑖𝑥₀;𝑢<0, (45) which contradicts the fact of (35). Therefore 𝑥₀ is an optimal solution of problem (𝑃).

Since any global minimal is a local minimal, applying Theorems 6 and 7, we can obtain the necessary and sufficient conditions for problem (𝑃).

Theorem 8. Suppose that 𝑓, −𝑔 and ℎ_𝑖,  𝑖=1,2,…,𝑚 are arc-directionally differentiable at at 𝑥₀∈𝐾 and semi-preinvex on 𝐾 with respect to a continuous arc 𝛽 defined as in Definition 5. If 𝑥₀ minimizes globally for the semi-preinvex programming problem (𝑃) if and only if there exists (𝜆,𝛾_𝑖)∈ℝ⁺×(ℝ⁺∪{0}), 𝑖=1,2,…,𝑚, such that 𝑓^′𝑥₀;𝑢−𝜆𝑔^′𝑥₀;𝑢+^𝑚_𝑖=1𝛾_𝑖ℎ^′_𝑖𝑥₀;𝑢≥0, (46) where 𝑢=𝛽^′(0⁺) and ^𝑚_𝑖=1𝛾_𝑖ℎ_𝑖𝑥₀=0. (47)

Remark 9. Our results also hold for preinvex functions.

Acknowledgments

The research of Wei-Shih Du was supported partially under Grant no. NSC 101-2115-M-017-001 by the National Science Council of the Republic of China.