B GIO DC V O TO TRNG I HC s PHM H NI NGUYN VN TUYấN IU KIN cc TR V N NH TRONG TI U VẫCT VI TH T SUY RNG LUN N TIN S TON HC NGUYN VN TUYấN IU KIN cc TR V N NH H NI - 2016 B GIO DC V O TO TRNG I HC s PHM H NI TRONG TI U VẫCT VI TH T SUY RNG Chuyờn ngnh: Toỏn Gii tớch Mó s: 62.46.01.02 LUN N TIN S TON HC NGI HNG DN KHOA HC: PGS TS NGUYN QUANG HUY H NI - 2016 Li cam oan Lun ỏn c hon thnh ti Trng i hc S phm H Ni 2, di s hng dn ca PGS TS Nguyn Quang Huy Cỏc kt qu lun ỏn ny l mi v cha tng cụng b bt k cụng trỡnh khoa hc no ca khỏc Tỏc gi lun ỏn Nguyn Vn Tuyờn Túm tt Lun ỏn trỡnh by mt s kt qu mi v iu kin cc tr v n nh ti u vộct vi th t suy rng Lun ỏn gm chng Chng nghiờn cu mt s c trng ca nghim ti u theo th t suy rng nh: mi quan h ca khỏi nim nghim ny vi cỏc khỏi nim nghim c in, s tn ti nghim v mt s tớnh cht tụpụ ca nghim Chng nghiờn cu v cỏc iu kin cc tr cho ti u theo th t suy rng Chng nghiờn cu tớnh cht n nh ca nghim hu hiu Pareto tng i Cỏc kt qu chớnh ca lun ỏn bao gm: 1) a cỏc phõn tớch chi tit v khỏi nim nghim ti u theo th t suy rng 2) Thit lp cỏc iu kin cho s tn ti nghim ti u vi th t suy rng 3) Thit lp cỏc iu kin cho tớnh úng v tớnh liờn thụng ca nghim ca bi toỏn ti u vộct vi th t suy rng; cỏc iu kin cc tr cho nghim ti u theo th t suy rng i vi lp bi toỏn ti u vộct li 4) Mt s tớnh cht tụpụ nh tớnh úng, tớnh trự mt ca im hu hiu Pareto tng i 5) Thit lp cỏc iu kin cho s hi t trờn v s hi t di theo ngha Kuratowski-Painleve ca im hu hiu Pareto tng i; cho tớnh na liờn tc di theo ngha Berge ca ỏnh x im hu hiu Pareto tng i Abstract This thesis presents some new results on the optimality conditions and the stability analysis in vector optimization with generalized order The thesis consists of three chapters Chapter investigates some characterizations of the optimal solution with generalized order optimality such as: compares this notion with the traditional notions, the existence solution and some topological properties of solution set Chapter establishes some optimality conditions for vector optimization problems with generalized order The goal of Chapter is to deal with the stability analysis of a vector optimization problem using the notion of relative Pareto efficiency The main results of the thesis include: 1) A detailed analysis of the notion of generalized order optimality 2) Existence theorems in vector optimization with generalized order 3) Some criteria for the closedness and connectedness of the set of generalized order solutions and some sufficient optimality conditions in convex vector optimization problems 4) Some topological properties of the relative Pareto efficient set 5) Some sufficient conditions for the upper convergence and the lower convergence in the sense of Kuratowski-Painleve of the relative Pareto efficient sets; some criteria for the lower semicontinuity in the sense of Berge of the relative Pareto efficient point multifunction Mc lc Mt s ký hiu N R cỏc s t nhiờn cỏc s thc := Ru {00} R n cỏc s thc m rng khụng gian Euclide n-chiu cỏc vộct khụng õm ca Rn cỏc vộct khụng Rn_ X* dng ca Rn khụng gian i ngu tụpụ ca khụng gian X (x\x) cp i ngu gia X * v X \\x chun ca vộct X oo Rn+ vộct khụng gian X s 0, hoc vộct khụng gian cho trc F :X =Y ỏnh x a tr t X vo Y dom F xỏc nh ca F gph F { x n } , th ca F (xn) Mx B dóy s thc, hoc dóy vộct hỡnh cu n v úng X hỡnh cu n v úng khụng gian nh chun cho trc đp(z), B( x , p ) hỡnh cu úng tõm X , bỏn kớnh p hỡnh cu m tõm X , Bp{x), B(x,p) bỏn kớnh p l tt c cỏc lõn cn ca im X l J\ớ(x) tt c cỏc lõn cn cõn ca im X NB{X) Lim gii hn trờn theo ngha Painlevộ - Kuratowski sup N(x] iù) ẹ{x,n) gii hn di theo ngha Painlevộ - Kuratowski nún phỏp tuyn Mordukhovich ca ti X V/Or) nún phỏp tuyn Frộchet ca ti X o hm Frộchet ca / ti X di vi phõn Mordukhovich ca / ti X df(x) di vi phõn suy bin ca / ti X df( di vi phõn Frộchet ca / ti X x) i o hm Frộchet ca F ti ( x , ) df(x) i o hm Mordukhovich ca F ti ( x , ) D ' F ( x , s)(-) ò^Oi.SX-) _ X a óAc B X > X v X G X > X v f ( x ) > f ( x ) An B giao ca hai hp A v B Au B hp ca hai A v B Ax B tớch Descartes ca hai A v B a > ó v a ^ ó A l ca B hiu ca hai A v B tng vộct ca hai A A\B A + B int A v B phn ca hp A ri A phn tng i ca hp A A , cl A bd bao úng ca hp A biờn ca hp A (A) Ac phn bự ca hp A aff { A ) conv () cone () M u bao aphin ca hp A bao li ca hp A bao nún ca hp A kt thỳc chng minh Ti u vộct (Vector optimization) hay cũn gi l Ti u a mc tiờu (Multicriteria optimization) c hỡnh thnh t nhng ý tng v cõn bng kinh t, lý thuyt giỏ tr ca F Edgeworth (1881) v V Pareto (1906) C s toỏn hc ca lý thuyt ny l nhng khụng gian cú th t c G Cantor a nm 1897, F Hausdorff nm 1906 v nhng ỏnh x n tr cng nh a tr cú giỏ tr mt khụng gian cú th t tha nhng tớnh cht no ú T nhng nm 1950 tr li õy, sau nhng cụng trỡnh v iu kin cn v cho ti u ca H w Kuhn v A w Tucker nm 1951, v giỏ tr cõn bng v ti u Pareto ca G Debreu nm 1954, lý thuyt ti u vộct mi thc s c cụng nhn l mt ngnh toỏn hc quan trng v cú nhiu ng dng thc t Lỳc u ngi ta mi nghiờn cu nhng bi toỏn cú liờn quan ti ỏnh x n tr t khụng gian Euclide ny sang khụng gian Euclide khỏc m th t nú c sinh bi nún orthant dng Sau ú ngi ta m rng cho cỏc bi toỏn khụng gian cú s chiu vụ hn vi nún li bt kỡ Khỏi nim im hu hiu ca mt hp khụng gian cú th t sinh bi nún li ó c a theo nhiu cỏch khỏc da vo cỏc tớnh cht tụpụ, i s ca nún nh: hu hiu Pareto, hu hiu Pareto yu, hu hiu lý tng, hu hiu thc s Nhiu nh toỏn hc cú tờn tui nh J M Borwein, M I Henig, J Jahn, D T Luc ó cú nhng úng gúp quan trng v s tn ti ca cỏc im hu hiu loi ny, v iu ny dn ti vic nghiờn cu cỏc lp bi toỏn ti u khỏc Sau ú lý thuyt ny c phỏt trin cho nhng bi toỏn liờn quan ti ỏnh x a tr khụng gian vụ hn chiu Khỏi nim v ỏnh x a tr ó c nhiu ngi a t nhng nm ca na u th k 20 nhu cu phỏt trin ca chớnh bn thõn toỏn hc v nhiu lnh vc khoa hc khỏc Nhng nh ngha, tớnh cht ca ỏnh x n tr dn dn c m rng cho ỏnh x a tr c Berge ó a cỏc khỏi nim khỏc v tớnh na liờn tc trờn v na liờn tc di ca ỏnh x a tr Tng t nh vy cỏc khỏi nim li trờn, li di, Lipschitz trờn v Lipschitz di cng c a Tip theo l tớnh kh di vi phõn ca hm s, di vi phõn ca hm li, di vi phõn ca hm Lipschitz a phng theo ngha ca F H Clarke T cỏc khỏi nim ny ngi ta tỡm c nhng iu kin cn v iu kin cc tr cho cỏc lp bi toỏn ti u khỏc Nghiờn cu s tn ti nghim l mt nhng quan trng nht nghiờn cu cỏc bi toỏn quy hoch toỏn hc v cỏc bi toỏn ti u vộct S tn nghim ca bi toỏn ti u vộct cỏc khụng gian vụ hn chiu ó c nhiu tỏc gi quan tõm v nghiờn cu (xem [2,19,26-28,37,41,42,61,64,71,73] v cỏc ti liu trớch dn c trớch dn ú) Theo hiu bit ca chỳng tụi, hu ht cỏc kt qu v s tn ti nghim ti u vộct u c xột cỏc khụng gian vộct tụpụ vi th t sinh bi mt nún li Mt kt qu c in (xem, p L Yu [71]) ch rng cỏc im hu hiu Min ( A \ , C ) khỏc rng nu c l nún li úng v A l compact Tuy nhiờn, gi thit v tớnh compact l khỏ cht gii bi toỏn khụng gian vụ hn chiu Sau ú, cú nhiu kt qu nghiờn cu t c v s tn ti im hu hiu ó loi b c hn ch v tớnh compact Chng hn, nh lý 3.3 [41] s dng tớnh C - y ( C - c o m p e t e ) thay cho tớnh compact Mt quan trng khỏc lý thuyt ti u ú l vic nghiờn cu cỏc iu kin cn v cc tr a cỏc iu kin ti u cho cỏc bi toỏn ti u vộct khụng trn, ngi ta s dng cỏc khỏi nim o hm suy rng Chng hn, M Pappalardo v w Stừcklin [54] ó s dng o hm suy rng ca Dini - Hadamard a mt s iu kin ti u cho nghim Pareto yu, trng hp hu hn chiu (i) Nu int c c thỡ aff ( ) = z Vỡ vy tớnh cht r-usc (r-H-usc) v usc (H-usc) trựng (ii) Nu F na liờn tc di tng i, thỡ F na liờn tc di iu ngc li núi chung khụng ỳng Chng hn, cho F : R =4 K2 xỏc nh bi F ( p ) = {(^1, z ) Ê M2 I < z2 < 1} Vp Ê M \ {0}, F(0) = {0R2} v c = M+ X {0} Khi ú, F l lsc ti mi im nhng khụng r-lsc ti p = (iii) Nu F l na liờn tc trờn Hausdorff tng i, thỡ F l na liờn tc trờn Hausdorff Tuy nhiờn, iu ngc li núi chung khụng ỳng Chng hn, ly p = M, z= M2, c = M+ X {0} Cho F : M =4 M2 c xỏc nh bi ^(0) = {(^è z ỡ) I Z = 0} \ {O R } v F ( p ) = {(^1, z ) I - |p| < Z1 < |p|} V p Ê R \ {0} Khi ú, vi mi > 0, ta cú F ( p ) c F ( ) + B ( R , ) Vp G R, |p| < 2' Vỡ vy F l H-usc ti Po = Tuy nhiờn, vi mi > 0, ta cú F ( 0) + [Ê(0R2,) n aff ()] = { { z u z ) \ \ z i \ < v Z 0}Vỡ vy F ( p ) ầ F ( 0) + [B(0R2, ặ) n aff ()] vi mi P G R \ {0} iu ny cú ngha l F khụng l r-H-usc ti Cho F v l hai ỏnh x a tr t p ti z nh ngha sau l mt dng yu hn ca khỏi nim v tớnh cht bao hm u (uniform containment property) ca mt ỏnh x a tr c trỡnh by [11] nh ngha 3.8 Ta núi rng tớnh cht (RCP) ỳng cho cp (F, G) u quanh Po nu vi mi lõn cn w G M ( Z ) tn ti V G M ( Z ) v u G M ( p o ) cho [F{p) \ (G{p) + W ) ] + [V n W{C)} c G{p) + c Vp G u (3.30) D thy rng nu tớnh (C P ) ỳng u cho cp ( F , G) ti Po5 thỡ (RCP ) cng ỳng u cho cp (F, G) ti im ny Tuy nhiờn, iu ngc li núi chung khụng ỳng (xem Vớ d 3.5 bờn di) nh lý 3.7 Gi s rng ri c v (RCP ) ỳng cho cp (F, 1Z) u quanh p Nu F r-H-usc v r-sc ti p, thỡ TZ sc ti p Chng minh Ly G 'F.(po) v w G M(0 Z ) Chng minh s kt thỳc nu ta ch rng tn ti u w G M (po) cho (z + IV) n 7Z(p) ^ V p E w Ly W i G MB(0Z) (3.31) tựy ý tha W i + W i c w Do (RCP) ỳng cho cp ( F , T l ) u quanh P0ỡ nờn tn ti w G J \ f ( z) v u G M ( p o ) cho vi mi p e Uo v y e [F(p) \ (7Ê(p) + W)} chỳng ta cú th tỡm c )y e 7Ê (p) v Cy e c tha y = ]y + Cy, (Cy + VKo) n aff (c ) c c (3.32) Chn w G A/s(0^) tha VK2 + w c VK0 T tớnh na liờn tc di tng i ca F ti P o , suy tn ti lõn cn U ca P o , U c u cho [z + {w l r\W ) r\ ^E{ C) ]r\ F{p) ^^ VpG/i Vi mi p G U \ ly y p [* + ( m n JV2) n óff()] n F(p) (3.33) Do tớnh na liờn tc trờn Hausdorff tng i ca F ti P o , ta suy tn ti u G M ( p o ) , u c U , cho F { p ) c F(po) + [{Wi n w ) n ĂF()] Vp u (3.34) Trc ht, gi s rng tn ti u e M ( p o ) , u c U Q , cho y p 7Ê(p) + VKi Vp u (3.35) Vi mi p G I n , t (3.33) v (3.35) ta suy tn ti W p e VKi n w , ] p E TZ(p) v Wp E Wi tha y p = z + Wp = Tp + Wp Vỡ vy T p = +W p Wp + + e VKi VKi c + w iu ny cú ngha l (z + w)n n(p) Vpeu n Vỡ vy, trng hp ny (3.31) nghim ỳng vi u w = I n Tip theo, gi s rng vi mi u e M(po), u c u ỡ tn ti p e u cho Vp K{p) + Wi (3.36) Kt hp (3.36) vi (3.32) ta cú y p G [F(p) \ (7Ê(p) + W\)] Do (3.32), tn ti Tp c tha Vp r ip G 7Ê(p) v Cp e = + C P1 icP + Wo) n aff (C) c c (3.37) T (3.34) v ) p G 7Ê(p) c F(p) ta suy tn ti z e F(p ) v Wo G (W n w ) n aff (c) cho P = z0 + w0 T (3.33), ta suy tn ti Wp e (Wi (3.38) n w ) n aff (c) tha y p = Z + Wp (3.39) Kt hp (3.39), (3.37) v (3.38) ta c z + Wp p + Cp ZQ + WQ + Cp iu ny kộo theo z = z0 + Cp + w0 - Wp Hn na, C P + W 0- W P n w ) n ĩ'(C')] - [ ( W i n w ) n ọf (C')] c Cp + [W n w ()] + [W n ấL(C)] c (Cp +W o ) n ọ f f ( C ' ) c c EC P + [(Wi t k := C p + W Q Wp Ta c k e ri v Z Q = z k Vỡ vy F(po) n {z r i ) 05 mõu thun vi Ê i z ( p ) Chỳ ý rng, nu ri c v k Ê ri c, thỡ k 0- T z k E F(p ) v k suy Z (z k ) Ê F(p )n(z ri C) Do ú, F(p0)n(z C) {} Vỡ vy, bng cỏch thay TZ bi F chỳng ta cú kt qu sau nh lý 3.8 Gi s rng c l mt nún li vi C ^ C v (RCP) ỳng cho cp (F, F) u quanh p Nu F l r-H-usc v r-lsc ti Po, thỡ T l Isc ti p H qu 3.6 (xem [11, Theorem 4]) Gi s rng int c c l mt nún li, nhn vi v (CP) ỳng cho cp (F,F) u quanh p Nu F l H-usc v Isc ti Po, thỡ T l Isc ti p Nhn xột 3.8 (i) nh lý 3.7 m rng [11, Theorem 4] t ỏnh x im hu hiu Pareto n ỏnh x im hu hiu Pareto tng i (ii) Chỳng tụi nhn mnh thờm rng iu kin i vi nún 3.8 yu hn [11, Theorem 4] Hn na, int c c nh lý 05 thỡ nh lý 3.8 tr thnh [11, Theorem 4] Vớ d 3.5 Ly p = [0,1], z = R2, c = R+ X {0} Cho F: : p =4 K c xỏc nh nh sau F(p) = { { z u z ) I f { z i ) < z < - Z i + 1} vi mi p E [0,1], ú t+p nu t < p nu p < t vi mi t E M Vi mi p E [0,1] ta cú F(p) = {{zi,z ) I z2 = - Z i + p , i < p} u {{zi,z ) I z2 = - Z i + Mi > !} D dng kim tra c tớnh (locCP) (xem [23, Definition 3.1]) khụng ỳng cho F u quanh p = Chỳ ý rng F l r-H-usc v r-lsc at p Bng mt tớnh toỏn trc tip ch rng tớnh (RCP) ỳng cho (F, T) u quanh p Vỡ vy T l lsc ti p Cui cựng, chỳng ta nhc li kt qu gn õy ca Chuụng, Yao v Yen [23, Theorem 3.2] Bng cỏch s dng cỏch tip cn ca Bednarczuk [11,13] v a cỏc khỏi nim mi gi l tớnh cht bao hm a phng , kớ hiu bi (locCP), cỏc tỏc gi ó nhn c cỏc kt qu v tớnh na liờn tc di ca ỏnh x im hu hiu Pareto Trong [23], cỏc tỏc gi ó ch rng nu (CP) ỳng u cho cp (F, F) quanh Po, thỡ (locCP) cng ỳng u cho cp (F, F) quanh im n/ Tuy nhiờn, tớnh cht (locCP) v (RCP) l c lp vi thy iu ny, chỳng ta xột cỏc vớ d sau Vớ d 3.6 Cho (F, p, z, c) nh Vớ d 3.5 D thy rng (locCP) (xem [23, Definition 3.1]) khụng ỳng u cho (F,F) quanh p = Trong ú, (RCP) ỳng u cho (F, F) quanh im ny Vớ d 3.7 (xem [23, Example 3.5]) Ly p = [0,1], z = R2, c = R Cho F: p =4 R2 c xỏc nh nh sau F(0) = {(21,2:2) I Z < z < 2i + 2} v F{p) = {(2i, 22) I f{zi) < z < Z\ + 2} vi mi p e p \ {0}, ú t + p nu < - t' p r p - nu /(0 - p - - + 2- p ) D thy rng (RCP) ỳng cho (F, F ) ti Po = Tuy nhiờn (RCP) khụng ỳng cho cp (F, F ) ti mi im p e p \ {0} Vỡ vy, (RCP) khụng ỳng u cho (F,F) quanh p Trong ú, d rng kim tra c rng tớnh (locCP) ỳng u cho (F,F) quanh p Chỳng ta cng ý rng, mc dự tớnh (locCP) v (RCP) c lp vi nhau, nhng iu kin i vi nún c nh lý 3.8 l yu hn [23, Theorem 3.2] Kt lun ca Chng Cỏc kt qu chớnh ca chng ny bao gm: - Thit lp cỏc iu kin cho tớnh úng v tớnh trự mt ca im hu hiu Pareto tng i Thit lp cỏc iu kin cho s hi t trờn v s hi t di theo ngha Kuratowski-Painlevộ ca im hu hiu Pareto tng i; cho tớnh na liờn tc di theo ngha Berge ca ỏnh x im hu hiu Pareto tng i Kt lun Cỏc kt qu chớnh ca lun ỏn ny bao gm: a cỏc phõn tớch chi tit v khỏi nim nghim ti u theo th t suy rng Thit lp cỏc iu kin cho s tn ti nghim ti u vi th t suy rng Thit lp cỏc iu kin cho tớnh úng v tớnh liờn thụng ca nghim ca bi toỏn ti u vộct vi th t suy rng; cỏc iu kin cc tr cho nghim ti u theo th t suy rng i vi lp bi toỏn ti u vộct li Mt s tớnh cht tụpụ nh tớnh úng, tớnh trự mt ca im hu hiu Pareto tng i Thit lp cỏc iu kin cho s hi t trờn v s hi t di theo ngha Kuratowski-Painleve ca im hu hiu Pareto tng i; cho tớnh na liờn tc di theo ngha Berge ca ỏnh x im hu hiu Pareto tng i Mt s cn tip tc c nghiờn cu: Cỏc iu kin cc tr cho nghim ti u theo th t suy rng ca cỏc bi toỏn ti u vộct khụng li Tớnh cht liờn thụng ca nghim ca bi toỏn ti u vi th t suy rng Cỏc iu kin cc tr bc cao cho nghim ti u theo th t suy rng Cỏc c trng cn v cho tớnh na liờn tc trờn, na liờn tc di v tớnh gi Lipschitz ca ỏnh x nghim ca bi toỏn ti u vộct theo th t suy rng cú tham s CC CễNG TRèNH LIấN QUAN EN LUN N Tuyen, N V., Yen, N D.: On the concept of generalized order optimality, Nonlinear Anal 75 (2012), 1592-1601 Tuyen, N V., Some characterizations of solution sets of vector optimization problems with generalized order, Acta Math Vietnam, (accepted) Huy, N Q., Kim, D S., Tuyen, N V.: Existence theorems in vector optimization with generalized order, Vietnam J Math, (submited) Tuyen, N V., Convergence of the relative Pareto efficient sets, Taiwanese J Math, (submited) Ti lieu tham khõo [1] Aubin, J P., Frankowska, H.: Set-Valued Analysis, Birkhauser, Boston, Massachusetts, 1990 [2] Bao, T Q., Mordukhovich, B S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions, Math Program 122 (2010), 101-138 [3] Bao, T Q., Mordukhovich, B S.: Extended Pareto optimality in multiobjective problems, Chapter 13 of the book Recent Advances in Vector Optimization (Q H Ansari and J.-C Yao, eds.), pp 467- 516, Springer, Berlin, 2011 [4] Bao, T Q., Mordukhovich, B S.: Sufficient conditions for global weak Pareto solutions in multiobjective optimization, Positivity 16 (2012), 579 602 [5] Bao, T Q., Mordukhovich, B S.: To dual-space theory of set-valued optimization, Vietnam J Math 40 (2012), 131-163 [6] Bao, T Q., Tammer, C.: Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications, Nonlinear Anal 75 (2012), 1089 1103 [7] Bao, T Q., Mordukhovich, B S.: Necessary nondomination conditions in set and vector optimization with variable ordering structures, J Optim Theory Appl 162 (2014), 350 370 [8] Bao, T Q., Mordukhovich, B S.: Sufficient optimality conditions for global Pareto solutions to multiobjective problems with equilibrium constraints, J Nonlinear Convex Anal 15 (2014), 105-127 [9] Bao, T Q.: Subdifferential necessary conditions in set-valued optimization problems with equilibrium constraints, Optimization 63 (2014), 181-205 [10] Bao, T Q., Pattanaik, S R.: Necessary conditions for ee-minimizers in vector optimization with empty interior ordering sets, Optimization (2014), DOI: 10.1080/02331934.2014.926358 [11] Bednarczuk, E M.: Berge-type theorems for vector optimization problems, Optimization 32 (1995), 373-384 [12] Bednarczuk, E M.: Some stability results for vector optimization problems in partially ordered topological vector, in: Proceedings of the First World Congress of Nonlinear Analysts, Volume III, pp 2371-2382, Tampa, Florida, 1996 [13] Bednarczuk, E M.: A note on lower semicontinuity of minimal points, Nonlinear Anal 50 (2002), 285-297 [14] Bednarczuk, E M.: Upper Holder continuity of minimal points, J Convex Anal (2002), 327 338 [15] Bednarczuk, E M.: Continuity of minimal points with applications to parametric multiple objective optimization, European J Oper Res 157 (2004), 59 67 [16] Bednarczuk, E M.: Stability analysis for parametric vector optimization problems, Diss Math 442 (2007) [17] Berge, C.: Topological Spaces, New York, 1963 [18] Bishop, E., Phelps, R.R.: The support functionals of a convex set, Proceedings of the Symposium in Pure Mathematics, vol 7, Convexity Amer Math Soc., 1963, pp 27-35 [19] Borwein, J M.: On the Existence of Pareto Efficient Points, Math Oper Res (1983), 64-73 [20] Borwein, J M., Lewis, A S.: Partially finite convex programming, Part I: Quasi relative interior and duality theory, Math Program 57 (1992), 15-48 [21] Borwein, J M., Goebel, R.: Notions of relative interior in Banach spaces, J Math Sci 115 (2003), 2542-2553 [22] Chicco, M., Mignanego, F., Pusillo, L., Tijs, S.: Vector optimization problems via improvement sets J Optim Theory Appl 150 (2011), 516-529 [23] Chuong, T D., Yao, J C., Yen, N D.: Further results on the lower semicontinuity of efficient point multifunctions, Pacific J Optim 6, 405-422 (2010) [24] Dolecki, S., Malivert, C.: Stability of efficient sets: Continuity of mobile polarities, Nonlinear Anal 12 (1988), pp 1461-1486 [25] Dolecki, S., El Ghali, B.: Some old and new results on lower semi- continuity of minimal points, Nonlinear Anal 39 (2000), 599 609 [26] Ferro, F.: An optimization result for set-valued mappings and a stability property in vector problems with constraints, J Optim Theory Appl 90 (1996), 63 77 [27] Ferro, F.: Optimization and Stability Results Through Cone Lower Semicontinuity, Set-Valued Anal (1997), 365-375 [28] Flores-Bazỏn F., Hernỏndez E., Novo V.: Characterizing efficiency without linear structure: a unified approach, J Glob Optim 41 (2008), 4260 [29] Gong, X.H.: Efficiency and Henig efficiency for vector equilibrium problems, J Optim Theory Appl 108 (2001), 139-154 [30] Gutiộrrez, C., Jimộnez, B., Novo, V.: Improvement sets and vector optimization, European J Oper Res 223 (2012), 304-311 [31] Ha, T X D., Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems, Nonlinear Anal 75 (2012), 1305 1323 [32] Henig, M I., The domination property in multicriteria optimization, J Math Anal Appl 114 (1986), 7-16 [33] Holmes, R B.: Geometric Functional Analysis and Its Applications, Grad Texts in Math 24, Springer-Verlag, New York, 1975 [34] Huy, N Q., Mordukhovich, B S., Yao, J C.: Coderivatives of frontier and solution maps in parametric multiobjective optimization, Taiwanese J Math 12 (2008), 2083-2111 [35] Huy, N Q., Kim, D S., Tuyen, N V.: Existence theorems in vector optimization with generalized order, Vietnam J Math, (submited) [36] Huy, N Q., Tuyen, N V.: New second-order optimality conditions for C , optimization problems, J Optim Theory Appl (submited) [37] Jahn, J.: Vector Optimization Theory, Application, and Extensions, Springer, Berlin-Heidelberg-New York, 2004 [38] Kruger, A.Y., Mordukhovich, B.S.: Extremal points and the Euler equation in nonsmooth optimization, Dokl Akad Nauk BSSR 24(1980), 684-687 (in Russian) [39] Kruger, A Y.: Weak stationarity: eliminating the gap between nec- essary and sufficient conditions, Optimization 53 (2004), 147-164 [40] Luc, D T.: Structure of the efficient point set, Proc Amer Math Soc 95(1985), pp 433-440 [41] Luc, D T.: Theory of Vector Optimization , Lecture Notes in Econom and Math Systems 319, Springer-Verlag, Berlin, Heidelberg, 1989 [42] Luc, D T.: An existence theorem in vector optimization, Math Oper Res 14 (1989), 693-699 [43] Luc, D T.: Contractibility of Efficient Point Sets in Normed Spaces, Nonlinear Anal 15(1990), 527-535 [44] Luc, D T., Lucchetti, R., Malivert, C.: Convergence of the efficient sets, Set-Valued Anal (1994), pp 207-218 [45] Lucchetti, R., Miglierina, E.: Stability for convex vector optimization problems, Optimization (2004), pp 517 528 [46] Makarov, E K., Rachkovski, N N.: Efficient sets of convex compacta are arcwise connected, J Optim Theory Appl 110 (2001), 159 172 [47] Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization, Math Methods Oper Res 58 (2003), pp 375 385 [48] Miglierina, E., Molho, E.: Convergence of the minimal sets in convex vector optimization SIAM J Optim 15 (2005), 513 526 [49] Mordukhovich, B S.: Variational Analysis and Generalized Differentiation, Vol I: Basic Theory, Springer, Berlin, 2006 [50] Mordukhovich, B S.: Variational Analysis and Generalized Differentiation, Vol II: Applications, Springer, Berlin, 2006 [51] Mordukhovich, B S., Necessary and sufficient conditions for linear suboptimality in constrained optimization, J Global Optim 40 (2008), 225244 [52] Mordukhovich, B.S.: Methods of variational analysis in multiobjective optimization, Optimization 58 (2009), 413-430 [53] Naccache, P H.: Stability in multicriteria optimization, J Math Anal Appl 68 (1979), pp 441-453 [54] Pappalardo, M., Stocklin, W., Necessary optimality conditions in nondifferentiable vector optimization, Optimization 50 (2001), 233 - 251 [55] Penot, J P., Sterna-Karwat, A.: Parametrized multicriteria opti- mization: Continuity and closedness of optimal multifunction, J Math Anal Appl., 120 (1986), pp 150-168 [56] Rockafellar, R T.: Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970 [57] Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of Multiobjective Optimization, Mathematics in Science and Engineering, 176 Academic Press, Inc., Orlando, FL, 1985 [58] Song, W.: A note on connectivity of efficient point sets, Arch Math 65 (1995), 540 545 [59] Song, W.: Connectivity of efficient solution sets in vector optimiza- tion of set-valued mappings, Optimization 39 (1997), 1-11 [60] Sonntag, Y., Zalinescu, C.: Set convergences: A survey and a clas- sification, Set-Valued Anal (1994), 339-356 [61] Sonntag, Y., Zalinescu, C.: Comparison of existence results for effi- cient points, J Optim Theory Appl 105 (2000), 161-198 [62] Stadler, W.: Initiators of multiobjective optimization In: Stadler, W (ed.) Multicriteria Optimization in Engineering and in Sciences, pp 3-25 Series in Mathematical Concepts and Mathematics in Science and Engineering 37 Plenum Press, New York (1988) [63] Stoer, J., Witzgall, C., Convexity and Optimization in Finite Dimensions /, Springer-Verlag, New York, 1970 [64] Sterna-Karwat, A.: On existence of cone-maximal points in real topological linear spaces, Israel J Math 54 (1986), 33-41 [65] Tanino, T., Sawaragi, Y.: Stability of nondominated solutions in multicriteria decision-making, J Optim Theory Appl 30 (1980), 229-253 [66] Tolstonogov, A A.: Differential Inclusions in a Banach Space, Mathematics and Its Applications, vol 524, Kluwer Academic, Dordrecht, 2000 [67] Tuyen, N V., Yen, N D.: On the concept of generalized order opti- mality, Nonlinear Anal 75 (2012), 1592-1601 [68] Tuyen, N V.: Some characterizations of solution sets of vector op- timization problems with generalized order, Acta Math Vietnam, (accepted) [69] Tuyen, N V.: Convergence of the relative Pareto efficient sets, Tai- wanese J Math, (submited) [...]... 0} = [0,1] X {0,1} Mnh sau c suy ra trc tip t nh ngha ca im hu hiu suy rng Mnh 1.3 Cho A l mt tp con khỏc rng trong z v 0 c z l mt tp cha 0zNu 0 D 0, thỡ T (1.9) v A c A + 0 ta suy ra A n GMin (A + 0 I 0) = A n [(A + 0) n bd (A + 0)] = A n bd (A + 0) = GMin (A I 0) Mnh c chng minh Trong nh ngha 1.1 chỳng ta khụng ũi hi 0 l mt nún li v cng khụng ũi hi 0 phi cú phn trong khỏc rng Nu 0 l mt nún li... s kt qu v s tn ti nghim ti u theo th t suy rng Mc 1.3 kho sỏt mt s tớnh cht tụpụ (tớnh úng v tớnh liờn thụng) ca tp nghim ca bi toỏn ti u vộct theo th t suy rng 1 3 Chng 2 nghiờn cu cỏc iu kin ti u cho bi toỏn ti u vộct vi th t suy rng Mc 2.1 nhc li mt s kin thc c s ca gii tớch bin phõn Cỏc kin thc ny l c s a ra cỏc iu kin ti u trong cỏc mc tip theo ca chng ny Trong Mc 2.2, bng cỏch tip cn trờn khụng... trỡnh by mt s c trng ca nghim ti u theo th t suy rng Mc 1.1 trỡnh by mt s tớnh cht ca nghim ti u theo th t suy rng v mi liờn h gia khỏi nim nghim ny vi cỏc khỏi nim nghim c in trong ti u vộct Mc 1.2 trỡnh by mt s kt qu v s tn ti nghim ti u theo th t suy rng Mc 1.3 kho sỏt mt s tớnh cht tụpụ (tớnh úng v tớnh liờn thụng) ca tp nghim ca bi toỏn ti u vộct theo th t suy rng Chng ny c vit trờn c s cỏc bi bỏo... Q) (1.4) Vỡ 2* G 0* nờn ( z * , 6 ) < 0 iu ny v (1.4) suy ra (z\z) < ,z), trỏi vi nh ngha ca Z Mnh ó c chng minh Mnh sau ch ra rng, nu A + 0 l mt tp li vi phn trong khỏc rng, thỡ mi im hu hiu suy rng ca tp A cng l im ta ca tp hp ny Mnh 1.2 Cho A l mt tp con khỏc rng trong khụng gian Banach z, v 0 c z cha 0z Gi s rng, A + 0 l mt tp li v cú phn trong khỏc rng Khi ú, GMin (A I 0) = 1J{^(^) I z* e Q... thỡ iu kin 11A + 0 cú phn trong khỏc rn trong Mnh 1.2 cú th b c H qu 1.1 Cho A l mt tp con khỏc rng trong Rm r0c Rm l mt tp bt kỡ cha gc Nu A + 0 li, thỡ GMin (A I 0) = \ J { A ) I 0*, z * 0}Chỳ ý rng cỏc kt qu trờn khụng ũi hi rng 0 phi l mt nún vi 0 \ (0) 0- H qu 1.1 l mt m rng ca [71, Lemma 4.5] t im hu hiu (xem nh ngha 1.3 bờn di) sang im hu hiu suy rng Vớ d 1.2 Trong R2, cho A = A u A2 u... nim nghim c in trong ti u vộct nh nghim Pareto, nghim Pareto tng i (hay nghim ti u theo ngha Slater) (xem [50,67]) Cn nhn mnh rng, tp sinh th t 0 khụng nht thit l tp li hay l nún iu ny ỏp ng ũi hi ngy cng tng trong thc t v c trong lý thuyt ỏp dng 1 2 ca ti u vộct; c bit l trong cỏc mụ hỡnh kinh t (xem [62]) Ngoi khớa cnh m rng phm vi ỏp dng ca cỏc khỏi nim nghim, nghim ti u theo th t suy rng cũn l mt... theo th t suy rng (locally generalized optimal solution) ca F tng ng vi tp sinh th t 0 trờn ớ, nu z e GMin (F(f n u ) I 0), vi u l mt lõn cn no ú ca X Nu trong nh ngha 1.7 cú th ly u = X, thỡ (x, z) c gi l nghim ti u ton cc (hay nghim ti u) theo th t suy rng Tp tt c cỏc nghim ti u theo th t suy rng ca F tng ng vi 0 trờn ớ c kớ hiu l GS (ớ, F) Khi F = f : X > z l mt ỏnh x n tr, chỳng ta b qua Z trong nghim... vy, Z zk Ê (A + 0)c vi mi k Ê N Ly u l mt lõn cn tựy ý ca Z Vỡ Z Ê A v Oz Ê 0 nờnta suy ra Z G { A + 0) Do ú, U n { A + 0) 0- T lim (z Zk) = ta Cể Z Zk G U vi k ln Vỡ vy, Z Zk G U n {A Z + 0)c vi k ln Suy ra U n { A + 0)c 0- Vỡ vy Z G bd ( + 0) iu ny kộo theo GMin ( A I 0) c A n bd ( A + 0) chng minh bao hm thc ngc li ly Z G A n bd ( A + 0) tựy ý T Z G bd ( A + 0) ta cú B z , n ( A + 0)c... k = 1, 2, Suy ra c lim Zfc = 0 v k>00 Z Zk = Xk Ê {A -\- 0)c VA; G N, hay l Z - Zk { A + 0) VfeeN iu ny ch ra rng Z l mt im hu hiu suy rng ca A tng ng vi 0 Vỡ vy, GMin ( A I 0) = A n bd ( A + 0) Cui cựng, nu A l mt tp con úng ca z , t tớnh úng ca bd (+0), ta suy ra GMin ( A I 0) úng nh lý c chng minh Nhn xột 1.2 (i) T nh lý 1.1, ta cú GMin(j4|0) c bdA Tht vy, gi s tn ti mt im hu hiu suy rng ca... Mc 2.2, bng cỏch tip cn trờn khụng gian nh chỳng tụi ó t c mt s iu kin cn, iu kin cho mt im hu hiu suy rng Cỏc kt qu v iu kin cn cú th coi l trng hp c bit ca cỏc kt qu trong [9,50] Tuy nhiờn kt qu v iu kin l mi Trong mc cui ca chng ny, chỳng tụi trỡnh by mt s iu kin cho im l nghim ti u theo th t suy rng di cỏc gi thit v tớnh li Chng 3 trỡnh by cỏc kt qu nghiờn cu v tớnh n nh ca bi toỏn ti u vộct