trong â Mβ = β 0 0 0 v N :R2×R2 →R3 l ma trên 2ì2ì3 N = 0 0 1 0 0 0 0 0 0 0 0 0 , tùc l ,N(u, v) = (0, u1v1,0), vỵi u, v ∈R2. Vẳ thá, clBf(0,0) ={Mβ|β∈ {−1} ∪[1,∞)}, clB(f,g)(0,0) ={(Mβ, N)|β ∈ {−1} ∪[1,∞)}, Bf(0,0)∞={Mβ|β ≥0}, B(f,g)(0,0)∞ ={(Mβ,02×2×3)|β ≥0}. Chånu= (1,0)∈S2. Ta câ, vỵi måi (c∗, k∗)∈C∗×K(g(0,0))∗,
hc∗, f0(0,0)ui+hk∗, g0(0,0)ui= 0. Do â, i·u ki»n (i) trong nh lỵ 4.1 khổng thọa.
Cho u= (u1, u2)∈S2 sao cho (f, g)0(0,0)u∈ −[C×clK(g(0,0))]. Khi â, u= (u1,0)
vỵi u1 =±1. Ta câ
T2(−K, g(0,0), g0(0,0)u) =A2(−K, g(0,0), g0(0,0)u),
v do â, vỵik∗ = (0,0,−1)∈N(−K, g(0,0)),
supk∈T2(−K,g(0,0),g0(0,0)u)hk∗, ki=−4
(quan sĂt rơng hiằn tữủng envelope-like xÊy ra). BƠy gií, vỵi måi(Mβ, N)∈clB(f,g)(0,0),
tỗn tƠi(c∗, k∗) = (1,0,0,−1)∈Λ1(0,0)thäa
hc∗,2Mβ(u, u)i+hk∗,2N(u, u)i= 2β >supk∈T2(−K,g(0,0),g0(0,0)u)hk∗, ki
v , vỵi måi(Mβ, N)∈ B(f,g)(0,0)∞\ {0}, tỗn tÔi c∗ = 1 ∈C∗\ {0}vỵi hc∗, f0(0,0)ui= 0
thäa
hc∗, Mβ(u, u)i=β >0.
Vẳ thá, (a0
) cõa Nhªn xt 4.2 v do õ (ii) (a) trong nh lỵ 4.1 thäa. Hìn núa, cho
w = (w1, w2) ∈ v⊥ \ {(0,0)}, tùc l , w1 = 0 v w2 6= 0, n¸u g0(0,0)w = (0,0, w2) ∈
clcone[cone(−K−g(0,0))−g0(0,0)u] ={(k1, k2, k3) ∈R3|k3 ≥0}, th¼ w2 >0. Vẳ thá,
vỵi måiMβ ∈Bf(0,0)∞, tỗn tƠi c∗ = 1 ∈C∗ \ {0} vỵi hc∗, f0(0,0)ui= 0 thäa
hc∗, f0(0,0)w+Mβ(u, u)i=w2+β >0,
v , vỵi måiMβ ∈Bf(0,0)∞\ {0}, tỗn tÔi c∗ = 1∈ C∗ \ {0} vỵi hc∗, f0(0,0)ui= 0 thäa
hc∗, Mβ(u, u)i =β > 0. Vẳ thá, bi Nhên xt 4.2 (ii), iÃu kiằn (ii) (b) cừa nh lỵ 4.1
thäa. H» qu£ l ,(0,0)∈ LFE(2, f, S).
V¼ f 6∈ C1 tÔi (0,0), c¡c H» qu£ 7, 8 cõa [7], nh lỵ 4.5 cừa [25] v cĂc hằ quÊ 4.4
v 4.5 ð tr¶n khỉng ¡p dửng ữủc. Hỡn nỳa, vẳd2(f, g)((0,0), u) =∅, nh lỵ 3 cừa [7]
Kát luên v hữợng nghiản cựu mð rëng · t i
Trong · ti nghiản cựu ny, Ưu tiản, chúng tổi giợi thiằu khĂi niằm và cĂc têp tiáp xóc c§p mët v c§p hai v kh£o s¡t mët số tẵnh chĐt cừa chúng. Tiáp theo, chúng tổi à xuĐt khĂi niằm Ơo hm suy rëng kiºu x§p x¿ c§p mët v c§p hai v ữa ra cĂc tẵnh chĐt cừa chúng. Cuối cũng, dũng cĂc Ôo hm suy rởng kiu xĐp x ny dữợi giÊ thiát khÊ vi cht (trong cĂc iÃu kiằn tối ữu cƯn) hay kh£ vi (trong c¡c i·u ki»n tèi ÷u õ), chúng tổi thiát lêp cĂc iÃu kiằn tối ữu cĐp hai mợi cho cĂc nghiằm u àa ph÷ìng v c¡c nghi»m chc chưn a phữỡng, vợi tẵnh chĐt envelope-like ữủc lm ró hìn, cõa b i to¡n tèi ÷u vectỡ khổng trỡn trong cĂc khổng gian vổ hÔn chiÃu (P).
Trong ká hoÔch nghiản cựu tữỡng lai, chúng tổi s m rởng hữợng nghiản cựu cõa · t i b¬ng c¡ch x²t bi toĂn tối ữu vectỡ khổng trỡn vợi rng buởc bao h m thùc kh¡ têng qu¡t sau ¥y:
(P1) minCf(x), sao chox∈S, 0∈F(x),
trong â f :X → Y l Ănh xÔ ỡn tr v F : X → 2Z l Ănh xƠ a tr, X v Z l c¡c
khỉng gian Banach,Y l khỉng gian ành chu©n, S ⊂X, v C ⊂Y l nõn lỗi õng.
Chúng tổi s thiát lêp cĂc iÃu kiằn tối ữu cƯn v õ c§p mët v cĐp hai cho cĂc nghiằm yáu v nghiằm chưc chưn cừa bi toĂn (P1) bơng cĂc quy tưc nhƠn tỷ Fritz-
John-Lagrange. Chúng tổi dũng cĂc Ơo h m suy rëng kiºu x§p x¿ chof, Ôo hm theo
hữợng a tr choF, v c¡c nõn tiáp xúc v têp tiáp xúc cĐp mởt v cĐp hai dữợi cĂc giÊ
T i li»u tham kh£o
[1] Allali, K., Amahroq, T.: Second-order approximations and primal and dual necessary optimality conditions, Optimization 40 (1997) 229-246.
[2] Bednar½k, D., Pastor, K.: On second-order optimality conditions in constrained mul- tiobjective optimization, Nonlinear Anal. 74 (2011) 1372-1382.
[3] Bonnans, J. F., Shapiro, A.: Perturbation Analysis of Optimization Problems, Springer, New York (2000).
[4] Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley Interscience, New York (1983).
[5] Cominetti, R.: Metric regularity, tangent sets and second order optimality conditions, Appl. Math. Optim. 21 (1990) 265-287.
[6] Dontchev, A. L., Rockafellar, R. T.: Regularity and conditioning of solution mappings in variational analysis, Set-valued Anal. 12 (2004) 79-109.
[7] Guti²rrez, C., Jim²nez, B., Novo, V.: On second order Fritz John type optimality con- ditions in nonsmooth multiobjective programming, Math. Program. (Ser. B) 123 (2010) 199-223.
[8] Hiriart-Urruty, J. B., Strodiot, J. J., Nguyen, V. H.: Generalized Hessian matrix and
second-order optimality conditions for problems withC1,1 data, Appl. Math. Optim. 11
(1984) 43-56.
[9] Jeyakumar, V., Luc, D. T.: Nonsmooth Vector Functions and Continuous Optimiza- tion, Springer, Berlin (2008).
[10] Jim²nez, B., Novo, V.: Second order necessary conditions in set constrained differ- entiable vector optimization, Math. Meth. Oper. Res. 58 (2003) 299-317.
[11] Jim²nez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets, Appl. Math. Optim. 49 (2004) 123-144.
[12] Jourani, A.: Metric regularity and second-order necessary optimality conditions for minimization problems under inclusion constraints, J. Optim. Theory Appl. 81 (1994) 97-120.
[13] Jourani, A., Thibault, L.: Approximations and metric regularity in mathematical programming in Banach spaces, Math. Oper. Res. 18 (1992) 390-400.
[14] Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second order necessary conditions for minimization problems, Math. Program. 41 (1988) 73-96.
[15] Khanh, P. Q., Tuan, N. D.: First and second-order optimality conditions using ap- proximations for nonsmooth vector optimization in Banach spaces, J. Optim. Theory Appl. 130 (2006) 289-308.
[16] Khanh, P. Q., Tuan, N. D.: Optimality conditions for nonsmooth multiobjective op- timization using Hadamard directional derivatives, J. Optim. Theory Appl. 133 (2007) 341-357.
[17] Khanh, P. Q., Tuan, N. D.: First and second-order approximations as derivatives of mappings in optimality conditions for nonsmooth vector optimization, Appl. Math. Optim. 58 (2008) 147-166.
[18] Khanh, P. Q., Tuan, N. D.: Optimality conditions using approximations for non- smooth vector optimization problems under general inequality constraints, J. Convex Anal. 16 (2009) 169-186.
[19] Khanh, P. Q., Tuan, N. D.: Corrigendum to Optimality conditions using approxima- tions for nonsmooth vector optimization problems under general inequality constraints", J. Convex Anal. 18 (2011) 897-901.
[20] Khanh, P. Q., Tuan, N. D.: Second-order optimality conditions with the envelope-
like effect in nonsmooth multiobjective mathematical programming, I: l-stability and
set-valued directional derivatives, J. Math. Anal. Appl. 403 (2013) 695-702.
[21] Khanh, P. Q., Tuan, N. D.: Second-order optimality conditions with the envelope- like effect in nonsmooth multiobjective mathematical programming, II: Optimality con- ditions, J. Math. Anal. Appl. 403 (2013) 703-714.
[22] Khanh, P. Q., Tuan, N. D.: Second-order optimality conditions with envelope-like ef- fect for nonsmooth vector optimization in infinite dimensions, Nonlinear Anal. 77 (2013) 130-148.
[23] Maruyama, Y.: Second-order necessary conditions for nonlinear optimization prob- lems in Banach spaces and their applications to an optimal control problem, Math. Oper. Res. 15 (1990) 467-482.
[24] Penot, J. P.: Optimality conditions in mathematical programming and composite optimization, Math. Program. 67 (1994) 225-245.
[25] Penot, J. P.: Second order conditions for optimization problems with constraints, SIAM J. Control Optim. 37 (1998) 303-318.
[26] Penot, J. P.: Recent advances on second-order optimality conditions, in Optimiza- tion, V. H. Nguyen, J. J. Strodiot, P. Tossings eds., Springer, Berlin, (2000) 357-380. [27] Rockafellar, R. T.: Convex Analysis, Princeton University Press, Princeton, New Jersey (1970).
[28] Taa, A.: Second-order conditions for nonsmooth multiobjective optimization prob- lems with inclusion constraints, J. Global Optim. 50 (2011) 271-291.
[29] Ward, D. E.: Calculus for parabolic second-order derivatives, Set Valued Anal. 1 (1993) 213-246.