1. Introduction
The study of multi-objective programming problems was very active in recent years. The minimum (efficient, Pareto) solution is an important concept in mathematical models, economics, decision theory, optimal control and game theory (see, for example, [1]). In most works, an assumption of convexity was made for the objective functions. Very recently, some generalized convexity has received more attention (see, for example, [2-6]). A significant generalization of convex functions is invex function introduced first by Hanson [7], which has greatly been applied in nonlinear optimization and other branches of pure and applied sciences.
The concept of B-invex functions was proposed by [8] as generalization of convex functions; these functions were extended to quasi B-invex, and pseudo B-invex functions. Many functions seem to be B-invex, but they are not, and many non B-invex functions are able to get B-invex by choosing a suitable condition. Based on the previous discussion, Tarek [9] introduced a new class of B-invex functions, this class called roughly B-invex functions.
Inspired and motivated by above works, the purpose of this paper is to formulate a multi-objective programming problem which it involves roughly B-invex functions. An efficient solution for considered problem is characterized by weighting and ε-constraint approaches. In the end of the paper, we obtain sufficient and necessary conditions for a feasible solution to be an efficient or properly efficient solution for this kind of problems. Let us survey, briefly, the definitions and some results of roughly B-invexity.
Definition 1 [10]
Let
. The set M is said to be B-invex with respect to 
at 
if there exists
such that
, for each
, and
.
M is said to be B-invex set with respect to η if M is B-invex at each 
with respect to the same η.
Note that, as in convex set, the intersection of finite (or infinite) family of B-invex sets is B-invex but the union is not necessarily B-invex set. Also, the sum of B-invex sets and the multiplying a B-invex set by a real number are again B-invex sets. Every B-invex set with respect to 
is an invex set when b = 1; but the converse is not necessarily true.
Definition 2 [9]
A numerical function f , defined on a B-invex subset M of Rn, is said to be roughly B-invex with respect to 
with roughness degree r at 
if there exists
, such that
for each
, and 
such that
.
f is said to be roughly B-invex on M with respect to 
if it is roughly B-invex at each 
with respect to the same
.
Every invex function, with respect to η is roughly B-invex function with respect to same η, where b (x, y) = 1; but the converse is not necessarily true. If the functions 
are all roughly B-invex with respect to 
with roughness degree 
on a B-invex set
, then the function
is roughly B-invex with respect to same η with roughness degree
on M for ai ≥ 0. If 
is roughly B-invex with respect to 
with roughness degree r, on B-invex set
, then for any real number 
the level set 
is B-invex set. A numerical function f defined on a B-invex set 
is roughly B-invex function with respect to 
with roughness degree r if and only if epi(f) is a B-invex set. If 
is a family of numerical functions, which are roughly B-invex with respect to 
with roughness degree r and bounded from above on a B-invex set
, then the numerical function
is a roughly B-invex with respect to same η with roughness degree r on M. If 
is a differentiable roughly B-invex function with respect to 
with roughness degree r, at
, then there exists a function
, such that
for each 
such that
.
Definition 3 [9]
A numerical function f, defined on a B-invex subset M of Rn, is said to be quasi roughly B-invex with respect to 
with roughness degree r at
, if there exists
, such that
for each
, and 
such that
.
f is said to be quasi roughly B-invex on M with respect to 
if it is roughly B-invex at each 
with respect to the same
.
A 
is quasi roughly B-invex with respect to η: M × M → Rn with roughness degree r, on
, if and only if the level set
is B-invex set. A roughly B-invex function, with respect to 
with roughness degree r is quasi roughly B-invex function with respect to same η with roughness degree r. Let 
be B-invex set, if 
is differentiable quasi roughly B-invex with respect to 
with roughness degree r, at
, then there exists a function
, such that
for each 
such that
.
Definition 4 [9]
A numerical function f, defined on a B-invex subset M of Rn, is said to be pseudo roughly B-invex with respect to 
with roughness degree r at
, if there exists
, and, there exists a strictly positive function a: Rn × Rn → R such that
for each 
such that
.
If 
is roughly B-invex function with respect to 
with roughness degree r on B-invex set
, then f is pseudo roughly B-invex function with respect to same η with roughness degree r on M. Let 
be B-invex set and 
be a differentiable pseudo roughly B-invex with respect to
with roughness degree r, at
, then there exists a function
, such that
for each 
such that
.
2. Problem Formulation
Let
, and 
i = 1, 2, ···, m are real valued roughly B-invex functions on Rn. A roughly B-invex multi-objective programming problem is formulated as follows:
(P) 
Definition 5 [11]
A feasible solution x for (P) is said to be an efficient solution for (P) if and only if there is no other feasible x for (P) such that, for some
,
.
Definition 6 [11]
An efficient solution 
for (P) is a properly efficient solution for (P) if there exists a scalar 
such that for each
, and each 
satisfying
, there exists at least one 
with 
and
.
Lemma 1 [9]
If 
is roughly B-invex with respect to 
with roughness degree r, on Rn, i = 1, 2, ···, m, then the set
is B-invex set.
Lemma 2 [9]
If 
is quasi roughly B-invex with respect to 
with roughness degree r, on Rn, i = 1, 2, ···, m, then the set
is B-invex set.
Lemma 3
Let
. If
is a roughly B-invex function with respect to 
with roughness degree r on a B-invex set
, then the set
is convexwhere
.
Proof. Let 
then for 
and 
we have
Since f is a roughly B-invex function on a B-invex set M. Then
, and hence A is convex set.
For a feasible point
, we denote 
as the index set for binding constraints at x*, i.e.,
.
3. Characterizing Efficient Solutions by Weighting Approach
To characterizing an efficient solution for problem (P) by weighting approach [11] let us scalarize problem (P) to become in the form.
(Pw) 
, s.t.
where 
and
, j = 1, 2, ···, k are roughly B-invex with respect to 
with roughness degree rj on B-invex set M.
Theorem 1
If 
is an efficient solution for problem (P), then there exist
such that 
is an optimal solution for problem (Pw).
Proof. Let 
be an efficient solution for problem (P), then the system 
j = 1, 2, ···, k has no solution
. Upon Lemma 3 and by applying the generalized Gordan theorem [12], there exist 
such that
and 
.
Denote 
then 
and 
.
Hence 
is an optimal solution for problem (Pw).
Theorem 2
If 
is an optimal solution for 
corresponding to
, then 
is an efficient solution for problem (P) if either one of the following two conditions holds:
(i) 
for all
; or (ii) 
is the unique solution of
.
Proof. To proof see V. Chankong, Y. Y. Haimes [11].
4. Characterizing Efficient Solutions by ε-Constraint Approach
An ε-constraint approach is one of the common approaches for characterizing efficient solutions of multiobjective programming problems [11]. In the following we shall characterizing an efficient solution for multiobjective roughly B-invex programming problem (P) in term of an optimal solution of the following scalar problem.
where 
are roughly B-invex with respect to 
with roughness degree rj on B-invex set M.
Theorem 3
If 
is an efficient solution for problem (P), then 
is an optimal solution for problem 
corresponding to
.
wang#title3_4:spProof.
Let 
be not optimal solution for 
where 
So there exists 
such that 
,
.
Thus, 
is inefficient solution for problem (P) which is a contradiction. Hence 
is an optimal solution for problem
.
Theorem 4
Let 
is an optimal solution of 
for all q = 1, 2, ···, k, where
, j = 1, 2, ···, k . Then 
is an efficient solution for problem (P).
wang#title3_4:spProof.
Since 
is an optimal solution for
, for all q = 1, 2, ···, k. So, for each
, we get 
q = 1, 2, ···, k.
This implies that the system 
j = 1, 2, ···, k has no solution
, i.e. 
is an efficient solution for problem (P).
5. Sufficient and Necessary Conditions for Efficiency
In this section, we discuss the sufficient and necessary conditions for a feasible solution x* to be efficient or properly efficient for problem (P) in the form of the following theorems.
Theorem 5
Suppose there exists a feasible solution x* for (P), and scalars
, such that
. (1)
If
, j = 1, 2, ···, k are roughly B-invex with respect to 
with roughness degree r at x* and gi, 
is roughly B-invex with respect to same η with roughness degree ri at x*. Then x* is a properly efficient solution for problem (P).
Proof.
Since
, j = 1, 2, ···, k and 
are roughly B-invex with respect to same η, then there exists a function 
such that
by (1) for each 
such that
.
Thus, 
for all
, which implies that x* is the minimizer of
such that 
under the constraint 
where
. Hence, from Theorem (4.11) of [11], x* is a properly efficient solution for problem (P).
Theorem 6 Let x* be a feasible solution for (P). If there exist scalars
,
such that the triplet (x*, wi, ui) satisfies (1) of Theorem (5),
is strictly roughly B-invex with respect to η: M × M → Rn with roughness degree r at x* and gi, 
is roughly B-invex with respect to same η with roughness degree ri at x*. Then x* is an efficient solution for problem (P).
Proof.
Suppose that x* is not an efficient solution for (P). Then, there exists a feasible
, and index v such that 
Since 
is strictly roughly B-invex with respect to η with roughness degree r at x*, then there exists a function 
such that
. (2)
Also, roughly B-invexity of gi, 
with respect to same η at x* with roughness degree ri implies
i.e. 
, (3)
such that
. Adding (2) and (3), contradicts (1). Hence, x* is an efficient solution for problem (P).
Remark 1
Similarly as in Theorem (5), it can be easily seen that x* becomes properly efficient solution for (P), in the above theorem, if wj > 0, for all j = 1, 2, ···, k.
Theorem 7
Suppose there exists a feasible solution x* for (P), and scalars wi > 0, j = 1, 2, ···, k, 
such that (1) of Theorem (5) holds. If
is pseudo roughly B-invex with respect to 
with roughness degree r at x* and gI is quasi roughly B-invex with respect to same η with roughness degree rI at x*. Then x* is a properly efficient solution for problem (P).
Proof.
Since
, and gI are quasi roughly B-invex with respect to η with roughness degree rI at
, then there exists a function 
such that
for all 
such that
. By using (1)we have 
which implies
since 
is pseudo roughly B-invex with respect to same η with roughness degree r at x* which implies that x* is the minimizer of
such that 
under the constraint 
where 
≥ max(r, ri). Therefore, x* is a properly efficient solution for problem (P).
Theorem 8
Suppose that there exist a feasible solution x* for (P) and scalars
,
, 
such that (1) of Theorem (5) holds. Let
be strictly pseudo roughly B-invex with respect to 
with roughness degree r at 
and gI be quasi roughly B-invex with respect to same η with roughness degree rI at x*. Then x* is an efficient solution for problem (P).
Proof. Suppose that x* is not an efficient solution for (P).Then, there exists a feasible x for (P), and index v such that 
then there exists a function 
such that,
Strictly pseudo roughly B-invexity of 
implies that 
for all 
such that
. Since gI is quasi roughly B-invex with respect to same η with roughness degree rI at 
and
then 
for all 
such that
.
The proof now similar to the proof of Theorem (6).
Remark 2
Similarly as in Theorem (7), it can be easily seen that x* becomes properly efficient solution for (P), in the above theorem, if
.
Theorem 9
Suppose there exists a feasible solution x* for (P), and scalars 
such that (1) of Theorem (5) holds. Let
be pseudo roughly B-invex with respect to 
with roughness degree 
at x* and 
be quasi roughly B-invex with respect to same η with roughness degree 
at x*. Then x* is a properly efficient solution for problem (P).
Proof. The proof is similar to the proof of Theorem (7).
Theorem 10
Suppose that there exist a feasible solution x* for (P) and scalars
,
such that (1) of Theorem (5) holds. If
,
is quasi roughly B-invex with respect to 
with roughness degree r at x* and 
is strictly pseudo roughly B-invex with respect to same η with roughness degree rI at x*. Then x* is an efficient solution for problem (P).
Proof. The proof is similar to the proof of Theorem (8).
Remark 3
Similarly as in Theorem (7), it can be easily seen that x* becomes properly efficient solution for (P), in the above theorem, if
.
Theorem 11 (Necessary Optimality Criteria)
Assume that x* is a properly efficient solution for problem (P). Assume also that there exist a feasible point 
for (P) such that
, and each 
is roughly B-invex with respect to η: M × M → Rn with roughness degree ri at x*. Then, there exists scalars 
and
, such that the triplet 
satisfies
. (4)
Proof. Let the following system
. (5)
has a solution for every
. Since by the assumed Slater-type condition,
and then from roughly Bi-invexity of gi at 
with respect to η, there exists a function 
such that
. (6)
Therefore from (5) and (6)
for all
. Hence for some positive λ small enough
Similarly, for
, and for 
small enough
.
Thus, for λ sufficiently small and all
is feasible for problem (P). For sufficiently small
, (5) gives
. (7)
Now, for all 
such that
(8)
Consider the ratio (see Equation (9))
From (5),
. Similarly,
; but, by (8)
.
Thus, the ratio in (9) becomes unbounded, contradicting the proper efficiency of x* for (P). Hence, for each q = 1, 2, ···, k, the system (5) has no solution. The result then follows from an application of the Farkas Lemma as in [12], namely
.
Theorem 12
Assume that x* is an efficient solution for problem (P) at which the Kuhn-Tucker constraint qualification is satisfied. Then, there exist scalars
,
such that
,
.
Proof.
Since every efficient solution is a weak minimum, then by applying Theorem (2.2) of Weir and Mond [13] for x*, we get, there exists 
such that
. (9)
, where
.