Bổ đề farkas tiệm cận cho hệ lồi

Untitled Science & Technology Development, Vol 19, No T6 2016 Trang 160 Asymptotic Farkas lemmas for convex systems  Nguyen Dinh University International, VNU – HCM  Tran Hong Mo Tien Giang Universi[.]

Science & Technology Development, Vol 19, No.T6-2016 Asymptotic Farkas lemmas for convex systems  Nguyen Dinh University International, VNU – HCM  Tran Hong Mo Tien Giang University, Tien Giang (Received on June 5th 2015, accepted on November 21th 2016) ASTRACT In this paper we establish characterizations of the containment of the set {x  X : x C, g(x)K}  {x  X : f (x)  0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g : X  Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function Here, no constraint qualification condition or qualification condition are assumed The characterizations are often called asymptotic Farkas-type results The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function It is also shown that under some closedness conditions, these asymptotic Farkas lemmas go back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature The results can be used to study the optimization Keywords: Farkas lemma, sequential Farkas lemma, limit inferior, limit superior INTRODUCTION AND PRELIMINARIES Farkas-type results have been used as one of the main tools in the theory of optimization [8] Typical Farkas lemma for cone-convex systems characterizes the containment of the set where is a closed convex subset of a locally convex Hausdorff topological vector space (brieftly, lcHtvs), is a closed convex cone in another lcHtvs and is a convex mapping, in a reverse convex set, define by the proper, lower semi-continuous, convex function If the characterization holds under some constraint qualification condition or qualification condition then it is called non-asymptotic Farkas-type result (see [6], [10-12]) Otherwise (i.e., without any qualification condition), such characterizations often hold in the limit forms and they are called asymptotic Farkas-type results (see [7, 5, 9, 13] and references therein) In this paper, we mainly established several forms of asymptotic Farkas-type Trang 160 results for convex systems in the two means: systems convex with respect to a convex cone (called -convex systems) and systems convex w.r.t an extended sublinear function S (called S -convex systems) The results can be used to establish the optimality conditions and dulaity for optimization problems where constraint qualification conditions failed, such as classes of semidefinite programs, or scalarized multi-objective programs, scalarized vector optimization problems We shoned also that under some closedness conditions, these asymptotic Farkas lemmas came back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature Let X and Y be lcHtvs, with their topological dual spaces X  and Y  , respectively The only topology we consider on X , Y * is the weak* topology For a set A  X  , the closure of A w.r.t TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 19, SỐ T6- 2016 the weak * -topology is denoted by cl A The indicator function of A is denoted by iA , i.e., x  X \ A Let iA  x = if x  A, iA  x =  if ̅ 𝑓: 𝑋 → ℝ ∪ {±∞} The effective domain of f is the set dom f := x  X : f  x <  The function f is proper if dom f   The set of all proper, lower semi-continuous (lsc in short) and convex functions on X will be denoted by   X  The epigraph of f is ̅ : 𝑓(𝑥) ≤ 𝛼} epi 𝑓 ≔ {(𝑥, 𝛼)𝜖𝑋 × ℝ The Legendre-Fenchel conjugate of f is the ∗ ̅ ≔ℝ ̅ ∪ {±∞} defined by function 𝑓 : 𝑋 ∗ → ℝ   f   x  = sup x, x  f  x , x  X  xX It is clear that for any x  X and x  X , the Young-Fenchel inequality always holds: f *(x*)  x*, x  f (x ) * * Moreover, for any 𝛽 ∈ ℝ one ( f   )*(x*) = f *(x*)   for all * x  X * has Now let K be a closed convex cone in Y and let K be the partial order on Y generated by K , i.e., y1 K y2 if and only if y2  y1  K We add to Y a greatest element with respect to K , denoted by K , which does not belong to Y , and let Y  = Y  {K}  Then one has y K K forevery y  Y We assume by convention: y  K = K  y = K , for  all y  Y , and  K = K if   The dual cone of K , denoted by K , is defined by K  := {y  Y  : y, y  0,  y  K} A mapping h : X  Y  is called K -convex if x1, x2  X , 1, 2 > 0, 1  2 =  h(1x1  2x2) K 1h(x1)  2h(x2), where " K " is the binary relation (generated by K ) extended to Y  by setting y K K forall y  Y  The domain of h , denoted by dom h, is defined to be the set dom h := {x  X : h(x)  Y} The of is h K -epigraph epiK h : {(x, y)  X  Y : y  h(x)  K} the set space Then, h1(K ) is closed (see [6]) It is worth observing that if h is K -convex, then h1(K ) is convex Moreover, for any y  Y  and  g : X  Y , we define the composite function 𝑦 ∗ 𝑜 g ∶ 𝑋 → ℝ ∪ {+∞} as follows  y, g(x) , if x  dom g,  y o g  (x)   else , The function 𝑆: 𝑌 → ℝ ∪ {+∞} (extended) sublinear if S( y  y)  S( y)  S( y), is called and S( y) =  S( y), y, y  Y,  > By convention, we set S(0Y ) = (this convention is appropriate to the assumption that S is lsc) Such a function S can be extended to Y  by setting S(K ) =  An extended sublinear function 𝑆: 𝑌 → ℝ ∪ {+∞} allows us to introduce in Y  a binary relation which is reflexive and transitive: y1 S y2 if y1 K y2 , where K  y  Y : S( y)  0 and y S K for all y  Y  We consider also the extension of S as 𝑆: 𝑌  → ℝ ∪ {+∞} By setting S (K ) =  Given an extended sublinear function 𝑆: 𝑌 → ℝ ∪ {+∞} , we adapt the notion S  convex ( i.e., convex with respect to a sublinear function) in [6] which generalized the one in [16] It is clear that h is K -convex if and only if epiK h is convex In addition, h : X  Y  is said to be K -epi closed if epiK h is a closed set in the product A mapping h : X  Y  is said to be if for all S  convex x1, x2  X , 1, 2 > 0, 1  2 = , one has h(1x1  2x2) S 1h(x1)  2h(x2) It is worth observing that, as mentioned in [15, Remark 1.10], " S  convex means different things under different circumstances" such as, when 𝑌 = ℝ, if S( y) :=| y | , S( y) := y , S( y) : = y, or S( y) = , respectively, then " S -convex" means Trang 161 Science & Technology Development, Vol 19, No.T6-2016 "affine", "convex", "concave" or "arbitrary", respectively Moreover, the equalities hold whenever one of the nets is convergent It is clear that if h is S -convex then h is K convex with K := {y  Y : S( y)  0} Conversely, if h is K -convex with some convex cone K then h is S  convex with S = i (see [6]) K Definition 1.1 [2, p.5] [1, p.32], [14, p.217] Let (ai)iI be a net of extended real numbers defined on a directed set ( Ι, ≫) e define limit inferior of the net (ai)iI as follows liminf : liminf a j = sup inf a j iI iI iI j? i j? i Similarly, limit superior of the net (ai)iI is defined by limsup : limsup a j  inf sup a j iI iI iI j? j? i We say that (ai)iI converges to 𝑎 ∈ ℝ denoted by lim  a or  a, if for any  > 0, there iI exists i0  I such that |  a |<  for all 𝑖 ≫ 𝑖0 The following properties were given in [2, p.9] and [14, p.221] Lemma 1.1 Let (ai)iI and (bi)iI be nets of extended real numbers Then the following statements hold: (i) limsup(ai)  liminf and limsup  liminf iI iI iI iI (ii) lim  a  ¡ if and only if iI liminf  limsup  a iI iI (iii) If  bi for all i  I , then liminf  liminf bi and limsup  limsup bi iI iI iI iI liminf (ai  bi)  liminf  liminf bi, iI iI iI and limsup (ai  bi)  limsup  limsupbi, iI iI iI provide that the right side of the inequalities are defined Trang 162 Approximate systems Farkas lemma for cone-convex In this section we will establish one of the main result of this paper: the asymptotic version of Farkas lemma for convex systems, which holds without any qualification condition Let X , Y lcHtvs, K be a closed convex cone in Y , C be a nonempty closed convex subset of X and 𝑓: 𝑋 → ℝ ∪ {+∞} be a proper lsc and convex function Let further g : X  Y  be a K -convex and K -epi closed mapping Let and assume that A := C  g 1(K ) (dom f )  A =  Theorem 2.1 [Asymptotic Farkas lemma 1] The following statements are equivalent: (i) x  C, g(x)  K  f (x)  , (ii) there exist ∗ ∗ (𝑥𝑙𝑖∗ , 𝑥2𝑖 , 𝑥3𝑖 , 𝜀𝑖 ) 𝑖∈𝛪 nets ∗ ∗ ( yi*)iI  K  and ⊂ 𝑋 × 𝑋 × 𝑋 × ℝ such such that i  f *(x1*i)  ( yi* o g)*(x2*i)  iC* (x3*i), i  I and (x1*i  x2*i  x3*i, i)  (0 *, 0), ∗ X (iii) there exist nets ( yi*)iI  K  and (𝑥𝑖∗ , 𝜀𝑖, )𝑖∈𝛪 ⊂ 𝑋 ∗ × ℝ such that i  ( f  yi* o g  iC)*(xi*), i  I and (xi*, i)  (0 *, 0), X (iv) there exists a net ( yi*)iI  K  such that f (x)  liminf( yi o g)(x)  0, x  C iI Proof [(i)  (ii)] Assume that (i) holds Observe firstly that A is closed and convex Secondly, (i) is equivalent to  ( f  iA)*(0 *), X or equivalently, (0 *, 0)  epi( f  iA)* X Since we also have [4, p 328]   epi( f  iA)* = cl  epi f *  U epi( g)*  epiiC*  ,   K  and so, (i) is equivalent to   (0 *, 0)  cl  epi f *  U epi( g)*  epiiC*  , X   K  and the equivalence between (i) and (ii) follows TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 19, SỐ T6- 2016 [(ii)  (iii)] Assume that (ii) holds, i.e., there exist nets ( yi*)iI  K  and We D := epi f *  ∗ ∗ (𝑥𝑙𝑖∗ , 𝑥2𝑖 , 𝑥3𝑖 , 𝜀𝑖 ) 𝑖∈𝛪 ⊂ 𝑋 × 𝑋 × 𝑋 ∗ × ℝ such * * that i  f (x1i)  ( yi* o g)*(x2*i)  iC* (x3*i), i  I , (2.1) F := ∗ ∗ and (x1*i  x2*i  x3*i, i)  (0 *, 0) X now U epi( y o g)  epi i * * * C yK  set and U epi( f  y o g  i ) * * C yK  (2.2) From the proof of Theorem 2.1, we get By the definition of the conjugate function, (2.1) Corollary 2.1 [Farkas lemma for cone-convex implies that * * * * systems] Consider the following conditions: i  x1i  x2i  x3i, x  ( f  yi o g  iC)(x), x  X , i  I (2.3) (0 *,0)  clD  (0 *,0)  D, X X Set xi* := x1*i  x2*i  x3*i for all i  I Then the above (2.4) (0 *,0)  clF  (0 *,0)  F, inequality gives rise to X i  ( f  yi* o g  iC)*(xi*), i  I and (2.2) becomes (xi*, i)  (0 *, 0) X [(iii)  (iv)] Assume that (iii) holds, i.e., there exist nets ( yi*)iI  K  and (𝑥𝑖∗ , 𝜀𝑖, )𝑖∈𝛪 ⊂ 𝑋 ∗ × ℝ such that i  ( f  yi* o g  iC)*(xi*), i  I and (xi*, i)  (0 *, 0) X Again by the definition of the conjugate function, one has i  xi*, x  ( f  yi* o g  iC)(x), x  X , i  I , or equivalently, f (x)  ( yi* o g)(x)  xi*, x  i, x  C, i  I , (which still holds even in case x  dom f and x  dom g ) Taking liminf in both sides of the last inequality, we get (iv) [(iv)  (i)] Assume that (iv) holds, i.e., there exists * i iI a net ( y )   K such that f (x)  liminf ( yi* o g)(x)  0, x  C Observe that if x  C such that g(x)  K, then Thus, * i The proof is complete Remark 2.1 The equivalence [(i)  (iv)] was established in [5] involved the space Y (instead of Y  ), under the assumption that y* o g   ( X ) for   (vi) there exists y*  K  such that f (x)  ( y* o g)(x)  0, x  C Then one has: (a) (2.3) is equivalent to [(i)  (v)], (b) (2.4) is equivalent to [(i)  (vi)] Proof As in the proof of Theorem 2.1, one has (i) is equivalent to (0 *,0)  clD X Moreover, it is easy to check that (v) is equivalent to (0 *,0) D Thus we get (a) X As shown in the proof of Theorem 2.1, (i) is equivalent to (0 *, 0)  epi( f  iA)* X As epi( f  iA)* = clF (see [3, Theorem 8.2]) we have (i) is equivalent to (0 *,0)  clF Moreover, it is clear that (vi) is equivalent to (0 *,0) F X Therefore, one also gets (b) The proof is complete for f (x)  f (x)  liminf ( y o g)(x)  iI (v) there exist y*  K , x1*  X * and x2*  X * such that f *(x1*)  iC* (x2*)  ( y* o g)*(x1*  x2*)  0, X iI for all ( yi* o g)(x)  i  I x  C such that g(x)  K, one gets X and the following statements: (i) x  C, g(x)  K  f (x)  , all y  K , which is much stronger the S -epi closedness of g used in Theorem 2.1 Corollary 2.1 [Farkas lemma for cone-convex systems] Consider the following conditions: (2.3) (0 *,0)  clD  (0 *,0)  D, X (0 *,0)  clF X X  (0 *,0)  F, X (2.4) and the following statements: (i) x  C, g(x)  K  f (x)  , (v) there exist y*  K , x1*  X * and x2*  X * such that f *(x1*)  iC* (x2*)  ( y* o g)*(x1*  x2*)  0, Trang 163 Science & Technology Development, Vol 19, No.T6-2016 (vi) there exists y*  K  such that f (x)  ( y o g)(x)  0, x  C Then one has: (a) (2.3) is equivalent to [(i)  (v)], (b) (2.4) is equivalent to [(i)  (vi)] * Proof As in the proof of Theorem 2.1, one has (i) is equivalent to (0 *,0)  clD X Moreover, it is easy to check that (v) is equivalent to (0 *, 0) D Thus we get (a) X As shown in the proof of Theorem 2.1, (i) is equivalent to (0 *, 0)  epi( f  iA)* X As epi( f  iA)* = clF (see [3, Theorem 8.2]) we have (i) is equivalent to (0 *, 0)  clF Moreover, it X is clear that (vi) is equivalent to (0 *,0) F X Therefore, one also gets (b) The proof is complete Corollary 2.2 [Stable Farkas lemma for coneconvex systems] Consider the following conditions: epi f   U epi( y* o g)*  epi iC is weak* - closed yK  in 𝑋 ∗ ∈ ℝ (2.5) U epi( f  y* o g  iC)* is weak* - closed in X *  ¡ yK  (2.6) Then we have (c) (2.5) holds if and only if for any x  X  and any 𝛽 ∈ ℝ, (x  C, g(x)  K  f (x)  x , x   ) c *  * * (y  K , x1  X and x2*  X * such that f *(x1*)  iC* (x2*)  ( y* o g)*(x*  x1*  x2*)   )  (d) (2.6) holds if and only if for any x  X  and any 𝛽 ∈ ℝ, (x  C, g(x)  K  f (x)  x, x   ) c *   y  K : f (x)  x , x  ( y* o g)(x)   , x  C Proof The proof is similar to that of Theorem 2.1 Remark 2.2 It is worth noting that (d) was given in [6] Moreover, if we replace (2.5) by epi( f  iA)* = epi f *  U epi( y* o g)*  epi iC* , and yK  Trang 164 (2.6) by epi( f  iA)* = U epi( f  y o g  i ) * * C y K  where  A := C  g 1(K), then the conclusion of Corollary 2.2 still holds, and the assumptions on the closedness and the convexity of C , f , and g can be removed Asymptotic Farkas lemma for sublinear-convex systems Let X , Y be lcHtvs, C be a nonempty closed convex subset of X , 𝑆: 𝑌 → ℝ ∪ {+∞} be an lsc sublinear function and g : X  Y  be an S -convex mapping such that the set {(𝑥, 𝑦, 𝜆) ∈ 𝑋 × 𝑌 × ℝ ∶ 𝑆(𝑔(𝑥) − 𝑦) ≤ 𝜆} (3.1) is closed in the product space X × Y × ℝ Let us consider 𝑓: 𝑋 → ℝ ∪ {+∞} and 𝜓: ℝ → ℝ ∪ {+∞} be proper convex lsc functions We now establish an asymptotic Farkas lemma for systems that are convex w.r.t the sublinear function 𝑆: 𝑌 → ℝ ∪ {+∞} Theorem 3.1 [Asymptotic Farkas lemma 2] Assume that the following condition holds: (dom f ) {x  C :   dom s t (S o g)(x)  }   (3.2) Then the following statements are equivalent: (a) x  C,   ¡ , (S o g)(x)    f (x)  ( )  ∗ (b) there exist nets (𝑦𝑖∗ , 𝛾𝑖 ) 𝑖∈𝛪 ⊂ 𝑌 × ℝ+ and ∗ ∗ ∗ ∗ (𝑥𝑙𝑖∗ , 𝑥2𝑖 , 𝑥3𝑖 , 𝜀𝑖 ) 𝑖∈𝛪 ⊂ 𝑋 × 𝑋 × 𝑋 ∗ × ℝ × ℝ such with yi*   iS on Y for all i  I such that i  f *(x1*i)  ( yi* o g)*(x2*i)  iC* (x3*i)  *(i   i), i  I and (x1*i  x2*i  x3*i,i, i)  (0 *, 0, 0), X ∗ (c) there exist nets (𝑦𝑖∗ , 𝛾𝑖 ) 𝑖∈𝛪 ⊂ 𝑌 × ℝ+ and ∗ (𝑥𝑙𝑖∗ , 𝜂𝑖 , 𝜀𝑖 ) 𝑖∈𝛪 ⊂ 𝑋 × ℝ × ℝ with yi*   iS on Y for all i  I such that i  ( f  yi* o g  iC)*(xi*)  *(i   i), i  I (3.3) and (xi*,i, i)  (0 *, 0, 0) X (3.4) ̃ Proof Let us set 𝑌̃ = 𝑌 × ℝ, 𝑌 = 𝑌  × ℝ, 𝐶̃ = 𝐶 × ℝ and set 𝑆̃ : 𝑌̃ → ℝ ∪ {+∞} defined by S°( y, ) = S( y)   for all (𝛾, 𝛼) ∈ 𝑌̃ Then 𝐶̃ is nonempty closed convex subset of 𝑋 × ℝ, 𝑆̃ is an ̃ =𝑋×ℝ lsc sublinear function Let also 𝑋: TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 19, SỐ T6- 2016 𝑔̃ = 𝑋̃ × ̃𝑌  and 𝑓̃: 𝑋̃ → ℝ ∪ {+∞} be mappings defined by 𝑔̃(𝑥, 𝛼): = (𝑔(𝑥), 𝛼), ∀∈ 𝑋̃ 𝑎𝑛𝑑 𝑓̃(𝑥, 𝛼) = 𝑓(𝑥) + 𝜓(𝑥), ∀(𝑥, 𝛼) ∈ 𝑋̃ [(a)  (b)] Assume that (a) holds Since f ,  are proper lsc, convex functions, so is °f Moreover, g° ° be the is S° -convex as g is S -convex Now let K closed convex cone defined by ° ° ° ° ° K := {( y, )  Y : S( y, )  0} Then, g is K convex as well The assumption (3.1) ensures that ° -epi closed while (3.2) guarantees ° is K g °g ° (K °) =  (dom ° f ) C 1 have °, g °(x,  ) = (g(x), )  K °  (x,  )  C ° f (x, )  0, which shows that (i) in Theorem 2.1 holds, and * ° hence, there exist nets (% and yi )iI  K ∗ ∗ ̃ ̃ ∗ ∗ ∗ ∗ ̃𝑙𝑖 , 𝑥̃ ̃ ̃ (𝑥 2𝑖 , 𝑥3𝑖 , 𝜀𝑖 ) 𝑖∈𝛪 ⊂ 𝑋 × 𝑋 × 𝑋 × ℝ such that %° * % °* (x% i  ° f (x% 1i )  ( yi o g) (x2i )  iC 3i), i  I and * * * % % (x% 1i  x2i  x3i, i )  (0 *, 0) X * * * * (3.5) (3.6) ° there exist such Since X = X*  ¡ , * * * * * * (x1i, x2i, x3i, i, i, i)iI  X  X  X  ¡  ¡  ¡ * * * % such that (x% 1i )iI = (x1i, i)iI , (x2i )iI = (x2i, i)iI and * * * (x% 3i)iI = (x3i, i )iI This and (3.6) imply that x1*i  x2*i  x3*i  * and i  i  i  * X (3.7) ° , by Lemma 3.5 in Moreover, since (% yi )iI  K * (3.8) * (%° yi o g)*(x2*i, i) = sup xX , ¡  ∗ [6], there exists a net (𝑦𝑖∗ , 𝛾𝑖 ) 𝑖∈𝛪 ⊂ 𝑌 × ℝ such * % that and yi = ( yi*,  i),  i  yi*   i S on Y for all i  I By the definition of the conjugate function, for any i  I , one has ([17, p.76]) x , x    ( y o g)(x)    * 2i * i i i = sup  x2*i, x  ( yi* o g)(x)  sup (i   i)  ¡ xX ( y o g)(x2*i) if i =  i, = i otherwise,  (3.9) and i* (x3*i,i) = sup ° C x , x    i (x) = sup x , x  i (x)  sup  * 3i xX , ¡ xX °, We now try to apply Theorem 2.1 with ° X , Y° , C ° playing the roles of X , Y , C , g, g°, °f , and K f , and K , respectively ° , we °, K From (a) and the definition of °f , g° , C * * ° f (x1*i, i) = f *(x1*i)  *(i), i * 3i C C  ¡ i i (x ) if i = 0, = otherwise  * C * 3i (3.10) Combining (3.5), (3.8), (3.9), and (3.10) we get (note that i  ¡ for all i  I ): i  f *(x1*i)  ( yi* o g)*(x2*i)  iC* (x3*i)  *(i) and i =  i, i = for all i  I , which together with (3.7) gives i  i  i = i   i  Set i := i   i for all i  I Then i  and the last inequality becomes i  f *(x1*i)  ( yi* o g)*(x2*i)  iC* (x3*i)  *(i   i), i  I Thus (b) is satisfied [(b)  (c)] The same as the proof of [(ii)  (iii)] in Theorem 2.1 [(c)  (a)] Assume that (c) holds, i.e., there exist nets ∗ (𝑦𝑖∗ , 𝛾𝑖 ) 𝑖∈𝛪 ⊂ 𝑌 × ℝ ∗ and (𝑥𝑖∗ , 𝜂𝑖 , 𝜀𝑖 ) 𝑖∈𝛪 ⊂ 𝑋 × ℝ × ℝ with yi*   iS on Y for all i  I such that (3.3) and (3.4) hold It follows from (3.3) that i  xi*, x  ( f  yi* o g)(x)  (i   i)  ( ), (x,  )  C  ¡ , i  I , or equivalently, f (x)  ( )  ( yi* o g)(x)  xi*, x  i   i   i , (x,  )  C  ¡ , i  I (3.11) (which still holds even in case x  dom f , x  dom g and   dom ) Trang 165 Science & Technology Development, Vol 19, No.T6-2016 Since yi*   iS on Y for all i  I , if x  C and such that then (S o g)(x)  ,  ¡ ( yi* o g)(x)   i(S o g)(x)   i for all i  I (note that  i  for all i  I ) So, for any x  C and   ¡ with (S o g)(x)  , (3.11) gives f (x)  ( )   i  xi*, x  i   i   i , i  I , which means that if x  C and   ¡ with (S o g)(x)  , one has f (x)  ( )  xi*, x  i   i , i  I iI Passing to the limit both sides of the last inequality and taking the fact that (3.4) into account, we get (a) The proof is complete Theorem 3.2 [Asymptotic Farkas lemma 3] Assume that (3.2) holds Then the following statements are equivalent: (a) x  C,   ¡ , (S o g)(x)    f (x)  ()  , (d) there exists a net ( yi*,  i,i)iI  Y *  ¡   ¡ with yi*   iS on Y for all i  I such that i  0, (i   i)iI  dom * and iI (3.12) Proof [(a)  (d)] Assume that (a) holds It follows from Theorem 3.1 that (c) holds i.e., there exist and nets ( yi*,  i)iI  Y *  ¡  (xi*,i, i)iI  X *  ¡  ¡ with yi*   iS on Y for all i  I and such that i  ( f  yi* o g  iC)*(xi*)  *(i   i), i  I , (3.13) [(c)  (a)] Assume that (c) holds, i.e., there nets ( yi ,  i )iI * (xi*,i ,i )iI  X *  ¡  ¡  Y * ¡ with  and yi*   i S on for all i  I such that (3.3) and (3.4) hold It follows from (3.3) that and (3.14) (xi*,i, i)  (0 *, 0, 0) X Y By the definition of the conjugate function, (3.13) This means that (d) holds [(d)  (a)] Assume that (d) holds, i.e., there exists a net ( yi*,  i,i)iI  Y *  ¡   ¡ y   iS on Y for all i  I such that i  0, (i   i)iI  dom * and (3.12) holds Then from the definition of the conjugate function and (3.12), one gets f (x)  liminf (( yi* o g)(x)  (i   i)  ())  0, x  C,   ¡ , iI which implies f (x)  ( )  liminf (( yi* o g)(x)    i )  0, i iI   i ) liminf (( yi* o g)(x)  i iI = liminf (( yi* o g)(x)   i )  liminf ( i ) iI iI = liminf (( yi* o g)(x)   i ), x  C,   ¡ iI Combining this and (3.16), one gets * f (x)  ( )  liminf (( yi o g)(x)   i )  0, x  C, iI   dom (Note that the last inequality still holds even   dom ) Hence, f (x)  ( )  liminf (( yi* o g)(x)   i )  0, x  C,   ¡ iI (3.17) On the other hand, as yi*   iS on Y for all i  I , it follows that if x  C and   ¡ such that (S o g)(x)  , then ( yi* o g)(x)   i(S o g)(x)   i, i  I , and hence, ( yi* o g)(x)   i  0, i  I Trang 166 with * i x  C,   dom (3.16) According to Lemma 1.1 (iv) and the fact that i  0, we have f (x)  liminf (( yi* o g)(x)  *(i   i))  0, x  C exist gives rise to i  xi*, x  ( f  yi* o g  iC)(x)  *(i   i), x  X , i  I (3.15) * * Moreover, (i   i)iI  dom , i.e.,  (i   i) attains finite value for all i  I So (3.15) is equivalent to f (x)  ( yi* o g)(x)  *(i   i)  xi*, x  i , x  C, i  I Taking the liminf in both sides of the last inequality (note also that (3.14) holds), we get f (x)  liminf (( yi* o g)(x)  *(i   i))  0, x  C TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 19, SỐ T6- 2016 ° as in the proof °, g So, for any x  C and   ¡ with (S o g)(x)  , ° , °f , and K Proof Set ° X , Y° , C we obtain from (3.17) of Theorem 3.1 The conclusion follows from f (x)  ( )  f (x)  ( )  liminf (( yi* o g)(x)   i )  0, Corollary 2.1 with ° ° °, g ° , °f , and K X , Y° , C iI playing the roles of X , Y , C , g , f and K , which is (a) and the proof is complete Set respectively M := {(x*, 0, r) : (x*, r)  epi f }  {(0 ,, r) : (, r)  epi } X Similar to Corollary 2.2, we get the following     result  U (x ,  , r) : (x , r)  epi( y o g)  yY ,  y  S Corollary 3.2 [Stable Farkas lemma for sublinear-convex systems] Assume that (3.2) holds Consider the following statements:  {(x*,0, r) : (x*, r)  epi iC} N := {(0 ,, r) : (, r)  epi } (d) M is weak* -closed in X *  ¡  ¡ (e) N is weak* -closed in X *  ¡  ¡  (f) For any x%= (x*,)  X   ¡ and any   ¡ , X  U {(x,  , r) : (x, r)  epi( f  y o g  iC )} yY ,  y  S Corollary 3.1 [Farkas lemma for sublinear-convex (x  C,   ¡ , (S o g)(x)    f (x)  ( )  x, x     ) systems] Assume that (3.2) holds Consider the c following conditions: (3.18) (( y*,  )  Y *  ¡ , x*  X * and x*  X * such that y*   S on Y (0 *,0,0)  clM  (0 *,0,0) M,  X X (0 *,0,0)  clN  (0 *,0,0)  N, X (3.19) X and the following statements: (a) x  C,   ¡ , (S o g)(x)    f (x)  ()  (b) there exist (y , ) Y  ¡ * *  and (x , x )  X  X with y   S on Y such that * * * * *  f *(x1*)  iC* (x2*)  ( y* o g)*(x1*  x2*)  *( ), (c) there exist ( y*,  )  Y *  ¡  with y*   S on Y such that f (x)  ( y* o g)(x)   *( ), x  C Then one gets (i) (3.18) is equivalent to [(a)  (b)], (ii) (3.19) is equivalent to [(a)  (c)] and f *(x1*)  iC* (x2*)  ( y* o g)*(x*  x1*  x2*)   *(  )   )  (g) For any x%= (x*,)  X   ¡ and any   ¡ , (x  C,   ¡ , (S o g)(x)    f (x)   ( )  x, x     ) c * * (( y ,  )  Y  ¡  such that y*   S on Y and f (x)  x, x  ( y* o g)(x)    *(  )   , x  C) Then we have [(d)  (f)] and [(e)  (g)] Remark 3.1 It is worth noting that [(e)  (g)] was given in [6] Các bổ đề Farkas dạng tiệm cận cho hệ lồi  Nguyễn Định Trường Đại học Quốc tế, ĐHQG-HCM  Trần Hồng Mơ Trường Đại học Tiền Giang TÓM TẮT Trong báo thiết lập điều kiện tương đương (gọi đặc trưng) bao hàm thức {x  X : x C, g(x) K}  {x  X : f (x)  0}, C tập lồi, đóng khơng gian lồi địa phương (kgldp) X, K nón lồi đóng Trang 167 Science & Technology Development, Vol 19, No.T6-2016 chứng minh rằng, số điều kiện quy kgldp Y, g : X  Y ánh xạ K- lồi, cịn tập thích hợp, kết đạt cho lại lồi đảo bên phải bao hàm thức xác định kết dạng Farkas dạng ổn định Farkas hàm lồi, nửa liên tục f Các đặc (stable Farkas lemmas) thiết lập nhiều trưng thiết lập mà khơng có điều tác giả năm gần đây, cho kiện quy thường gọi kết phiên định lý Các kết đạt dạng Farkas tiệm cận (hay dạng xấp xỉ) Phần được sử dụng để nghiên cứu thứ hai báo dành cho thiết lập biến thể tốn tối ưu mà điều kiện quy khơng khác bổ đề Farkas dạng tiệm cận cho ánh xạ g thỏa mãn lồi theo hàm tuyến tính mở rộng S (thay lồi theo nón K trên) Chúng tơi Từ khóa: Bổ đề Farkas, Bổ đề Farkas theo dãy, giới hạn trên, giới hạn REFERENCES [1] C.D Aliprantis, K.C Border, Infinite dimensional analysis, Springer-Verlag, Berlin, Germany (2006) [2] H.H Bauschke, P.L Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York (2010) [3] R.I Bot, Conjugate Duality in Convex Optimization, Springer-Verlag, Berlin (2010) [4] R.I Bot, S.M Grad, G Wanka, New regularity conditions for strong and total FenchelLagrange duality in infinite dimensional, Nonlinear Analysis 69, 1, 323–336 (2008) [5] N Dinh, M.A Goberna, M.A López, M Volle, Convex inequalities without constraint qualification nor closedness condition, and their applications in optimization, Set-Valued Anal, 18, 2540–2559 (2010) [6] N Dinh, M.A Goberna, M.A López, T.H Mo, From the 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extebded Farkas’s lemmas and Lagrangian dualities in convex infinite programming, SIAM J Optim 20, 1311–1332 (2010) [13] V Jeyakumar, G.M Lee, N Dinh, New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programming, SIAM J Optim., 14, 2, 534-547 (2003) [14] R.E Megginson, An Introduction to Banach Space Theory, Springer, New York (1998) [15] S Simons, From Hahn-Banach to monotonicity, Springer-Verlag, Berlin (2007) [16] S Simons, The Hahn-Banach-Lagrange theorem, Optimization, 56, 149–169 (2007) ( [17] Z a linescu, C., Convex Analysis in General Vector Spaces World Scientific, River Edge, NJ (2002)  ... điều kiện quy không khác bổ đề Farkas dạng tiệm cận cho ánh xạ g thỏa mãn lồi theo hàm tuyến tính mở rộng S (thay lồi theo nón K trên) Chúng tơi Từ khóa: Bổ đề Farkas, Bổ đề Farkas theo dãy, giới... [(e)  (g)] Remark 3.1 It is worth noting that [(e)  (g)] was given in [6] Các bổ đề Farkas dạng tiệm cận cho hệ lồi  Nguyễn Định Trường Đại học Quốc tế, ĐHQG-HCM  Trần Hồng Mơ Trường Đại... điều tác giả năm gần đây, cho kiện quy thường gọi kết phiên định lý Các kết đạt dạng Farkas tiệm cận (hay dạng xấp xỉ) Phần được sử dụng để nghiên cứu thứ hai báo dành cho thiết lập biến thể tốn