Approximate quasi efficient solutions in multiobjective ... - SSMR

Bull. Math. Soc. Sci. Math. Roumanie Tome 51(99) No. 2, 2008, 109–121

Approximate quasi efficient solutions in multiobjective optimization by Miruna Beldiman, Eugenia Panaitescu and Liliana Dogaru

Abstract In this paper we consider unitar concepts for some optimality and efficiency or approximate solutions in scalar optimization and vectorial optimization respectively. In this cases some necessary and/or sufficient conditions for these approximate solutions in the multiobjective case are derived via scalarization and an alternative theorem. Thus are generalized and refined some corresponding results in multiobjective optimization.

Key Words: Multiobjective programming, optimization, approximate solution. 2000 Mathematics Subject Classification: Primary 90C29, Secondary 90C30, 49M37. 1

Introduction

The general goal in any optimization is to identify a single or all best solution within a set of feasible points. While it is theoretically possible to identify the complete set of best solutions, finding an exact description of this set often turns out to be practically impossible or at least computationally too expensive, and thus many research efforts focus on approximation concepts and procedures. Interest in getting approximate solutions of optimization problems has spread greatly during the past 20 years. The first concepts of approximate solution or ε-efficient solution in multiobjective optimization appear in the works of Kutateladze [6], Loridan [10, 11], and White [21]. Loridan [11] define ε-efficient point for a set and, starting from this concept, he extend the ε-optimal solution concept from scalar optimization problems to multiobjective optimization problems defining ε-efficient solution in a Pareto context. This definition is the same as Kutateladze’s notion introduced in [6] and is the concept more used in the literature to study approximate solutions for multiobjective optimization problems. White [21] analyze six different concepts of

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ε-efficient solution in the context of five solution methods of multiobjective optimization problems through scalarization. Tanaka [19] proposed a new approximate solution concept and Yokoyama [22] explored relations among three types of approximate solution set. Li [8], Lui [12] and Valyi [20] discussed ε-vector Lagrangians, generalized ε-saddle points and ε-problems for nonsmooth nonconvex multiobjective programming problems. Tammer [18] studied approximate solutions to vector optimization problems via generating the well known Ekeland’s variational principle. Related to general result - the Theorem on Nonconvex Functions of Kaliszewski [5] (a counterpart of the Fundamental Theorem on Convex Functions [15] in the case the convexity assumption does not hold), in this paper we discuss different conditions for (ε, ε)-quasi weak efficiency, (ε, ε)-quasi proper efficiency and (ε, ε)quasi efficiency for a multiobjective and nonconvex optimization problem. These results generalize and refine some known results in the literature of multiobjective programming. Further we can apply these results in statistics or information theory [1, 2]. We organize this paper as follows: in Section 2 we present the terminology used in this paper, derive necessary and/or sufficient conditions for the (ε, ε)-quasi weak efficiency, the (ε, ε)-quasi proper efficiency and the (ε, ε)-quasi efficiency in Sections 3, 4 and 5 respectively. Finally, in Section 6 we give a few concluding remarks. 2

Preliminaries

Let ϕ : X → R and the scalar problem (P )

½

inf ϕ(x) x ∈ X.

where X is a nonempty set in a normed linear space.

Definition 1. [4, 9] Let α > 0. A point x0 ∈ X is called i1) α-optimal solution of the scalar problem (P )if ϕ(x0 ) ≦ ϕ(x)+α,for all x ∈ X; i2) α-quasi optimal solution of the scalar problem (P ) if ϕ(x0 ) ≦ ϕ(x) + αkx − x0 k, for all x ∈ X. We see that an optimal solution is an α-optimal solution and an optimal solution is an α-quasi solution. When α = 0, an α-optimal solution (quasi solution) of (P ) is an optimal solution.

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Definition 2. Let (α0 , α0 ) ∈ R+ × R+ and x0 ∈ X. We say that x0 is (α0 , α0 )quasi optimal solution to (P ) if ϕ(x0 ) ≦ ϕ(x) + α0 kx − x0 k + α0 , ∀x ∈ X. We see that for α = α0 = 0 we get the concept of optimal solutions; if α0 = 0 we get α0 -optimal solution and when α0 = 0 we obtain α0 -quasi optimal solution concept. Consider the following multiobjective optimization problem ½ minimize f (x) = (f1 (x), ..., fm (x))T (V P ) subject to x ∈ X, where fi : X → R, i = 1, 2, ..., m, and m ≥ 2. Definition 3. Let x0 ∈ X and ε ∈ Rn+ . Then i) the point x0 is called an ε -quasi efficiency solution of (V P ) if f (x) + εkx − x0 k − f (x0 ) · 0, ∀x ∈ X ii) the point x0 is called an ε -quasi weakly efficiency solution of (V P ) if f (x) + εkx − x0 k − f (x0 ) ≮ 0, ∀x ∈ X If ε = ε0 em where ε0 ∈ R∗+ and em = (1, ..., 1)T ∈ Rm , then this definition is reduced to Definition 5.1 [4]. If ε = 0 we get the concepts of efficient solution and weak efficient solution respectively. Similar to ε-properly efficiency we consider ε-quasi properly efficiency. Definition 4. x0 ∈ X is called ε-quasi properly efficient solution of (V P ) if it is ε-quasi efficient and if there exists M > 0 such that for any i ∈ M = {1, 2, ..., m} and x ∈ X satisfying fi (x) < fi (x0 ) − εi kx − x0 k there exists j ∈ M with fj (x) > fj (x0 ) − εj kx − x0 k and fi (x0 ) − εi kx − x0 k − fi (x) ≦ M. fj (x) − fj (x0 ) + εj kx − x0 k Remark 1. (1) ε-quasi proper efficiency ⇒ ε-quasi efficiency ⇒ α-quasi weakly efficiency (2) when ε = 0, an ε-(quasi properly, quasi weakly) efficient solution of (V P ) is a (properly, weakly) efficient solution. m Definition 5. Let (ε, ε) ∈ Rm + × R+ and x0 ∈ X. We say that x0 is:

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i1) (ε, ε)-quasi efficient to (V P ) if f (x) + εkx − x0 k + ε − f (x0 ) · 0, ∀x ∈ X i2) (ε, ε)-quasi weakly efficient to (V P ) if f (x) + εkx − x0 k + ε − f (x0 ) ≮ 0, ∀x ∈ X i3) (ε, ε)-quasi properly efficient to (V P )if exists M > 0 such that for any i ∈ M = {1, 2, ..., m} and x ∈ Xsatisfying fi (x) < fi (x0 ) − εi kx − x0 k − εi there exists j ∈ M with fj (x) > fj (x0 ) − εj kx − x0 k − εj and fi (x0 ) − εi kx − x0 k − εi − fi (x) ≦ M. fj (x) − fj (x0 ) + εj kx − x0 k + εj As in [7], to prove our main results, we quote an alternative theorem for nonconvex functions given by Kaliszewski [5, Theorem 3.1]. Theorem Kaliszewski One and only one of the following alternatives holds: (a) there exists some x ∈ X such that fi (x) < 0, i ∈ M; (b) for any negative numbers δ 1 , δ 2 , ..., δ m there exist positive numbers λi = −δ −1 i , i ∈ M such that maxi∈M λi [fi (x) − δ i ] ≧ 1, for any x ∈ X. 3

Case of (ε, ε)-Quasi Weak Efficiency

In this section, relate to (ε, ε)-quasi weak efficiency we give a necessary and sufficient condition. Theorem 1. Let x0 ∈ X. Then x0 is an (ε, ε)-quasi weakly efficient solution of (V P ) if and only if for any yi∗ < fi (x0 ), i ∈ M, x0 is (ε0 , ε0 )-quasi optimal solution to the following scalar optimization problem ½ minimize maxi∈M λi [fi (x) − yi∗ ], (SP1 ) subject to x ∈ X, where λi = [fi (x0 ) − yi∗ ]−1 , i ∈ M, ε0 = maxi∈M λi εi and ε0 = maxi∈M λi εi . Proof: Firstly, let x0 ∈ X be a (ε, ε)-quasi weakly efficient solution (q.w.e.s.) of (V P ) and we prove that x0 is (ε0 , ε0 )-quasi optimal for (SP1 ). Hence the system of inequalities fi (x) < fi (x0 ) − εi kx − x0 k − εi , i ∈ M

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has no solution x ∈ X. Thus by Theorem [Kaliszewski, 5] for any negative numbers δ 1 , δ 2 , ..., δ m there exist positive numbers λi = −δ −1 i , i ∈ M such that maxi∈M λi [fi (x) − fi (x0 ) + εi kx − x0 k + εi − δ i ] ≧ 1, for any x ∈ X. Since maxi∈M λi [fi (x) − yi∗ + εi kx − x0 k + εi ] ≦ maxi∈M λi [fi (x) − yi∗ ] + ε0 kx − x0 k + ε0 for all x ∈ X and λi [fi (x0 ) − yi∗ ] = 1, i ∈ M, we obtain easily that x0 is ε0 -quasi optimal for (SP1 ). Conversely, let x0 be a (ε0 , ε0 )-quasi optimal solution for (SP1 ). We have to prove that x0 is (ε, ε)-quasi weakly efficient to (V P ). We suppose contrary, i.e. x0 is not (ε, ε)-quasi weakly efficient to (V P ). Therefore, there would exist some x0 ∈ X such that fi (x0 ) < fi (x0 ) − εi kx0 − x0 k − εi , i ∈ M. = fi (x0 ) − yi∗ = fi (x0 ) − fi (x0 ) > Let yi∗ = fi (x0 ), i ∈ M. Then yi∗ < fi (x0 ), λ−1 i εi kx0 − x0 k + εi and hence λi εi kx0 − x0 k + λi εi < 1 for i ∈ M. By assumption that x0 is (ε0 , ε0 )-quasi optimal to (SP1 ), for any x ∈ X we have maxi∈M λi [fi (x) − fi (x0 )] ≧ ≧ maxi∈M λi [fi (x0 ) − fi (x0 )] − maxi∈M λi (εi kx0 − x0 k + εi ) = = maxi∈M λi [fi (x0 ) − fi (x0 )] − ε0 kx0 − x0 k − ε0 = = 1 − ε0 kx0 − x0 k − ε0 > 0. Taking x = x0 in above inequality, we get 0 > 0. This contradiction implies that x0 is (ε, ε)-quasi weakly efficient to (V P ). This proof is completed.

4

Case of (ε, ε)-Quasi Proper Efficiency

As in [7] we can prove a similar theorem for (ε, ε)-quasi proper efficiency. Lemma 1. If x0 ∈ X is (ε, ε)-quasi proper efficient to (V P ) then the system αi fi (x) + ρeT f (x) < < αi fi (x0 ) + ρeT f (x0 ) − αi εi kx − x0 k − αi εi − ρeT εkx − x0 k − ρeT ε with i ∈ M admits no solution x ∈ X for some ρ > 0, where αi > 0, i ∈ M.

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Proof: We suppose that x0 is (ε, ε)-quasi properly efficient to (V P ). By the (ε, ε)-quasi proper efficiency of x0 , the system fi (x) < fi (x0 ) − εi kx − x0 k − εi , i ∈ M

(1)

has no solution in X. Therefore, the system αi fi (x) < αi fi (x0 ) − αi εi kx − x0 k − αi εi , i ∈ M has no solution in X for αi > 0, i ∈ M. Let x b0 ∈ X be fixed. We discuss in the following two cases: Case 1 : We prove that if eT f (x0 ) − eT εkb x0 − x0 k − eT ε ≤ eT f (b x0 ) then the system of m inequalities αi fi (b x0 ) + ρeT f (b x0 ) < x0 − x0 k − ρeT ε < αi fi (x0 ) + ρeT f (x0 ) − αi εi kb x0 − x0 k − αi εi − ρeT εkb with i ∈ M is inconsistent for any ρ > 0. This is because if it was not the case, we would have αi fi (x0 ) − αi εi kb x0 − x0 k − αi εi − αi fi (b x0 ) > > ρ[eT f (b x0 ) − eT f (x0 ) + eT εkb x0 − x0 k + eT ε] with i ∈ M a contradiction to (1). Case 2 : If eT f (x0 ) − eT εkb x0 − x0 k − eT ε > eT f (b x0 ), then x0 )} 6= ∅ I = {i ∈ M|fi (x0 ) − εi kb x0 − x0 k − εi > f (b By Remark 1, x0 is (ε, ε)-quasi efficient to (V P ). There exists some j0 ∈ M such x0 − x0 k − εj0 . x0 ) > fj0 (x0 ) − εj0 kb that fj0 (b Let x0 ) − fi (x0 ) + εi kb x0 − x0 k + εi ]. fl (b x0 ) − fl (x0 ) + εl kb x0 − x0 k + εl = maxi∈M [fi (b It is clear that fl (b x0 ) − fl (x0 ) + εl kb x0 − x0 k + εl > 0. By the (ε, ε)-quasi properly efficiency of x0 , there exists an M > 0, such that for any i ∈ I, there exists an j ∈ M satisfying fj (b x0 ) > fj (x0 ) − εj kb x0 − x0 k − εj and x0 ) fi (x0 ) − εi kb x0 − x0 k − εi − fi (b ≦ M. fj (b x0 ) − fj (x0 ) + εj kb x0 − x0 k + εj Thus, x0 ) ≦ M[fj (b x0 ) − fj (x0 ) + εj kb x0 − x0 k + εj ] ≦ fi (x0 ) − εi kb x0 − x0 k − εi − fi (b ≦ M[fl (b x0 ) − fl (x0 ) + εl kb x0 − x0 k + εl ]

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and hence [fl (b x0 ) − fl (x0 ) + εl kb x0 − x0 k + εl ]−1

X

x0 )] ≦ [fi (x0 ) − εi kb x0 − x0 k − εi − fi (b

i∈I

≦ M(m − 1) Since 0 < eT [f (x0 ) − εkb x0 − x0 k − ε − f (b x0 )] ≦

X

x0 )] [fi (x0 ) − εi kb x0 − x0 k − εi − fi (b

i∈I

we have x0 −x0 k−ε−f (b x0 )] ≦ M(m−1). [fl (b x0 )−fl (x0 )+εl kb x0 −x0 k+εl ]−1 eT [f (x0 )−εkb Let ρ = (mini∈M αi )[M(m − 1)]. Then ρ ≦ αl [M(m − 1)]−1 ≦ x0 − x0 k − ε − f (b x0 ))]−1 ≦ αl [fl (b x0 ) − fl (x0 ) + εl kb x0 − x0 k + εl ] · [eT (f (x0 ) − εkb i.e. x0 )] ≤ αl [fl (b x0 ) − fl (x0 ) + εl kb x0 − x0 k + εl ]. ρeT [f (x0 ) − εkb x0 − x0 k − ε − f (b Hence αl fl (x0 ) + ρeT f (x0 ) − αl εl kb x0 − x0 k − αl εl − ρeT εkb x0 − x0 k − ρeT ε < < αl fl (b x0 ) + ρeT f (b x0 ). Therefore, the system of m inequalities αi fi (b x0 ) + ρeT f (b x0 ) < < αi fi (x0 ) + ρeT f (x0 ) − αi εi kb x0 − x0 k − αi εi − ρeT εkb x0 − x0 k − ρeT ε with i ∈ M is inconsistent. Noting that x b0 can be any element of X, we conclude that the system  x0 ) + ρeT f (b x0 ) < αi fi (x0 ) + ρeT f (x0 ) − αi εi kb x0 − x0 k − αi εi −  αi fi (b − ρeT εkb x0 − x0 k − ρeT ε, i ∈ M  x ∈ X.

(2)

has no solution for some ρ > 0. This completes the proof.

From Theorem Kaliszewski and Lemma 1 we get the following necessary condition for a feasible solution to be an (ε, ε)-properly efficient solution to (V P ).

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Theorem 2. Let x0 be a (ε, ε)-quasi proper efficient solution of (V P ). Then x0 is an (ε0 , ε0 )-quasi optimal solution to the following scalar optimization problem ½ minimize maxi∈M λi [(fi (x) − yi∗ ) + ρeT (f (x) − y ∗ )] (SP2 ) subject to x ∈ X, for some ρ > 0, when ε0 = maxi∈M λi (εi + ρeT ε), ε0 = maxi∈M λi (εi + ρeT ε) and λi = [fi (x0 ) − yi∗ + ρeT (f (x0 ) − y ∗ )]−1 . Proof: Let αi = 1, i ∈ M. By Lemma 1, there exists a ρ > 0 such that the system (2) has no solution. By Theorem Kaliszewski, for any negative numbers δ 1 , δ 2 , ..., δ m , λi = −δ −1 > 0, i ∈ M and i maxi∈M λi [(fi (x) − fi (x0 )) + ρeT (f (x) − f (x0 )) + εi kx − x0 k + εi + +ρeT εkx − x0 k + ρeT ε − δ i ] ≧ 1 for any x ∈ X. Let yi∗ be a number such that δ i = (yi∗ − fi (x0 )) + ρeT (y ∗ − f (x0 )) < 0, i ∈ M. Denote λi = −δ −1 i . Then λi > 0, i ∈ M and for any x ∈ X, 1 ≦ maxi∈M λi [(fi (x) − yi∗ ) + ρeT (f (x) − y ∗ )+ +εi kx − x0 k + εi + ρeT εkx − x0 k + ρeT ε] ≦ ≦ maxi∈M λi [(fi (x) − yi∗ ) + ρeT (f (x) − y ∗ )]+ +maxi∈M λi [εi kx − x0 k + εi + ρeT εkx − x0 k + ρeT ε] = = maxi∈M λi [(fi (x) − yi∗ ) + ρeT (f (x) − y ∗ )]+ +maxi∈M λi (εi + ρeT ε)kx − x0 k + maxi∈M λi (εi + ρeT ε) = = maxi∈M λi [(fi (x) − yi∗ ) + ρeT (f (x) − y ∗ )] + ε0 kx − x0 k + ε0 . Hence, for any x ∈ X maxi∈M λi [(fi (x) − yi∗ ) + ρeT (f (x) − y ∗ )] ≧ ≧ maxi∈M λi [(fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )] − ε0 kx − x0 k − ε0 . The proof is completed. Theorem 3. Let x0 be a (ε, ε)-quasi properly efficient solution of (V P ). Then there exists ρ ∈ R∗+ such that x0 is (ε0 , ε0 )-quasi optimal to the following scalar optimization problem ½ minimize maxi∈M [λi (fi (x) − yi∗ ) + ρeT (f (x) − y ∗ )] (SP3 ) subject to x ∈ X,

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where ε0 = maxi∈M (λi εi +ρeT ε), ε0 = maxi∈M (λi εi +ρeT ε), λi = [fi (x0 )−yi∗ ]−1 and yi∗ is a fixed number such that λi > 0, i ∈ M. Proof: Assume that x0 ∈ X is an (ε, ε)-quasi properly efficient solution to (V P ). Let yi∗ be a fixed number such that λi = [fi (x0 ) − yi∗ ]−1 > 0, i ∈ M. By Lemma 1, there exists a ρ > 0 such that for any x ∈ X, the system λi [fi (x) − yi∗ ] + ρeT [f (x) − y ∗ ] < < λi [fi (x0 ) − yi∗ ] + ρeT [f (x0 ) − y ∗ ] − λi εi kx − x0 k − λi εi − ρeT εkx − x0 k − ρeT ε with i ∈ M is inconsistent. Hence, there exists some i0 ∈ M such that λi0 [fi0 (x) − yi∗0 ] + ρeT [f (x) − y ∗ ] ≧ ≧ λi0 [fi0 (x0 ) − yi∗0 ] + ρeT [f (x0 ) − y ∗ ]− −λi0 εi0 kx − x0 k − λi0 εi0 − ρeT εkx − x0 k − ρeT ε = = 1 + ρeT [f (x0 ) − y ∗ ] − [λi0 εi0 kx − x0 k + λi0 εi0 + ρeT εkx − x0 k + ρeT ε] = = maxi∈M [λi (fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )]− −[λi0 εi0 kx − x0 k + λi0 εi0 + ρeT εkx − x0 k + ρeT ε] ≧ ≧ maxi∈M [λi (fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )]− −maxi∈M [λi εi kx − x0 k + λi εi + ρeT εkx − x0 k + ρeT ε] ≧ ≧ maxi∈M [λi (fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )]− −maxi∈M [λi εi + ρeT ε]kx − x0 k − maxi∈M [λi εi + ρeT ε] = = maxi∈M [λi (fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )] − ε0 kx − x0 k − ε0 . Therefore, maxi∈M [λi (fi (x) − yi∗ ) + ρeT (f (x) − y ∗ )] ≧ ≧ maxi∈M [λi (fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )] − ε0 kx − x0 k − ε0 . That implies that x0 is (ε0 , ε0 )-quasi optimal to scalar optimization problem (SP3 ). The proof is completed.

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Case of (ε, ε)-Quasi Efficiency

Theorem 4. Let x0 be a (ε0 , ε0 )-quasi optimal solution of (SP2 ) where ε0 = maxi∈M λi (εi + ρeT ε), ε0 = maxi∈M λi (εi + ρeT ε), ρ ∈ R∗+ , yi∗ , i ∈ M such that λi = [fi (x0 ) − yi∗ + ρeT (f (x0 ) − y ∗ )]−1 > 0 with i ∈ M. Then x0 is (ε, ε)-quasi efficient for (V P ). Proof: Suppose that there exists some ρ > 0 such that, for any yi∗ such that λi = [fi (x0 ) − yi∗ + ρeT (f (x0 ) − y ∗ )]−1 > 0, i ∈ M, x0 is (ε0 , ε0 )-quasi optimal to (SP2 ). Assume that x0 was not (ε, ε)-quasi efficient to (V P ). By Definition 5, there would exist some x0 ∈ X satisfying fj (x0 ) ≦ fj (x0 ) − εj kx0 − x0 k − εj , j ∈ M with at least one strict inequality. Let yi∗ = fi (x0 ), i ∈ M. Then λ−1 = fi (x0 ) − yi∗ + ρeT (f (x0 ) − y ∗ ) = i = fi (x0 ) − fi (x0 ) + ρeT (f (x0 ) − f (x0 )) > > εi kx0 − x0 k + εi + ρeT εkx0 − x0 k + ρeT ε ≥ 0, i ∈ M. We get λi (εi kx0 − x0 k + εi + ρeT εkx0 − x0 k + ρeT ε) < 1, i ∈ M. Because ε0 = maxi∈M λi (εi + ρeT ε) and ε0 = maxi∈M λi (εi + ρeT ε), then ε0 kx0 − x0 k + ε0 < 1. Since x0 is (ε0 , ε0 )-quasi optimal to (SP2 ), we have 0 = maxi∈M λi [(fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )] ≥ ≥ maxi∈M λi [(fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )] − ε0 kx0 − x0 k − ε0 = = 1 − ε0 kx0 − x0 k − ε0 > 0 a contradiction. This implies that x0 is (ε, ε)-quasi efficient to (V P ). Theorem 5. Let x0 be a (ε0 , ε0 )-quasi optimal solution of (SP3 ) with ε0 = maxi∈M (λi εi + ρeT ε), ε0 = maxi∈M (λi εi + ρeT ε), ρ > 0, yi∗ ∈ R, i ∈ M such that for any i ∈ M, λi = [fi (x0 ) − yi∗ ]−1 > 0 then x0 is (ε, ε)-quasi efficient solution (V P ).

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Proof: Suppose that there exists some ρ > 0 such that, for any yi∗ satisfying λi = [fi (x0 ) − yi∗ ]−1 > 0, i ∈ M, x0 is (ε, ε)-quasi optimal to (SP3 ). If x0 was not (ε, ε)-quasi efficient to (V P ), there would exist an x0 ∈ X such that fi (x0 ) ≤ fi (x0 ) − εi kx0 − x0 k − εi , i ∈ M

(3)

with at least one strict inequality. Let yi∗ = fi (x0 ) − εi kx0 − x0 k − εi , i ∈ M. Then λ−1 = fi (x0 ) − yi∗ = εi kx0 − x0 k + εi > 0, i ∈ M, i and ε0 = maxi∈M (λi εi + ρeT ε), ε0 = maxi∈M (λi εi + ρeT ε), hence ε0 kx0 − x0 k + ε0 = maxi∈M (λi εi + ρeT ε)kx0 − x0 k + maxi∈M (λi εi + ρeT ε) = = maxi∈M (λi εi kx0 − x0 k + λi εi + ρeT [εkx0 − x0 k + ε]) = = 1 + ρeT [εkx0 − x0 k + ε]. By (3) and (ε0 , ε0 )-quasi optimality of x0 with respect to (SP3 ), we have 0 > maxi∈M [λi (fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )] > > maxi∈M [λi (fi (x0 ) − yi∗ ) + ρeT (f (x0 ) − y ∗ )] − ε0 kx0 − x0 k − ε0 = = maxi∈M (λi εi kx0 − x0 k + λi εi + ρeT εkx0 − x0 k + ρeT ε]) − ε0 kx0 − x0 k − ε0 = maxi∈M [1 + ρeT (εkx0 − x0 k + ε)] − ε0 kx0 − x0 k − ε0 = 0 a contradiction. Therefore, x0 is (ε, ε)-quasi efficient to (V P ). The proof is completed.

Remark 2. For ε = 0 and/or ε = 0 we get some results stated in [3, 4, 5, 7, 9, 13, 14, 16]. 6

Conclusions

In this paper we give necessary and sufficient conditions for the (ε, ε)-quasi weak efficiency, two necessary conditions for the (ε, ε)-quasi proper efficiency and two sufficient conditions for (ε, ε)-quasi efficiency in multiobjective optimization. No assumption of any convexity is made in this paper and the discussion is carried out in a very general framework. The results generalize the corresponding ones in this field. We remark that some other results could be obtained, for example in the convex case with ε-subdifferential calculus for scalar functions or locally Lipschitz case by using the generalized gradient of Clarke [17].

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Miruna Beldiman, Eugenia Panaitescu and Liliana Dogaru

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