Báo cáo hóa học: " Research Article Optimality Conditions for Approximate Solutions in Multiobjective Optimization Problems" pdf

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 620928, 17 pages doi:10.1155/2010/620928 Research Article Optimality Conditions for Approximate Solutions in Multiobjective Optimization Problems Ying Gao, 1 Xinmin Yang, 1 and Heung Wing Joseph Lee 2 1 Department of Mathematics, Chongqing Normal University, Chongqing 400047, China 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong Correspondence should be addressed to Ying Gao, gaoyingimu@163.com Received 18 July 2010; Accepted 25 October 2010 Academic Editor: Mohamed El-Gebeily Copyright q 2010 Ying Gao et al. 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 first- and second-order necessary and sufficient optimality conditions for approximate weakly, properly efficient solutions of multiobjective optimization problems. Here, tangent cone, -normal cone, cones of feasible directions, second-order tangent set, asymptotic second-order cone, and Hadamard upper lower directional derivatives are used in the characterizations. The results are first presented in convex cases a nd then generalized to nonconvex cases by employing local concepts. 1. Introduction The investigation of the optimality conditions is one of the most attractive topics of optimization theory. For vector optimization, the optimality solutions can be characterized with the help of different geometrical concepts. Miettinen and M ¨ akel ¨ a 1 and Huang and Liu 2 derived several optimality conditions for efficient, weakly efficient, and properly efficient solutions of vector optimization pro blems with the help of several kinds of cones. Engau and Wiecek 3 derived the cone characterizations for a pproximate solutions of vector optimization problems by using translated cones. In 4, Aghezzaf and Hachimi obtained second-order optimality conditions by means of a second-order tangent set which can be considered an extension of the tangent cone; Cambini et al. 5 and Penot 6 introduced a new second-order tangent set called asymptotic second-order cone. Later, second-order optimality conditions for vector optimization problems have been widely studied by using these second-order tangent sets; see 7–9. 2 Journal of Inequalities and Applications During the past decades, researchers and practitioners in optimization had a keen interest in approximate solutions of optimization problems. There are several important reasons for considering this kind of solutions. One of them is that an approximate solution of an optimization problem can be computed by using iterative algorithms or heuristic methods. In vector optimization, the notion of approximate solution has been defined in several ways. The first concept was introduced by Kutateladze 10 and has been used to establish vector variational principle, approximate Kuhn-Tucker-type conditions and approximate duality theorems, and so forth, see 11–20. Later, several authors have proposed other -efficiency concepts see, e.g., White 21;Helbig22 and Tanaka 23. In this paper, we derive di fferent characterizations for approximate solutions by treating convex case and nonconvex cases. Giving up convexity naturally means that we need local instead of global analysis. Some definitions and notations are given in Section 2.InSection 3, we derive some characterizations for global approximate solutions of multiobjective optimization problems by using tangent cone, the cone of feasible directions and -normal cone. Finally, in Section 3, we introduce some local approximate concepts and present some properties of these notions, and then, first and second-order sufficient conditions for local properly approximate efficient solutions of vector optimization problems are derived. These conditions are expressed by means of tangent cone, second-order tangent set and asymptotic second-order set. Finally, some sufficient conditions are given for local weakly approximate efficient solutions by using Hadamard upper lower directional derivatives. 2. Preliminaries Let R n be the n-dimensional Euclidean space and let R n  be its nonnegative orthant. Let C be a subset of R n , then, the cone generated by the set C is defined as coneC∪ α≥0 αC, and int C and cl C referred to as the interior and the closure of the set C, respectively. A set D ⊂ R n is said to be a cone if cone D  D. We say that the cone D is solid if int D /  ∅,and pointed if D ∩ −D ⊂{0}. The cone D is said to have a base B if B is convex, 0 / ∈ cl B and D  cone B. The positive polar cone and strict positive polar cone of D are denoted by D  and D s , respectively. Consider the following multiobjective optimization problem: min  f  x  : x ∈ S  , 2.1 where S ⊂ R n is an arbitrary nonempty set, f : S → R m . As usual, the preference relation ≤ defined in R m by a closed c onvex pointed cone D ⊂ R m is used, which models the preferences used by the decision-maker: y, z ∈ Y, y ≤ z ⇐⇒ y − z ∈−D. 2.2 We recall that x 0 ∈ S is an efficient solution of 2.1 with respect to D if fx 0  − D ∩ fS{fx 0 }. x 0 ∈ S is a weakly efficient solution of 2.1 with respect to D if fx 0  − int D ∩ fS∅ in this case, it is assumed that D is solid. x 0 ∈ S is a Benson properly efficient solution see 24 of 2.1 with respect to D if cl conefSD −fx 0  ∩ −D{0}. x 0 ∈ S is a Henig’ properly efficient solution see 24 of 2.1 with respect to D if x 0 ∈ Ef, D  , for some convex cone D  with D \{0}⊂int D  . Journal of Inequalities and Applications 3 Definition 2.1 see 18, 25.Letq ∈ D \{0} be a fixed element, and  ≥ 0. i x ∈ S is said to be a weakly q-efficient solution of problem 2.1 if fS − fx q ∩ − int D∅ in this case it is assumed that D is solid. ii x ∈ S is said to be a efficient q-solution of problem 2.1 if fS−fxq∩−D \ {0}∅. iii  x ∈ S is said to be a properly q-efficient solution of problem 2.1,ifclconefS q  D − fx ∩ −D{0}. The sets of q-efficient solutions, weakly q-efficient solutions, a nd properly q- efficient solutions of problem 2.1 are denoted by AEf, S, q,WAEf, S, q,and PAEf, S, q, respectively. Remark 2.2. If   0, then q-efficient solution, weakly q-efficient solution, and properly q- efficient solution reduce to efficient solution, weakly efficient solution and properly efficient solution of problem 2.1. Definition 2.3. Let Z ⊂ R m be a nonempty convex set. The contingent cone of Z at z ∈ Z is defined as T  z, Z    d ∈ R m :thereexistst j ↓ 0andd j −→ d such that z  t j d j ∈ Z  . 2.3 The cone of feasible directions of Z at z ∈ Z is defined as F  z, Z   { d ∈ R m :thereexistst>0suchthatz  td ∈ Z } . 2.4 Let  ≥ 0, the -normal set of Z at z ∈ Z is defined as N   z, Z    y ∈ R m : y T  x − z  ≤ , ∀x ∈ Z  . 2.5 Lemma 2.4 see 26. Let N, K ⊂ R m be closed convex cones such that N ∩ K  {0}. Suppose that K is pointed and locally compact, or int K  /  ∅,then,−N   ∩ K s /  ∅. 3. Cone Characterizations of Approximate Solutions: Convex Case In this section, we assume that fS is a convex set. Theorem 3.1. Let x ∈ S and  ≥ 0.If F  f  x  ,f  S   ∩  −q − D \ { 0 }   ∅, 3.1 then x ∈ AEf, S, q. Proof. Suppose, on the contrary, that x / ∈ AEf, S, q, then, there exist x ∈ S and p ∈ D \{0} such that fx − f xq  −p.Thatis,fxfx−q − p. T herefore, −q − p ∈ Ff x,fS, which is a contradiction to Ffx,fS ∩ −q − D \{0}∅. This completes the proof. 4 Journal of Inequalities and Applications Theorem 3.2. Let x ∈ S. i If Tf x,fS ∩ −D \{0}∅,thenx ∈ PA E f, S. ii Let >0,andD is solid set and q ∈ int D.IfTf x,fS ∩ −q − D \{0}∅,then x ∈ PAEf, S, q. Proof. i Suppose, on the contrary, that x / ∈ PAEf, S, then, there exists q ∈−D \{0} such that q ∈ cl conefS − f xD. Hence, there exist λ n ∈ R  , x n ∈ S and q n ∈ D, n ∈ N such that λ n fx n  − fxq n  → q.Sinceq /  0, there exists n ∈ N such that λ n > 0. Since fS is convex set, cl conefS − f x  Tfx,fS.Hence,clconefS − f x ∩ −D \{0}∅.FromLemma 2.4,thereexistsu ∈ D s such that u, y≥0, for all y ∈ cl conefS − f x. On the other hand, from u ∈ D s ,wehaveu, q < 0. Therefore, there exists n 1 ∈ N such that u, fx n 1  − fxq n 1  < 0, and so u, fx n 1  − fx < 0, which deduces a contradiction, and the proof is completed. ii Now, we let >0. From Tf x,fS ∩ −q − D \{0}∅,wehave T  f  x  ,f  S   ∩  − int D   ∅. 3.2 In fact, if there exists p ∈ R m such that p ∈ Tfx,fS ∩ − int D, then, from q ∈ int D and >0, there exists λ>0suchthatp 1  −λp − q ∈ D \{0}.Hence,−q − p 1  λp ∈ Tf x,fS ∩ −q − D \{0}, which is a contradiction to the assumption. Since fS is a convex set, cl conefS − f x  Tfx,fS.Hence, cl cone  f  S  − f  x   ∩  − int D   ∅. 3.3 By using the convex separation theorem, there exists u ∈ R m \{0} such that u, y≥0, for all y ∈−int D and u, y≤0, for all y ∈ cl conefS − f x. It is easy to get that u, y≥0, for all y ∈−D.Hence,u, y > 0, for all y ∈−int D. Suppose, on the contrary, that x / ∈ PAEf, S,q, then, there exists y ∈ R m such that y ∈ cl cone  f  S   q  D − f  x   ∩  −D \ { 0 } , 3.4 and there exist y n ∈ conefSq  D − fx,foralln ∈ N such that y n → y.Thatis,there exist λ n ≥ 0,x n ∈ S and p n ∈ D,foralln ∈ N such that y n  λ n fx n q  p n − fx,forall n ∈ N.Since y /  0, there exists n 1 ∈ N such that λ n > 0, for all n ≥ n 1 .From>0, q ∈ int D and p n ∈ D,foralln ∈ N,wehaveq  p n ∈ int D,foralln ∈ N. Therefore,  u, y n   λ n  u, f  x n  − f  x     u, q  p n  <λ n  u, f  x n  − f  x   ≤ 0, ∀n ≥ n 1 . 3.5 Which implies u, y < 0. On the other hand, from y ∈−D \{0},wehaveu, y≥0, which yields a contradiction. This completes the proof. Remark 3.3. If   0, then the conditions of Theorems 3.1 and 3.2 are a lso necessarysee 2. But for >0, these are not necessary conditions, see the following example. Journal of Inequalities and Applications 5 Example 3.4. Let D  R 2  , q 1 , 1 T , S  {x ∈ R 2 : x 1 ≥ 0,x 2 ≥ 0},f : S → R 2 , fxx,   1/2and x 1/2, 1/2 T , then, x ∈ AEf, S, q and x ∈ PAEf, S, q.ButFfx,fS  R 2  Tfx,fS.Hence,Ffx,fS ∩ −q − D \{0} /  ∅ and Tfx,fS ∩ −q − D \ {0} /  ∅. Theorem 3.5. Let x ∈ S,  ≥ 0, D be a solid set and q ∈ int D.Ifthereexistsu ∈−D  \{0} such that −u, q≥1 and u ∈ N  fx,fS,thenx ∈ WAEf, S,q. Conversely , if x ∈ WAE f, S, q, then there exists u ∈−D  \{0} such that −u, q  1 and u ∈ N  fx,fS. Proof. Assume that, there exists u ∈−D  \{0} such that −u, q≥1andu ∈ N  fx,fS. Suppose, on the contrary, that x / ∈ WAEf, S,q, then, there exist p ∈−int D and x ∈ S such that p  fx−f xq.Fromu ∈−D  \{0} and −u, q≥1, we have u, fx−fxq > 0. Hence,  u, f  x  − f  x   > −  u, q  ≥ . 3.6 On the other hand, from u ∈ N  fx,fS,wehaveu, fx − fx≤,whichisa contradiction to the above inequality. Hence, x ∈ WAEf, S, q. Conversely, let x ∈ WAEf, S,q, then, fS − fxq ∩ − int D∅.SincefS is convex and D is a convex cone, there exists u ∈−D  \{0} such that u, fx − fxq≤ 0, for all x ∈ S.Sinceq ∈ int D,thereexistsu ∈−D  \{0} such that −u, q  1and u, fx − f xq≤0, for all x ∈ S. Therefore, u, fx − fx≤−u, q  ,forallx ∈ S, which implies u ∈ N  fx,fS. This completes the proof. Theorem 3.6. Let x ∈ S and  ≥ 0.Ifthereexistsu ∈−D s such that −u, q≥1 and u ∈ N  fx,fS,thenx ∈ PAEf, S, q. Conversely, assume that D is a locally compact set, if x ∈ PA Ef, S, q, then there exists u ∈−D s such that −u, q  1 and u ∈ N  fx,fS. Proof. Assume that, there exists u ∈−D s such that −u, q≥1andu ∈ N  fx,fS. Suppose, on the contrary, that x / ∈ PAEf, S,q, then, there exists p ∈ R m such that p ∈ cl cone  f  S   q  D − f  x   ∩  −D \ { 0 } , 3.7 and there exists p n ∈ conefSq  D − fx,foralln ∈ N such that p n → p.From u ∈ D s and p ∈ −D \{0},wehaveu, p > 0. Hence, there exists n 1 ∈ N such that u, p n  > 0, for all n ≥ n 1 .Fromp n ∈ conefSq  D − fx,foralln ∈ N,thereexist λ n ≥ 0, x n ∈ S,andq n ∈ D such that p n  λ n fx n q  q n − fx,foralln ∈ N. Therefore, u, fx n qq n −fx > 0, for all n ≥ n 1 , which combing with q n ∈ D and −u, q≥1 yields u, fx n  − fx > −u, q≥,foralln ≥ n 1 , which is a contradiction to u ∈ N  fx,fS. Hence, x ∈ PAEf, S, q. Conversely, let x ∈ PAEf, S, q, then, cl cone  f  S   q  D − f  x   ∩  −D   { 0 } . 3.8 Since fS is a convex set, cl conefSq  D − f x is a closed convex cone. From Lemma 2.4,thereexists u ∈ −D s  −D s such that u ∈−cl conefSq  D − fx  . Since q ∈ int D, D s and cl conefSq  D − fx  are cone, there exists u ∈ −D s such that −u, q  1andu ∈−cl conefSq  D − f x  . 6 Journal of Inequalities and Applications Now, we prove that u ∈ N  fx,fS.Thatis,u, fx − fx≤,forallx ∈ S. From u ∈−cl conefSq  D − f x  ,wehave  u, f  x  − f  x   q  p  ≤ 0, ∀x ∈ S, p ∈ D. 3.9 Since 0 ∈ D and −u, q  1, we have  u, f  x  − f  x   ≤−  u, q   , ∀x ∈ S. 3.10 Which implies u ∈ N  fx,fS. This completes the proof. Example 3.7. Let D  R 2  , q 1 , 1 T , S  {x ∈ R 2 : x 1 ≥ 0,x 2 ≥ 0},f : S → R 2 , fxx,   1/2and x 1/2, 1/2 T , then, x ∈ WAEf,S,  and x ∈ PAEf, S, .Letu  −1/2, 1/2 T , then u, p  1andu ∈ N  fx,fS  {x ∈ R 2 : x 1  x 2 ≥−1,x 1 ≤ 0,x 0 ≤ 0}. Remark 3.8. i If   0andD  R m  , then Theorems 3.1 and 3.5 reduce to the corresponding results in 1. ii In 1, the cone characterizations of Henig’ properly efficient solution were derived. We know that Henig’ properly efficient solution equivalent to Benson properly efficient solution, when D is a closed convex pointed conesee 24. Therefore, if   0and D  R m  ,Theorems3.2 and 3.6 reduce to the corresponding results in 1. 4. Cone Characterizations of Approximate Solutions: Nonconvex Case In this section, fS is no longer assumed to be convex. In nonconvex case, the corresponding local concepts are defined as follows. Definition 4.1. Let q ∈ D \{0} be a fixed element and  ≥ 0. i x ∈ S is said to be a local weakly q-efficient solution of problem 2.1,ifthereexists a neighborhood V of x such that fS ∩ V  − fxq ∩ − int D∅ in this case, it is assumed that D is solid. ii x ∈ S is said to be a local q-efficient solution of problem 2.1,ifthereexistsa neighborhood V of x such that fS ∩ V  − fxq ∩ −D \{0}∅. iii x ∈ S is said to be a local properly q-efficient solution of problem 2.1,ifthere exists a neighborhood V of x such that cl conefS∩V qD−fx∩−D{0}. The sets of local q-efficient solutions, local weakly q-efficient solutions and local properly q-efficient solutions of problem 2.1 are denoted by LAEf, S,q,LWAEf, S, q and LPAEf, S, q, respectively. If   0, then, i, ii,andiii reduce to the definitions of local weakly effi cient solution, local efficient solution and local properly efficient solution, respectively, and the sets of local weakly, properly efficient solutions of problem 2.1 are denoted by LEf, SLWEf, S,LPEf, S, respectively. Journal of Inequalities and Applications 7 Definition 4.2 see 4, 5.LetZ ⊂ R m and y, v ∈ R m . i The second-order tangent set to Z at y, v is defined as T 2  Z, y, v    d ∈ R m : ∃t n ↓ 0, ∃d n −→ d such that y n  y  t n v  1 2 t 2 n d n ∈ Z, ∀n ∈ N  . 4.1 ii The asymptotic second-order tangent cone to Z at y, v is defined as T   Z, y, v    d ∈ R m : ∃  t n ,r n  ↓  0, 0  , ∃d n −→ d such that t n r n −→ 0,y n  x  t n v  1 2 t n r n d n ∈ Z, ∀n ∈ N  . 4.2 In 4–9, some properties of second-order tangent sets have been derived, see the following Lemma. Lemma 4.3. Let y ∈ cl Z and v ∈ R m ,then, i T 2 Z, y, v and T  Z, y, v are closed sets contained in cl cone cone Z − y − v,and T  Z, y, v is a cone. ii If v / ∈ Ty, Z,thenT 2 Z, y, vT  Z, y, v∅.Ifv ∈ Ty, Z,thenT 2 Z, y, v ∪ T  Z, y, v /  ∅.Ify ∈ int Z,thenT 2 Z, y, vT  Z, y, vR m ,andT 2 Z, y, 0 T  Z, y, 0Ty, Z. iii Let Z is convex. If v ∈ Ty, Z and T  Z, y, v /  ∅,thenT 2 Z, y, v ⊂ T  Z, y, v cl cone coneZ − y − vTv, TZ, y. Definition 4 .4 see 27.LetK ⊂ R n and φ : K → R be a nonsmooth function. The Hadamard upper directional derivative and the Hadamard lower directional derivative derivative of φ at x ∈ K in the direction d ∈ R n are given by φ    x, d   lim t↓0 sup h → d φ  x  th  − φ  x  t , φ  −  x, d   lim t↓0 inf h → d φ  x  th  − φ  x  t . 4.3 Lemma 4.5 see 7. Let Y be a finite-dimensional space and y 0 ∈ E ⊂ Y . If the sequence y n ∈ E \{y 0 } converges to y 0 , then there exists a subsequence (denoted the same) y n such that y n − y 0 /t n converges to some nonnull vector u ∈ Ty 0 ,E,wheret n  y n − y 0 , and either y n − y 0 − t n u/1/2t 2 n converges to some vector z ∈ T 2 E, y 0 ,u∩ u ⊥ or there exists a sequence r n → 0  such that t n /r n → 0 and y n − y 0 − t n u/1/2t n r n converges to some vector z ∈ T  E, y 0 ,u ∩ u ⊥ \{0}, where u ⊥ denotes the orthogonal subspace to u. In the following theorem, we derive several properties of local weakly, properly approximate efficient solutions. 8 Journal of Inequalities and Applications Theorem 4.6. i Let int D /  ∅, then, for any fixed q ∈ D \{0}, LWE  f, S  ⊂  >0 LWAE  f, S, q  . 4.4 Conversely, if x ∈ S, and there exists a neighborhood V of x such that fS ∩ V  − fx ∩ −q − int D∅, for all >0,thatis, x ∈ WAEf, S ∩ V,q, for all >0,thenx ∈ LWEf, S. ii For any fixed q ∈ D \{0},LEf, S ⊂  >0 LAEf, S, q. Conversely, if x ∈ S and there exists a neighborhood V of x such that for any fixed q ∈ D \{0} and >0, fS ∩ V  − fx ∩ −q − D \{0}∅,then x ∈ LEf, S. iii For any fixed q ∈ D \{0},LPEf, S ⊂  >0 LPAEf, S,q. Conversely, if x ∈ S and there exists a neighborhood V of x such that for any fixed q ∈ D \{0 } and >0, conefS ∩ V  − fxq  D is a closed set, and cl conefS ∩ V  − fxq  D ∩ −D{0},then x ∈ LPEf, S. Proof. i Let x ∈ LWEf, S, then, there exists a neighborhood V 1 of x such that fS ∩ V 1  − f x ∩ − int D∅.Fromq ∈ D \{0 },wehave f  S ∩ V 1  − f  x  ∩  −q − int D   ∅, ∀>0 . 4.5 Which implies x ∈  >0 LWAEf,S, q. Conversely, we assume that there exists a neighborhood V of x such that x ∈ WAEf,S ∩ V,q,forall>0. Suppose, on the contrary, that x / ∈ LWEf, S, then, for any neighborhood V of xfS ∩ V  − fx ∩ − int D /  ∅.TakeV  V , then, there exist p ∈ int D and x ∈ S ∩ V such that fx − fx−p. Therefore, if >0issufficiently small, we have fx − f x −p  −q − p − q ∈−q − int D, which is a contradiction to x ∈ WAEf, S ∩ V,q,forall>0. This completes the proof. ii It is easy to see that LEf, S ⊂  >0 LAEf, S, q. Conversely, we assume that there exists a neighborhood V of x such that for any fixed q ∈ D \{0} and >0, fS ∩ V  − fx ∩ −q − D \{0}∅.Suppose,onthecontrary,that x / ∈ LEf, S, then, for any neighborhood V of x,wehavefS ∩V −fx ∩−D \{0} /  ∅.Take V  V , then, there exist p ∈ D \{0} and x ∈ S ∩ V such that fx − fx−p.Takeq  p/2 and   1, then, fx − f x−p  −q − p/2 ∈−q − D \{0}, which is a contradiction to the assumption. This completes the proof. iii It is easy to see that LPEf, S ⊂  >0 LPAEf,S, q. Conversely, we assume that there exists a neighborhood V of x such that for any fixed q ∈ D \{0} and >0, conefS ∩ V  − fxq  D is a closed set, and cl cone fS ∩ V  − fxq  D ∩ −D{0}. Suppose, on the contrary, that x / ∈ LPEf,S, then, for any neighborhood V of x,wehaveclconefS ∩ V − fxD ∩ −D \{0} /  ∅.TakeV  V , then, there exist λ>0, p 1 ∈ D \{0}, p 2 ∈ D and x ∈ S ∩ V such that λfx − fxp 2 −p 1 .Take q  p 1 /2λ and   1, similar to the proof of ii we can complete the proof. Journal of Inequalities and Applications 9 Theorem 4.7. Let f be a continuous function on S, x ∈ S,and>0. i If Tf x,fS ∩ −q − D∅,thenx ∈ LAEf, S, q. ii If Tf x,fS ∩ −q − D /  ∅,andforeachv ∈ Tfx,fS ∩ −q − D T 2  f  S  ,f  x  ,v  ∩ v ⊥ ∩  −cl cone  D  q  v   ∅, T   f  S  ,f  x  ,v  ∩ v ⊥ ∩  −cl cone  D  q  v   { 0 } , 4.6 then x ∈ LAEf, S, q. Proof. i Let Tf x,fS ∩ −q − D∅. Suppose, on the contrary, that x / ∈ LAEf, S, q, then, there exists x n ∈ S and x n → x such that fx n  − fxq ∈−D \{0},foralln ∈ N. Since f is a continuous function and D is a pointed cone, fx n  /  fx,foralln ∈ N and fx n  → fx. Therefore, fx n  − fx/fx n  − fx→d ∈ Tfx,fS. On the other hand, for any n ∈ N,wehave f  x n  − f  x    f  x n  − f  x    ∈− 1   f  x n  − f  x     q  D \ { 0 }  ⊂−  q  D \ { 0 }   1   f  x n  − f  x    − 1  q  . 4.7 Since fx n  → fx and q ∈ D \{0},thereexistsn 1 ∈ N such that  1   f  x n  − f  x    − 1  q ∈ D, ∀n ≥ n 1 . 4.8 Hence, d ∈−q  D, which is a contradiction to the assumption. This completes the proof. ii Suppose, on the contrary, that x / ∈ LAEf, S, q. Similar to the proof of i,wehave there exists x n → x such that f  x n  − f  x    f  x n  − f  x    −→ d ∈ T  f  x ,f  S   ∩  −q − D  . 4.9 Let t n  fx n  − fx and z n 2/t n fx n  − fx/t n − d,foralln ∈ N. Similar to the proof of Lemma 4.3,wehavethereexistsz ∈ R m such that z ∈ T 2 fS,fx,d ∩ d ⊥ ∩ −cl coneq  D  d or z ∈ T  fS,fx,d ∩ d ⊥ \{0}∩−cl cone q  D  d,whichisa contradiction to the assumptions. This completes the proof. 10 Journal of Inequalities and Applications Corollary 4.8. Let f be a continuous function on S, x ∈ S and   0. i If Tf x,fS ∩ −D{0},thenx is a local efficient solution of problem 2.1. ii If Tf x,fS ∩ −D \{0} /  ∅,andforeachv ∈ Tfx,fS ∩ −D \{0} T 2  f  S  ,f  x  ,v  ∩ v ⊥ ∩  −cl cone  D  v   ∅, T   f  S  ,f  x  ,v  ∩ v ⊥ ∩  −cl cone  D  v   { 0 } , 4.10 then x is a local efficient solution of problem 2.1. Proof. The proof is similar to Theorem 4 .7. Remark 4.9. If fS is convex, then the condition ii of Theorem 4.7 is equivalent to the following condition ii  Tfx,fS ∩ −q − D /  ∅,andforeachv ∈ Tfx,fS ∩ −q − D 0 / ∈ T 2  f  S  ,f  x ,v  ,T   f  S  ,f  x ,v  ∩ v ⊥ ∩  −cl cone  D  q  v   { 0 } , 4.11 since T 2 fS,fx,v ⊂ T  fS,fx,v by Lemma 4.3iii. Theorem 4.10. Let f be continuous on S, x ∈ S,and ≥ 0. i Assume that D has a compact base B, p  αb for b ∈ B and α>0, and there exists δ>0 such that fS − f x ∩ δU ⊂ Tfx,fS.IfTfx,fS ∩ −q − D \{0}∅, then x ∈ LPAEf, S, q. ii Assume that Tf x,fS ∩ −q − D \{0} /  ∅, and there exists β>0 such that for each d ∈ Tf x,fS \{0} ∩ −q − D  βU the following conditions hold T 2  f  S  ,f  x  ,d  ∩ d ⊥ ∩  −cl cone  D  q  βU  d   ∅, T   f  S  ,f  x  ,d  ∩ d ⊥ ∩  −cl cone  D  q  βU  d   { 0 } , 4.12 then x ∈ LPAEf, S, q,where,U denotes the closed unit ball of R m . Proof. i Let Tf x,fS ∩ −q − D \{0}∅, then, Tfx,fS ∩ −λb − B∅,forall λ>0. The assumptions and the separation result 28,page9 implies that for any λ>0there exists a neighborhood V λ of 0 such that T  f  x  ,f  S   ∩  −λb − B  V λ   ∅. 4.13 Suppose, on the contrary, that x / ∈ LPAEf, S, q, then, for any neighborhood V of 0, we have cl cone  f  S ∩  x  V n  − f  x   q  D  ∩  −D \ { 0 } /  ∅. 4.14 [...]... “Second-order optimality conditions in multiobjective optimization problems,” Journal of Optimization Theory and Applications, vol 102, no 1, pp 37–50, 1999 5 A Cambini, L Martein, and M Vlach, “Second-order tangent sets and optimaity conditions, ” Tech Rep., Japan Advanced Studies of Science and Technology, Hokuriku, Japan, 1997 6 J.-P Penot, “Second-order conditions for optimization problems with constraints,”... new approach to approximation of solutions in vector optimization problems,” in Journal of Inequalities and Applications 24 25 26 27 28 17 Proceedings of APORS, M Fushimi and K Tone, Eds., vol 1995, pp 497–504, World Scientific, Singapore, 1994 Y Sawaragi, H Nakayama, and T Tanino, Theory of Multiobjective Optimization, vol 176 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla,... 85–106, 2009 9 G Bigi, “On sufficient second order optimality conditions in multiobjective optimization, ” Mathematical Methods of Operations Research, vol 63, no 1, pp 77–85, 2006 10 S S Kutateladze, “Convex ε-programming,” Soviet Mathematics Doklady, vol 20, pp 390–1393, 1979 11 I V´ lyi, Approximate saddle-point theorems in vector optimization, ” Journal of Optimization Theory a and Applications, vol 55,... “ε-properly efficient solutions to nondifferentiable multiobjective programming problems,” Applied Mathematics, vol 12, no 6, pp 109–113, 1999 13 S Bolintin´ anu, “Vector variational principles; ε-efficiency and scalar stationarity,” Journal of Convex e Analysis, vol 8, no 1, pp 71–85, 2001 14 J Dutta and V Vetrivel, “On approximate minima in vector optimization, ” Numerical Functional Analysis and Optimization, ... 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There are several important reasons for considering this kind of solutions. . Approximate Solutions in Multiobjective Optimization Problems Ying Gao, 1 Xinmin Yang, 1 and Heung Wing Joseph Lee 2 1 Department of Mathematics, Chongqing Normal University, Chongqing 400047, China 2 Department. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 620928, 17 pages doi:10.1155/2010/620928 Research Article Optimality Conditions for Approximate