非线性互补约束优化问题的可行性条件
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k k dy + Dw dw + Φ(y k , w k , µk ) = 0, Dy
h1
Qk ∈ R(n+2m)×(n+2m)
x
H&kGj K
y y k Dw
h≡
xF k Dy ≡
k k x h( x , y ) , ≡ x F (xk , y k ), k k y Φ(y , w , µk ),
SFD (x, y, S ) = LFD (x, y, S ),
[6]),
i a
(2.4)
\+96Jj (2.4) k'3?*U3 *E@l (2.1) s? g(x, y), h(x, y) k F (x, y) Rbiug 3*U bh d (A1), (A2). H\G? (x, y), L S0 l! &3P) ? ;
bgk
EH]5
d
0 ≤ y ⊥ F (x, y ) ≥ 0.
[ x g T , − x hT ]u + x F T v = 0 vo [ y g T , − y hT ]u + y F T v ≤ 0
−→ vo [
yg
T
,−
y
hT ]u +
yF
T
v = 0 (2.2)
'3 Rs o l Hadamard (~ (A2 ) ! S&3 ? ; S S,
2
"
E(>\! MPEC:
minimize f (x, y ) subject to g (x, y ) = 0, h(x, y ) ≥ 0, 0 ≤ y ⊥ F (x, y ) ≥ 0, (2.1)
&(
∗
2000 N 12 ` 12 o= 2003 N 7 ` 2 o=5YZ e ~Z) } (70271019 j) {w*I
λ µ
= 0,
(2.11) (2.12) (2.13) (2.14) (2.15)
y
F T (y − ζ ) + [ y ≥ 0,
yg
,−
hT ]
= θ − F (x, y ),
µ ≥ 0, θ ≥ 0, ζ ≥ 0,
h(x, y ) ≥ 0, F (x, y ) ≥ 0,
µ ◦ h(x, y ) = 0, θ ◦ y = 0, ζ ◦ F (x, y ) = 0, λ µ
minimize subject to f (x, y ) Ax + By = b, ω = N x + M y − q, ω T y = 0, x ∈ C, y ≥ 0, ω ≥ 0, (1.1)
-*
+24
Rs f : Rn+m → R 48* A ∈ Rp×n, B ∈ Rp×m, N ∈ Rm×n k M ∈ Rm×m D c2|0a m+6 C Rn s? JFz Om]s m&3r B! @8S&3r ? MPEC ?*13
D 26 $ D 4 Q 2003 M 10 _
ACTA MATHEMATICAE APPLICATAE SINICA
KNw::
Vol. 26 No. 4 Oct., 2003
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( )
4s ^tPw @+24 @ b 9Pn^t@v9du T'4s
1
0 + ? :T'4s ^tPw @+24 St^t /s z -DCzP e 7lDMzP y 7 h,#@S T'4^t
h≡ F ≡
k k y h( x , y ) , ≡ y F (xk , y k ), k k w Φ(y , w , µk ),
E lG d (A1) L_ l| (3.2) ?*13 ,X (3.2) ]s s? dw, e]s */'
k k x h dx + y h dy ≥ −h(x , y ), k k k (Dw x F ) dx + (Dy + Dw y F ) dy
yF
T
v ≤ 0,
+bgk (2.2) ?T 2.2. d yF ?&3P) ⎛
⎝−
kGj H\G? (x, y) ∈ Rn+m , ! `V 6 z ∈ Rn
xg xF x
⎞ h⎠ z = [
yg T
0 ,−
y
hT ]u
(2.6)
HBT u ∈ Rp+l oS e (A1 ) '3 d (u, v) B bgk (2.2) ?
R [8] s?G/ 2.7.8 *m
(u, v ) ≥ 0
T xh u + T yh u + T x F v = 0, k −1 k (Dw ) Dy +
(3.3)
yF
k`V 6 dx, dy S A V
T
v = 0, v = 0,
T
⎫ ⎬ ⎭
k −1 k =⇒ h(xk , y k )T u + (Dw ) s v ≥ 0.
( )
(
/ . # ') 5 7 849 qJ7 x X 8 \ %`
410082) SQP,
∗
n'
8Om]s?Ov ( & MPEC) mp]s s. $?Ak{ Ak ]s y g8#u?r {8#u? U Ak pg& Om]s r X. ?6"UvAU f %V a6CByO (d ) 6WCLyO (x ) 6 g+"VOm]s s cR;?Ak{ Ak]ss L gCL yO 6 (#' $ [1–4]). R] GcV N % MPEC f{uoPN (\Ha SQP PN) s^d?| ?*13 <f*1YS,? [5] CB3\+9\!? MPEC *13*U
(3.4)
qA
yh
T
k −1 k u + (Dw ) Dy +
yF
T yF
k k RV Dy , Dw & k?H j e S
c|N?DBF:!W
minmize s.t.
?\!L3
dx dy
y h dy yF
f (xk , y k )T
x x
⎛ ⎞ dx 1 + (dxT , dy T , dw T )Qk ⎝ dy ⎠ 2 dw (3.2)
h dx + F dx +
+ h(xk , y k ) ≥ 0,
dy − dw + F (xk , y k ) − w k = 0,
Q t;uq} T'4s ^tPw @+24 647 Rs f : Rn+m → R, g : Rn+m → Rl , h : Rn+m → Rp, F : Rn+m → Rm 48* x CByO (d ) 6 y CLyO (x ) 6 Om]s 8#u x ?S &3r
4
iH
(A1 )
(2.1),
⎧ T T xF y + xg λ − ⎪ ⎪ ⎪ T ⎪ ⎪ ⎨ F (x, y ) + x F y + µi ≥ 0, hi (x, y ) ≥ 0, ⎪ ⎪ ⎪ θj ≥ 0, yj ≥ 0, θj yj ⎪ ⎪ ⎩ ζk ≥ 0, Fk (x, y ) ≥ 0,
T T xh µ − x F ζ = 0, T T yh µ − θ − yF ζ =
[
xg T
6
;
eS
xF T
,−
T
xቤተ መጻሕፍቲ ባይዱ
hT ]u +
y T
v = 0,
T
(2.7) (2.8)
vo [
yg
,−
h ]u +
yF
v ≤ 0.
R (2.8) k*>
vT [
yg
T
,−
y
hT ]u +
yF
T
v ≤ 0,
vT [
yg
T
,−
y
hT ]u ≤ −v T
y
F T v ≤ 0, z,
xg x
RsCL AkL< y F ? kG3 Har u, RV&3P) (2.6) kS
650 minimize s.t. f (x, y ) h(x, y ) ≥ 0, F (x, y ) − w = 0 Φ(y, w, µ) = 0,
Rs µ SWHa#u iu Φ : R2m × [0, ∞) → Rm GF
⎛
⎞ 2 2 + w1 + µ ) 1/ 2 y1 + w 1 − ( y1 ⎜ ⎟ . . Φ(y, w, µ) = ⎝ ⎠, . 2 2 1/ 2 ym + w m − ( ym + w m + µ ) (3.1)
6
649
! (2.5)
0, i = 1, 2, · · · , p, j = 1, 2, · · · , m, k = 1, 2, · · · , m,
µi hi (x, y ) = 0, = 0, ξk Fk (x, y ) = 0,
x
F T (y − ζ ) + [
xg T
T
, −
y
x
hT ] λ µ
= sk ,
Rs
k S k ≡ −Φ(y k , w k , µk ) + Dw (F (xk , y k ) − w k ), x h dx
/'
(3.3)
+ y h dy ≥ −h(xk , y k ), k −1 k k −1 k Dy + y F dy = (Dw ) s . x F dx + (Dw )
