Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 527183, 5 pages http://dx.doi.org/10.1155/2013/527183
Research Article New Optimality Conditions for a Nondifferentiable Fractional Semipreinvex Programming Problem Yi-Chou Chen1 and Wei-Shih Du2 1 2
Department of General Education, National Army Academy, Taoyuan 320, Taiwan Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan
Correspondence should be addressed to Wei-Shih Du;
[email protected] Received 29 July 2012; Accepted 1 January 2013 Academic Editor: Jen Chih Yao Copyright Β© 2013 Y.-C. Chen and W.-S. Du. 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. We study a nondifferentiable fractional programming problem as follows: (π)minπ₯βπΎ π(π₯)/π(π₯) subject to π₯ β πΎ β π, βπ (π₯) β€ 0, π = 1, 2, . . . , π, where πΎ is a semiconnected subset in a locally convex topological vector space π, π : πΎ β R, π : πΎ β R+ and βπ : πΎ β R, π = 1, 2, . . . , π. If π, βπ, and βπ , π = 1, 2, . . . , π, are arc-directionally differentiable, semipreinvex maps with respect to a continuous map πΎ : [0, 1] β πΎ β π satisfying πΎ(0) = 0 and πΎσΈ (0+ ) β πΎ, then the necessary and sufficient conditions for optimality of (π) 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 [1β27] 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 SzilΒ΄agyi [9] studied the programming with convex set functions and proved that the alternative theorem is valid for convex set functions defined on convex subfamily π of measurable subsets in π and showed that if the system
π (Ξ©) βͺ π, π (Ξ©) < π
(1)
has on solution, where π stands for zero vector in a topological vector space, then there exists a nonzero continuous linear function (π¦β , π§β ) β πΆβ Γ π·β such that β¨π (Ξ©) , π¦β β© + β¨π (Ξ©) , π§β β© β₯ 0
βΞ© β π.
(2)
In this paper, we study the following optimization problem: min π₯βπΎ
π (π₯) π (π₯)
subject to
π₯ β πΎ β π,
βπ (π₯) β€ 0,
(π)
π = 1, 2, . . . , π, where πΎ is a semiconnected subset in a locally convex topological vector space π, π : πΎ β R, π : πΎ β R+ and βπ : πΎ β (ββ, 0], π = 1, 2, . . . , π, 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.
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2. Preliminaries Throughout this paper, we let π be a locally convex topological vector space over the real field R. Denote πΏ1 (π) by the space of all linear operators from π into R. Let π be a nonempty convex subset of π. Let π : π β R be differentiable at π₯0 β πΎ. Then there is a linear operator π΄ = πσΈ (π₯0 ) β πΏ1 (π), such that π ((1 β πΌ) π₯0 + πΌπ₯) β π (π₯0 ) = πσΈ (π₯0 ) (π₯ β π₯0 ) . (3) πΌβ0 πΌ
Example 2. Let π΄ := [4, 8], π΅ := [β8, β4] and πΎ := π΄ βͺ π΅ be bounded sets. Let π : πΎ Γ πΎ Γ [0, 1] β R be defined by π (π₯, π¦, πΌ) =
π₯βπ¦ , 1βπΌ
1 for (π₯, π¦, πΌ) β π΄ Γ π΄ Γ [0, ] , 2
π (π₯, π¦, πΌ) =
π₯βπ¦ , 1βπΌ
1 for (π₯, π¦, πΌ) β π΅ Γ π΅ Γ [0, ] , 2
π (π₯, π¦, πΌ) =
β8 β π¦ , 1βπΌ
1 for (π₯, π¦, πΌ) β π΄ Γ π΅ Γ [0, ] , 2
π (π₯, π¦, πΌ) =
4βπ¦ , 1βπΌ
1 for (π₯, π¦, πΌ) β π΅ Γ π΄ Γ [0, ] , 2
π (π₯, π¦, πΌ) =
π₯βπ¦ , πΌ
1 for (π₯, π¦, πΌ) β π΄ Γ π΄ Γ [ , 1] , 2
π (π₯, π¦, πΌ) =
π₯βπ¦ , πΌ
1 for (π₯, π¦, πΌ) β π΅ Γ π΅ Γ [ , 1] , 2
π (π₯, π¦, πΌ) =
β8 β π¦ , πΌ
1 for (π₯, π¦, πΌ) β π΄ Γ π΅ Γ [ , 1] , 2
π (π₯, π¦, πΌ) =
4βπ¦ , πΌ
1 for (π₯, π¦, πΌ) β π΅ Γ π΄ Γ [ , 1] . 2 (10)
lim
Recall that a function π : π β R is called convex on π, if π ((1 β πΌ) π₯0 + πΌπ₯) β€ (1 β πΌ) π (π₯0 ) + πΌπ (π₯)
(4)
or π ((1 β πΌ) π₯0 + πΌπ₯) β π (π₯0 ) β€ π (π₯) β π (π₯0 ) . πΌ
(5)
If π : π β R is convex and differentiable at π₯0 β πΎ, then by (3) and (5), we have πσΈ (π₯0 ) (π₯ β π₯0 ) β€ π (π₯) β π (π₯0 ) .
(6)
In 1981, Hanson [13, 14] introduced a generalized convexity on π, so-called invexity; that is, π₯ β π₯0 is replaced by a vector π(π₯0 , π₯) β π in (6), or πσΈ (π₯0 ) π (π₯0 , π₯) β€ π (π₯) β π (π₯0 ) .
Then πΎ is a bound semiconnected set with respect to π. Theorem 3 (see [6, Theorem 2.2]). Let πΎ β π be a semiconnected subset and π : πΎ β R a semipreinvex map. Then any local minimum of π is also a global minimum of π over πΎ. From the assumption in problem 9, there exists a positive number π such that
(7)
π (π¦) β₯π π (π¦)
So an invex function is indeed a generalization of a convex differentiable function. Definition 1 (see [6]). (1) A set πΎ β π is said to be semiconnected with respect to a given π : π Γ π β R if π₯, π¦ β πΎ, 0 β€ πΌ β€ 1 σ³¨β π¦ + πΌπ (π₯, π¦, πΌ) β πΎ.
π (π₯ + πΌπ (π₯, π¦, πΌ)) β€ (1 β πΌ) π (π₯) + πΌπ (π¦) , lim πΌπ (π₯, π¦, πΌ) = π,
(9)
πΌβ0
where π stands for the zero vector of π. The following is an example of a bounded semiconnected set in R, which is semiconnected with respect to a nontrivial π.
(11)
π (π¦) β ππ (π¦) β₯ 0. Consequently, we can reduce the problem 9 to an equivalent nonfractional parametric problem: π (π) := min (π (π¦) β ππ (π¦)) β₯ 0,
(8)
(2) A map π : π β R is said to be semipreinvex on a semiconnected subset πΎ β π if each (π₯, π¦, πΌ) β πΎΓπΎΓ[0, 1] corresponds a vector π(π₯, π¦, πΌ) β π such that
βπ¦ β π,
π¦βπ
(ππ )
where π β [0, β) is a parameter. We will prove that the problem (π) is equivalent to the problem (ππβ ) for the optimal value πβ . The following result is our main technique to derive the necessary and sufficient optimality conditions for problem (π). Theorem 4. Problem (π) has an optimal solution π¦0 with optimal value πβ if and only if π£(πβ ) = 0 and π¦0 is an optimal solution of (ππβ ). Proof. If π¦0 is an optimal solution of (π) with optimal value πβ , that is, πβ :=
π (π¦0 ) π (π§) π (π§) β€ = min π (π¦0 ) π§βπ π (π§) π (π§)
βπ§ β π.
(12)
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It follows from (12) that π (π§) β πβ π (π§) β₯ 0
βπ§ β π,
(13)
π (π¦0 ) β πβ π (π¦0 ) = 0. Thus, we have 0 β€ min (π (π§) β πβ π (π§)) β€ π (π¦0 ) β πβ π (π¦0 ) = 0. π§βπ
π½ (π‘) π½ (π‘) β π½ (0) = , π‘ π‘β0
(23)
then
β
β
β
π (π ) = min (π (π§) β π π (π§)) = π (π¦0 ) β π π (π¦0 ) = 0. π§βπ
(15) Therefore, π¦0 is an optimal solution of (ππβ ) and π(πβ ) = 0. Conversely, if π¦0 is an optimal solution of (ππβ ) with optimal value π(πβ ) = 0, then π (π¦0 ) β πβ π (π¦0 ) = min (π (π§) β πβ π (π§)) = 0. π§βπ
(16)
So π (π§) β πβ π (π§) β₯ 0 = π (π¦0 ) β πβ π (π¦0 )
βπ§ β π.
(17)
It follows from (17) that π (π§) β₯ πβ π (π§)
βπ§ β π, (18)
π (π¦0 ) = πβ , π (π¦0 ) and hence
limπ (π₯, π¦, π‘) = π½σΈ (0+ ) = π’, π‘β0
σ΅¨σ΅¨ π σ΅¨ = π½σΈ (0+ ) = π’. [π‘π (π₯, π¦, π‘)]σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨π‘=0+ ππ‘
(24)
Let π : π β R, βπ : π β Rβ and βπ : π β Rβ , π = 1, 2, . . . , π, be semipreinvex maps on a semiconnected subset πΎ in π. Consider a constrained programming problem as (π). The following Fritz John type theorem is essential in this section for programming problem (π). Theorem 6 (Necessary Optimality Condition). Suppose that π, βπ and βπ , π = 1, 2, . . . , π are arc-directionally differentiable at π₯0 β πΎ and semipreinvex on πΎ with respect to a continuous arc π½ defined as in Definition 5. If π₯0 minimizes locally for the semipreinvex programming problem (π), then there exist πβ β (0, β) and {πΎπ }π π=1 β [0, β) such that π
πσΈ (π₯0 ; π’) β πβ πσΈ (π₯0 ; π’) + βπΎπ βπσΈ (π₯0 ; π’) β₯ 0,
π (π§) β₯ πβ , π§βπ π (π§)
min
(25)
π=1
π (π§) π (π¦0 ) min = πβ . β€ π§βπ π (π§) π (π¦0 )
(19)
where π’ = π½σΈ (0+ ) and π
βπΎπ βπ (π₯0 ) = 0.
Therefore,
(26)
π=1
π (π¦0 ) π (π§) = πβ = π§βπ π (π§) π (π¦0 )
min
(20)
and we know π¦0 is an optimal solution of (π) with optimal value πβ .
βπ (π₯) β€ 0,
π½σΈ (0+ ) = π’
(in π) ,
(21)
that is, the continuous function π½ is differentiable from right at 0, and the limit π (π₯0 + π½ (π‘)) β π (π₯0 ) β
πσΈ (π₯0 ; π’) exists. π‘
π = 1, 2, . . . , π
(27)
has no solution in πΎ, then we have
Definition 5 (see [6]). A mapping π : πΎ β π β R is said to be arcwise directionally (in short, arc-directionally) differentiable at π₯0 β πΎ with respect to a continuous arc π½ : [0, 1] β πΎ β π if π₯0 + π½(π‘) β πΎ for π‘ β [0, 1] with π½ (0) = π,
Proof. By Theorem 4, the minimum solution to (π) is also a minimum to (ππβ ). Then π₯0 is the local minimal solution to (ππβ ). By Theorem 3, we have π₯0 is the global minimal solution to (ππ ). It follows that the system [π (π₯) β πβ π (π₯)] β [π (π₯0 ) β πβ π (π₯0 )] < 0,
3. The Existence of the Necessary and Sufficient Conditions for Semipreinvex Functions
π‘β0
π (π₯, π¦, π‘) :=
(14)
Then, by (14), we get
lim
Note that the arc directional derivative πσΈ (π₯0 ; β
) is a mapping from π into R. Moreover, how can we make πΎ to be a semiconnected set? Indeed, we can construct a function π concerned with π½ defined as follows. For any π₯, π¦ β πΎ and π‘ β [0, 1], we choose a vector
(22)
π
[π (π₯) β πβ π (π₯)] β [π (π₯0 ) β πβ π (π₯0 )] + βπΎπ βπ (π₯) < 0 π=1
(28) has no solution in πΎ for any {πΎπ }π π=1 β [0, β). Thus for any π₯ β πΎ, π
[π (π₯) β πβ π (π₯)] β [π (π₯0 ) β πβ π (π₯0 )] + βπΎπ βπ (π₯) β₯ 0 π=1
(29)
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Journal of Applied Mathematics
for some {πΎπ }π π=1 β [0, β). Putting π₯ = π₯0 in (29), we get
By Theorem 4, π₯0 was not optimal for problem (ππ ). Then there is an π₯ β π such that
π
βπΎπ βπ (π₯0 ) β₯ 0.
π (π₯) β ππ (π₯) < π (π₯0 ) β ππ (π₯0 ) ,
(30)
π=1
βπ (π₯) β€ 0
Since πΎπ β₯ 0 and βπ (π₯0 ) β€ 0, it follows that
for π = 1, 2, . . . , π. Moreover, we have
π
βπΎπ βπ (π₯0 ) = 0.
(31)
[π (π₯) β ππ (π₯)] β [π (π₯0 ) β ππ (π₯0 )] < 0,
π=1
So (26) is proved. As πΎ is a semiconnected set, for any π₯ β πΎ and π‘ β [0, 1], we have π₯0 + π‘π (π₯0 , π₯, π‘) β πΎ.
(32)
π
βπΎπ [βπ (π₯) β βπ (π₯0 )] β€ 0 π=1
π
(sinceβ πΎπ βπ (π₯0 ) = 0) π=1
for any {πΎπ }π π=1 β [0, β). Thus [π (π₯) β ππ (π₯)] β [π (π₯0 ) β ππ (π₯0 )] π
+ βπΎπ [βπ (π₯) β βπ (π₯0 )] < 0.
[π (π₯0 + π‘π (π₯0 , π₯, π‘)) β π (π₯0 )]
π
(33)
π=1
Since π and π are arc-directionally differentiable with respect to π½, choose a vector π(π₯0 , π₯, π‘) as (23), so that (24) hold. It follows that if we divide (33) by π‘ =ΜΈ 0 and take the limit as π‘ β 0, then we have
Since the semi-preinvex maps π, βπ and βπ , π = 1, 2, . . . , π are arc-directionally differentiable, it follows that for (π₯, π₯0 , π‘) β πΎ Γ πΎ Γ [0, 1] there corresponds a vector π(π₯, π₯0 , π‘) β π such that π (π₯0 + π‘π (π₯, π₯0 , π‘)) β€ (1 β π‘) π (π₯0 ) + π‘π (π₯) , βπ (π₯0 + π‘π (π₯, π₯0 , π‘)) β€ (1 β π‘) (βπ) (π₯0 ) + π‘ (βπ) (π₯) , βπ (π₯0 + π‘π (π₯, π₯0 , π‘)) β€ (1 β π‘) βπ (π₯0 ) + π‘βπ (π₯) , (42)
π
(34)
π=1
which proves (25) and the proof of theorem is completed. Theorem 7 (Sufficient Optimality Condition). Let π, βπ and βπ , π = 1, 2, . . . , π be arc-directionally differentiable at π₯0 β πΎ and semipreinvex on πΎ with respect to a continuous arc π½ defined as in Definition 5. If there exist π β (0, β) and {πΎπ }π π=1 β [0, β) satisfying π
πσΈ (π₯0 ; π’) β ππσΈ (π₯0 ; π’) + βπΎπ βπσΈ (π₯0 ; π’) β₯ 0,
(35)
π=1
with π’ = π½σΈ (0+ ) and (36)
π=1
Proof. Suppose to the contrary that π₯0 is not optimal for problem (π) and π = π(π₯0 )/π(π₯0 ). Then π(π₯0 ) β ππ(π₯0 ) = 0. Therefore, π₯βπ
thus π(π) = minπ₯βπ (π(π₯) β ππ(π₯)) = 0.
π (π₯0 + π‘π (π₯, π₯0 , π‘)) β π (π₯0 ) β€ π (π₯) β π (π₯0 ) , π‘ (βπ) (π₯0 + π‘π (π₯, π₯0 , π‘)) + π (π₯0 ) β€ (βπ) (π₯) + π (π₯0 ) , π‘ βπ (π₯0 + π‘π (π₯, π₯0 , π‘)) β βπ (π₯0 ) β€ βπ (π₯) β βπ (π₯0 ) . π‘
(43)
Letting π‘ β 0, we have limπ‘β0 π(π₯, π₯0 , π‘) = π½σΈ (0+ ) = π’ and the last inequalities imply
βπσΈ (π₯0 , π’) β€ β [π (π₯) β π (π₯0 )] ,
(44)
βπσΈ (π₯0 , π’) β€ βπ (π₯) β βπ (π₯0 ) .
then π₯0 is an optimal solution for problem (π).
0 β€ min (π (π₯) β ππ (π₯)) β€ π (π₯0 ) β ππ (π₯0 ) = 0,
and so
πσΈ (π₯0 , π’) β€ π (π₯) β π (π₯0 ) ,
π
βπΎπ βπ (π₯0 ) = 0,
(41)
π=1
+ βπΎπ (βπ (π₯0 + π‘π (π₯0 , π₯, π‘)) β βπ (π₯0 )) β₯ 0.
πσΈ (π₯0 ; π’) β πβ πσΈ (π₯0 ; π’) + βπΎπ βπσΈ (π₯0 ; π’) β₯ 0,
(39)
(40)
For π‘ =ΜΈ 0, the point π₯Μ = π₯0 + π‘π(π₯0 , π₯, π‘) =ΜΈ π₯0 does not solve the system (27). So substituting π₯Μ in (29) and using the result (26), we obtain
β πβ [π (π₯0 + π‘π (π₯0 , π₯, π‘)) β π (π₯0 )]
(38)
(37)
Consequently, from (41) and (44), we obtain π
πσΈ (π₯0 ; π’) β ππσΈ (π₯0 ; π’) + βπΎπ βπσΈ (π₯0 ; π’) < 0,
(45)
π=1
which contradicts the fact of (35). Therefore π₯0 is an optimal solution of problem (π).
Journal of Applied Mathematics
5
Since any global minimal is a local minimal, applying Theorems 6 and 7, we can obtain the necessary and sufficient conditions for problem (π). Theorem 8. Suppose that π, βπ and βπ , π = 1, 2, . . . , π are arcdirectionally differentiable at at π₯0 β πΎ and semi-preinvex on πΎ with respect to a continuous arc π½ defined as in Definition 5. If π₯0 minimizes globally for the semi-preinvex programming problem (π) if and only if there exists (π, πΎπ ) β R+ Γ (R+ βͺ {0}), π = 1, 2, . . . , π, such that π
πσΈ (π₯0 ; π’) β ππσΈ (π₯0 ; π’) + βπΎπ βπσΈ (π₯0 ; π’) β₯ 0,
(46)
π=1
where π’ = π½σΈ (0+ ) and π
βπΎπ βπ (π₯0 ) = 0.
(47)
π=1
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.
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Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
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Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
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Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014