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. .-. . . . ()
. .-. . . . ()
. . . . . ()
. .-. . . . ()
. .-. . . . ()
. .-. . . . ()
. .-. . . . ()
. . ()
. . ()
. . . . . ()
. .-. . . . ()
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. .-. . . . ()
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..-.. . . ()
. ..
, 7 14 2007
2007
22.1 482
-: (, 7 14 2007). : - , 2007. 192 .
ISBN 978-5-86134-134-9.
, (DOOR-07).
: . . , . . , . . , . . , . . , . . . . . , . . . . . .
:
( 07-01-06052)
20070382011602100000
)( .
ISBN 978-5-86134-134-9 . . . , 2007
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
. . . . . . . . . . . . . . . . 94
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
. . . . . . . . . . 142
. . . . . . . . . 149
. . . . . . . . . . . . 157
. . . . . . . . 173
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
. .
-
-
. ,
.
-
-
, ,
-
.
-
, -
() . -
.
-
( ), -
, .
-
[1, 2].
x
= (x1, , xn),
:
x
= argmaxP{X(x
, t) Dx
,t [0, T ]} (1) X(x
, t) ; Dx
-
; T . D
x
, , -
, :
aj y(x)j bj, j = 1,m y = {yj}mj=1 , yj = Fj(x1, , xn), Fj() , . , -
, -
-
.
-
.
(-) (-
-
) . N ,
4
, -
.
-
.
-
, , .
-
-
, , -
. ,
, -
, . ,
,
,
.
-
. -
,
, -
. (1)
- (master-slave) -
.
-
( ),
-
.
-
[3, 4]. -
.
, -
.
1. .. . -
. . : . 1992.
2. .. . -
// , 4, 2006. . 38.
3. .. , .. .
. // , 4, 2005. . 1926.
4. .. , .. . -
. // , 4, 2003. . 1115.
,
, . , 5, ,
690041, , (8-4232) 31-02-02, E-mail:[email protected]
5
1
. .
1
, -
= - - p = p , w = w -
, f(w) Min{, f(w) | g(w) p, w }, (1)
f(w) , 0, 0, (2)g(w) p, p p 0, p 0. (3) f(w), g(w)- , - , f(w) Rm, g(w) Rm1 , w Rn, Rm+ , p Rm1+ . (1) p 0 - ( ) ,
f(w) D(p) = {w Rn | g(w) p, w }, [1]
f(wy) ParetoMin{f(w) | w D(p)}. (4) f(wp) , - () K(f(wp)) f(wp), w
p D(p), .. K(f(wp)) = {f Rm | f f(w)} - f(D(p)) f(wp), .. K(f(w
p))
f(D(p)) = f(wp). f(wp) - . , (1)-(3) ,
wp , - f(wp) -
( p 0). , 0 p 0 , - , -
w D(p), f (w) Rm+ p
Rm1+ . - , 0, p 0 f (, p), p(, p). - ,
(1)-(3).
1
( 05-01-00242)
( -2240.2006.1)
6
2 -
,
- ,
- .
, (1)-(3) p 0 - (1) .
, f(w) Min{, f(w) | g(w) p, w }, (5)f(w) , 0, 0. (6) D = {w | g(w) p, w } (5) , f f(D) = {f = f(w), w D} f(w), w D - . - f(D) f , " " " ": fi = fi(w
) =min{fi(w) | w D}, i = 1, ...,m. wi , w f(w) " " f . , f 0, (5) , .. i 6= 0 , f < 0, . f , i, i = 1, 2, ...m , - . ,
.
. (5),(6) ,
fi(x), i = 1, 2, ...,m fi(w)
fi(w) > 0 w (7) , i = 1. (5) -
f(w) D. ( ) . , -
f(w) -, , f, f = f(w) w D f(w), , f , f f = f(w), w D, .. , f f 0. , -
-
.
(6) , -
f(w) , = 0, (8)
f(w) 0. (9)
7
, 6= 0, (8)
= f(w). (10)
(8) = 0, (9) f(w) 0. , f(w) ,.. fi(w) 0, i = 1, 2, ...,m, i 6= 0, i = 1, 2, ...,m. , . , (7) (6)
(10).
(5) -
,
( ) (
). w, f(w) ( ). () -. , -
(5),(6) ()
(10).
.
(6) ,
( ) -
argmax{, f(w) 12 | 0}. (11)
(5),(6)
, f(w) Min{, f(w) | g(w) p, w }, (12)
= argmax{12| f(w)|2 | 0}. (13) ,
.
3
, 0 (1)-(3) - .
, f(w) = (w), (1)-(3)
w = Argmin{(w) | g(w) p, w }, (14)
g(w) p, p p 0, p 0. (15)
8
(w) : Rn R, g(w) : Rn Rm1 - , Rn- , p Rm1+ . D(p) = {w | g(w) p, w } - p 0. p 0 -. -
( )
- () p 0
f(p) = min{(w) | g(w) p, w }, (16)
f(p), p 0 - [2]. :
1. f(p) - , - .
2. f(p) - , f(p) - - p Rm1+ . p , , , -.
3. f(p) p (14) p - .
(14) L(v, p) = (w) +p, g(w) p, w , p Rm1+ ( p ). , p 0 (14) ( ) , , p - w, p .
L(w, p) = (w) + p, g(w) p w , p Rm1+ . (17)
f(p)
f(p+4p) f(p),4p 0 p 0,4p 0.
f(p) - , .. p : Rm1+ Rm1+ . - . ,
-
p 0, f(p) : Rm1+ Rm1+ . p 0 - (14),(15).
. ,
(14), (15) -
, ,
p 0. -
w ,
9
f(p) (14). , (- p) , () , , .. -
( ) ,
-
. -
. - , ,
-
, ,
. , f(p), - p r(p), p. , p 0, f(p) = r(p). , () () .
, -
. ( p) ""f(p) r(p). -
f(p) r(p), (), .. [3]. r(p) r(p), - , .. f(p) r(p) f(p). (14),(15). L(w, p, y(s)) M(w, p, p)
M(w, p, p) = (w) + p, g(w) (1/2)p w , p Rm+ . (18) w ( -) p Rm1+ . , , . w p Rm1+ .
(w) + p, g(w) (1/2)p (w) + p, g(w) (1/2)p (w) + p, g(w) (1/2)p w , p Rm1+ . (19)
p Argmax{p, g(w) (1/2)p | p 0}. (20) , (19) (20) - p, w -
w Argmin{(w) | g(w) p, w }. (21)
4
(1)-(3), -
. (11) (20) (1)-(3)
10
, f(w) Min{, f(w) | g(w) p, w }, (22) argmax{, f(w) (1/2) | 0}, (23)p argmax{p, g(w) (1/2)p | p 0}. (24) (22)-(24) -
[4]:
()
wn = argmin{12|w wn|2 + (n, f(w) n+ pn, g(w) pn) | w }, (25)
()
n+1 = argmin{12| n|2 (, f(wn) (1/2) | 0}, (26)
pn+1 = argmin{12|p pn|2 (p, g(wn) (1/2)p | p 0}. (27)
wn+1 = argmin{12|w wn|2 + (n+1, f(w) n+ pn+1, g(w) pn) | w }. (28)
1 (1)-(3) ,
f(w),g(w) , - , wn, n, pn (25)-(28) > 0, , .. wn, n, pn w, , p n w0, 0, p0. , ,
.
1. .. , .. , .. . -
. .: , 1986.
2. S. Zlobec. Stable Parametric Programming. Dordrecht -London.: Kluwer Academic
Publishers. 2001.
3. .. . .
. .5. .: , 2005. . 148156.
4. .. . -
// . .
. . 1995. . 35, 5. . 688704.
, . .. -
, . 40, , 119991, . tel: (7-495) 135-
42-50, fax: (7-495) 135-61-59. E-mail: [email protected]
11
,
,
. .
-
. -
-
() ( ) (. [1-3]).
( 1904.) -, ( )
-. , , -
, (1874.) [2,3,6].
-
. Ax 0, .. A1. :
Ax 0 Ax y = 0, y 0 x = A1y =nj=1
ajyj, y 0,
.. Ax 0 {aj j 1, n} A1. , {x =nj
ajyj, yj 0} = C, C - Ax 0 Ax 0. , x 6= 0. m n Ax 0 (. [4]).
.
X Y Rm Rn ; A = (aij)mn X. v = max
XminY(x,Ay)
X ( ). v = minY
maxX
(x,Ay) Y
(A). v ( -
). , A Y . Y : w = max
YminX
(x,Ay) -
Y , w = minX
maxY
(x,Ay) X. C
w A X X., {v, v} {w,w} - max, min . v+ = max{c, x | Ax b} - v+ = max
xminu0
{(xu) = (c, x) (u,Ax b)}. - (.. )
v = min{(c, x) | Ax b} . , v+ x 0 max
x0, -
v x 0. (x 0) : v+ b;
12
, b. : max{(b, u) | uA = c, u 0} min{(b, u) | uA = c, u 0}. , , : max min . . 1. v v+ . (xu) Ax b. . Uk ={u Rm |
mj
uj = k, uj 0} (k > 0) : v+mink
maxx
minUk
(x, u).
, , ,
v = mink>0
max{(c, x) + kt | Ax+ tem b}.
em = (1 . . . 1) Rm. v+ v = max
k>0minx
maxUk
(x, u) = maxk>0
min{(c, x) +ht | Ax+ tem b} .
max{(c, x) | Ax em} = k min{(c, x) | Ax em} = k. f(k) ( max) v. v = min
k>0f(k). v = max{(c, x) | Ax b}. 2. k k f(k) k k k, v , v = v = f(k) k =
ni=1
ui, (u1 . . . um)
v. (ai,x) bi, i 1,m
v+ = maxx
mini{(ai, x) bi}, v = min
xmaxi{(ai, x) bi}
v+ = max{t | Ax b tem}, v = min{t | Ax b tem}.
v+ v (v+ > 0) (v+ < 0) Ax b. Ax b.
v+ = max{(b, u) | uA = 0, (u, e) = 1, u o}v = min{(b, u) | uA = 0, (u, e) = 1, u o} v v+. - [2,3]. , v+ v (x, u) = (u,Ax b) (ue) = 1 u 0 .
13
cj (j 1, s) bj (j 1, k), .
L : v = max
min
maxx
{(sj
jcj, x
)Ax kj
bj,
}
kj
j = 1 =sj
j, j 0, j 0. , L
v = max
{x0|Ax
kj
bj, j, x0 (cj, x) (j 1, s),kj
j = 1, j 0}.
L min
max
minx{ . . . } .
()
c cj j 1, s Ax b:1) x (c1, x), . . . , (cs, x). - C(x) = ((ci, x)/(cj, x))ss - ( )
. , , - . -
Ax b.2) Ax = b, x 0 s
x =s
j=1
xj xj 0 : (sj = 1 j 0).
C : minj
maxxj
{sj(cj, xj)
sAxj = b, xj 0
}.
: n = 1 {x1 = 1 x1 0} 1/c1, . . . , 1/cs . C
minj
maxxj
{sj
jxjcj
sxj 1 xj 0
}.
: j = ci/scj -
: (EA)x = c, x 0, A = (aij) 0, E nn, x, c 0 Rn. , c > 0 x 0 (aij i- j- ). [5]
(, A > 0).
14
hi(x) =nj=1
aijxj i- xT = (x1, . . . , xn)
(i 1, n);hi(x)/xi () i- x;maxi{hi(x)/xi} x;
mini{hi(x)/xi} x;
h = minx>0
maxi{hi(x)/xi} - ;
h = maxx>0
mini{hi(x)/xi} . , x0 > 0 (Ax0 = 0x0, 0 < 0 < 1). h = 0 =h = hi(x
0)/x0i , i 1, n. . ki = 1/x
0i , i 1, n K ki > 0. A(K) = (aij) = KAK
1; e =
(e1, . . . , en) A = (aij); e= (e
1, . . . , e
n) ( ) ei = kie
i.
aij = aijki/kj .. A(K) e.
A 1/x0i , - 0 < 1, .
, .
-
1) (EA)x 0 2). (EA)x 0 (A-). - - 1) C = {y = (E AT )u, u 0}. (2) K = {y = (EA)1x, x 0}. 3. A K C. -, .. (E A)1 0, K Rn+, (E AT )u = yu 0 y Rn (AT -). : (E AT )u = (E A)1x u 0, x 0, x 0. (EA)(EAT )u = x u 0 x 0. A = A+ AT AAT . 4 ( A). A-, 1)x 0 (E A)y = x, y 0;2)(EA)1 = BTB > 0;3) A 0, A ;4)E A . B = E + AAT , A = A+ AT . 5. A , BxAx c, x 0 , .. c 0; B 0 ,A .. A . 1) ATx A1y = 0, x 0 y > 0 ;2)AATx = y, x 0, y > 0 ;3) - Ax 0
Ax 0 .
15
.
3) .
. ,
AAT A + AT . aij = aijaji i- - i- j. A = (aij) = A AT . A . 6. A , A (E A) - .
, , i- A i- A2, : (E A) A .
, , -
, (-
) ,
. -
, (
-).
( 07-01-00399) -
( -5595.2006.1).
1. .. . . . , 1998, 248 .
2. . .. , 1959, 470 .
3. .. . . , 1968, 468 c.
4. . . .: , 1973, 470 .
5. P. . .: , 1989, 656 .
6. .. . . .: , 1982, 152 c.
, ,
. ., 16, , 620219, ,
. (343)375-34-28, (343)374-25-81, E-mail: [email protected]
16
. .
( )
. -
-
, () ,
. -
,
.
, , -
-
.
,
, .
,
.
,
.
:
1. .
2. , ,
.
3. ,
,
.
, , , -
,
, -
, -
.
-
-
-
:
max(xi),(xij)
{iI
fixi +jJ
iI
djxij jJ
iI
dj zij
};
iI
xij 1, j J ;
17
xi xij, i I; j J ;xi +
lji
xlj 1, i I; j J ;
xi, xij {0, 1}, i I; j J ;(zi), (zij) :
max(zi),(zij)
{iI
gizi +jJ
iI
djzij
};
iI
zij 1, j J ;
zi zij, i I; j J ;xi + zi +
lji
xlj 1, i I; j J ;
zi, zij {0, 1}, i I; j J.
dj j J ;fi i I -;
gi i I -;
j I, j J . i j l, j i l. i 4j l , i j l i = l.
-
.
(0, 1) w = (wi)(i I) I0(w) = {i I|wi =0}. j J ij(w) i0 I0(w), i0 4j i, i I0(w). I0(w) = , ij(w) i0 I, i0
u :
minu
{g(u, y) =
iI
giui +jJ
dj
iji0(y)ui
};
ui {0, 1}, i I.
(0, 1) y u. (0, 1) y, - u (0, 1) y u , f(y, u) f(y, u). , ,
( ) (0, 1)y, y u - (y, u) f(y, u).
-
(0, 1) y (0, 1) ys, s = 1, . . . , S, - -
. ys us - (ys, us), s = 1, . . . , S, (y0, u0) f(y, u). y0 -.
ys (0, 1) ws, u1, . . . , us1 . . -
. , -
, -
.
060100075.
1. .. -
. : , 2005.
. . . . 4,
, 630090, , . (383) 333-28-92, (383) 333-25-98,
E-mail: [email protected]
19
. .
-
, -
.
- -
.
1. , , -
(Q,), Q - , - - - (. [1]). , , [1]
, Q - d, = B, B - - (Q, d).
(. -
3 ). K- [2], [1,3,4].
, (Q, d) - d, B - -. V v : B R, v() = 0. - [4,5] = (Q,B, v) , Q -, e Q, B - ., v(e) e. , , () v.
, -
v(Q) Q . , (v), - 1953 . -
. , .
( ., , [4]) ,
v = f , f - [0, 1], - - B., - C(v). - v(Q)): C(v) e B. . . [6] , ,
, ,
. C(v) .
2. ,
. e - B. H(e) -
20
B- e H = eBH(e)., = {ei}m1 H v V v() = v({ei}m1 ) v,
v() =
(1)m||v(iei),
= {1, . . . ,m}, = {ei}i, || - . v().
1 [3]. vo = sup { |v()| H(Q)} v. , v V , vo
rV , o o , .. (o)- - o. -
rV, (. [7]).
4 [3]. v rV n, n + 1 : {ei}n+11 H v({ei}n+11 ) = 0. - rV n.
5 [3]. rpV = n=1rV n. rpV () .
6 [3]. , v rV n n, rV n1 (.. |v| |u| = 0 u rV n1). n rV (n).
rV n, rV (n) rpV .
1. rpV (-) rV (.. v rpV |v| o |u| u rpV ). 2. n 1 rV n rV (n) - () rV .
2 , , v rV m 1 v(m) rV (m):
v(m) = sup{u rV (m)+ | v+ o u} sup{w rV (m)+ | v o w};
( ) v rV+ v(m) rV (m)+ , rV
(m)+ = (rV
(m))+ - 1
rV (m). , v rV v(m), m = 1, . . . , - .
v rV, , v(m), v.
7 [3]. v rV () -, : v =
m=1 v(m), - o. rV raV.
-
, [3] -
, ..
1. ,
, .
1
W V rW+ =W rV+.
22
1 ( ), -
(
o) . -
.
3. , n- . v rV n - v B[n] B, . e B e[n] = { e
| | n}, , , | | . 8 [3]. Q[n], - {e[n] | e B}, B[n] ( n) B.
e[n] H: = {e1, . . . , em} H
[n] = { Q[n] mi=1ei, ei 6= , i = 1, . . . ,m}. B[n] .
3 [1]. B[n] - [n] ( H). E B[n] - H(Q) (, E) , ( ):
E =
(E,)()[n].
v - v B[n], -
v(E) =
(E,)v(),
(E, ) E ( v 3). - v Q, [3], v - - v - B
[n], - B[n] ( {f1, . . . , fm}[n] -, {fi}m1 H, fi F, i = 1, . . . ,m; . [3] ). [3], B[n] . -
, d[n](, ) = min { , }, , - -
, Q[n], , d[n] d Q[n], - Q[n] , , (Q[n], d[n]) - . , .
4 [3]. B[n] - - (Q[n], d[n]).
23
, Q Q(n) = { Q[n] | | = n} Q(n) B[n] \ B[n], n = 2, . . . , , n 2 B[n] B[n]. f - I(Q,B)
B- , c - , 2 m n, , c(< x1 >) = x1 x1 R. fnc : Q[n] R
fnc () = c(< f(t1), . . . , f(tm) >), = {t1, . . . , tm} Q[n]. c ( n), fc = f
nc - c- f., , .
9. f I(Q,B), v rV n(B), c n. , f (v, c)-, Icv(f) =
fcdv =
Q[n] f
nc dv ;
fcdv (v, c)- f.
-
, :
1) (< x1, . . . , xm >) =m
i=1 xi/m,2) (< x1, . . . , xm >) =
mi=1 xi,
3) s(< x1, . . . , xm >) = max{xi i = 1, . . . ,m}.4.
[3] ( ,
. 1953., , , [5]). -
[4], W rV , v W T v W , T - (Q,B), v v (e) = v((e)), T , e B. , rV n, rV (n), rpV - ( , , -
-
). , Supp v
v : Supp v = {R B v(e R) = v(e), e B}. 10. W rV : W rV 1, :
A1. (v) o 0, v W rV+;A2. ( v) = (v), T , v W ;A3. (v)(R) = v(Q), R Supp v, v W.
v(Q) (., , [5]), - - - v(Q), e B. C(v) v
C(v) = { rV 1 (Q) = v(Q), (e) v(e), e B}.2
, m - < x1, . . . , xm > xi R.
24
C(v) 1962 ... [6]. , , -
: v V , e, e B v(e e) + v(e e) v(e) + v(e). , -
Hv v, -
Hv(f) = sup {fd
C(v)}, f I(Q,B), .
11 [1]. rpV B Sh : rpV B R,
Sh(v, f) =fdv, v rpV, f B.
, , -
,
, -
.
12 [3]. -
rpV B P : rpV B R,
P (v, f) =fdv, v rpV, f B. [3], fV -: fV = {v V | R Supp v : (|R|
I(Q,B)n, -
P v (f, . . . , f) = Pv(f), f I(Q,B).
3. v rV (n). e B
(v)(e) = P v (e, Q, . . . , Q).
,
-
.
4. v rpV
Hv(f) =fsdv, f I(Q,B).
, -
, ( -
, , ..).
05-02-02005a 07-06-00363.
1. .. . -
. // . 1998. . 1, 2. . 24-67.
2. .. . . .: ,
1961.
3. .. . a . // -
. 1975. . 16(33). . 99-120.
4. . , . . . .: ,1977.
5. . . . .: , 1974.
6. .. . n . // , . ., ., .1962. . 13. . 141-142.
7. . , . . . .: ,1962.
,
. .. , - , 4,
, 630090, , . (383) 333-26-83, (383) 333-25-98,
E-mail: [email protected]
26
. .
1. . ,
Rk
.
NP-, ,
, -
.
-
.
, , , -
, , , -
. [2,4]
k- Rk,
. . x =x21 + . . .+ x
2k.
:
1: V = {~v1, ~v2, . . . , ~vn} - Rk m < n. V m, .
1 :
ki=1
( nj=1
vijxj)2 max; (1)
nj=1
xj = m; (2)
xj {0, 1}, 1 j n. (3)
2: V = {~v1, ~v2, . . . , ~vn} Rk m l, lm < n. V X = {~va1 , ~va2 , . . . , ~vam}, ai+1 ai l i = 1, 2, . . . ,m 1. k l , [2, 4].
2. 1 2.
1. [1] 1 2 NP-.
2. [1] 1 -
,
k18L2
O(nk2(2L+ 1)k1
),
27
L .
3. [1] k Rk 1 L = L(n), L(n) - n. k Rk - . ,
= k18L2
. L = (k18)1/2 -
O(nk2
(k 12
+ 1)k1)
.
4. [1] 2 -
,
k18L2
,
O(nk(k +m)(2L+ 1)k1
).
b - V.
5. [1] k Rk , 1 c
L = 0, 5kmb O(nk2(kmb)k1).
3.
1 c .
3.1. .
.
~B Zk1+ Bi =m
r=1 vi,r(i), 1 i < k, (i) , (vi1, vi2, . . . , vin) i-
(vij) ; B = {~ Zk1+ 0 ~ ~B}. 6. fmn(~) { n
j=1
vkjxj nj=1
vijxj = i, 1 i < k;nj=1
xj = m; xj {0, 1}, 1 j n}.
1 c -
S = max{k1i=1
2i + f2mn(
~) ~ B}.. 1 (1)-(3)
k1i=1
2i +( nj=1
vkjxj)2 max; (4)
nj=1
vijxj = i, 1 i < k; (5)
28
~ B; (6)nj=1
xj = m; (7)
xj {0, 1}, 1 j n, (8) .
A {fmn(~) ~ B}. < m,n; ~ > fmn(~)
{< , j; ~ > 1 j n; 0 ~ ~B} f,j(~).
f,j(~) < , j; ~ >, - ~v1, . . . , ~vj ~vj.
A.
:
a) f1,j(~) :=; f1,j(~) := j = 1, . . . , n ~ B.b) f1,j(~vj) := vkj j = 1, . . . , n.
c) f1,j(~) := max{f1,j(~); f1,j1(~)} j = 1, . . . , n ~ B. = 2, . . . ,m :
a) f,j(~) := j = , . . . , n ~ B.b)
~ B
f,j(~) =
{f,(~) j = ,
vk,j + f,j1(~ ~vj), < j n;
f,j(~) =
{f,(~) j = ,
max{f,j(~); f,j1(~)}, < j n.
A
7. k Rk , 1 c
O(mn
k1i=1 Bi
).
3.2. ~v1, . . . , ~vn - .
A 1 ~v1, . . . , ~vn .
Bi =m
r=1(vi,r(i) vi,r(ni+1)), 1 i < k,
~B Zk1+ ; B = {~ Zk1+ | 0 ~ ~B}; bi = min{vij |1 j n} 1 i k.
29
vij = vij bi, (vij) c . A 1 - (vij), (~v1, . . . , ~vn) :
S = max{k1i=1
(i mbi)2 +(fmn(~)mbk
)2 ~ B}., 6
. ,
~B b ( V ):
8. k Rk , 1 c
O(mn(mb)k1
)( ) O
(mn(2mb)k1
)(
).
.
1) -
2.
2) 1 2
Rk.3)
( 04-77-7173) .. ,
.. .. [1].
4)
.. [4].
( 05-01-00395, 07-07-00022, 07-
07-00168).
1. . . , . . , . . , . . . -
// . .
, 2, 2007, . 14, N 1.
2. .. , .. , .. , .. . -
-
// . . . .
2006. T. IX, N(25). . 5574.
3. . , . . M.:
. 1982.
4. .. , .. , .. . -
- -
// . . . . 2002. . V, N 2(10). . 94-108.
, . .. ,
. , 4, , 630090, , . (8-383-3) 33-21-89,
(8-383-3) 33-25-98, E-mail:[email protected]
30
. . , . .
(), , -
. (-,
, ) -
. - -
, . -
,
.
. ,
. -
, -
.
, -
. ,
, -
.
, , ,
. -
-
. -
MATLAB.
(
) ( ).
P-IV 2.6 -
. .
(, CPLEX) -
MATLAB
-
. .
8
30 .
06-01-00547
-2240.2006.1.
, .. ,
. 40, , 119991, , . (495) 135-00-20. E-mail: [email protected]
, .. ,
. 40, , 119991, , . (495) 135-61-61. E-mail: [email protected]
31
. . , . . , . .
-
(),
-
[1].
G = (V,E) V - E. (i, j) E lij, - () cij, rij Qij. S. s = 1, . . . , S V s V . -
. i V s csi . 0
s, -
rs0. [2].
T 0. Pk 0 k T , Tj T j, Cj =
(i,j)Tj cij +
iTj ci Tj. (i, j) T
dij = rij
(cij2+ Cj
). (1)
0 k T tk = r0C0 +
(i,j)Pk dij.
s T s, V s, (i, j) E Qij , . -
NP- [3].
.
MAD s Qs -. - Qs, s = 1, . . . , S, , -
.
.
.
,
, , -
. -
.
32
.
, ,
.
MAD
G = (V , E ), V . (i, j) E dij (1), G
, -
q, MAD.
MAD.
MAD
0. T = (0, ), t0 = 0. 1. (i, j) = arg min
(u,v)E; uT, v /T{tu(T {(u, v)}) + duv},
tu(T {(u, v)}) = tu(T ) + r0(cuv + cv) +ePu
re(cuv + cv).
T = T {(i, j)} tk, k T . j T , tj = ti+dij,
tk T . j , k T , - k T , - .
.
T , 1.
MAD ,
.
G, q dij. - G , , , G . dij - Cj Tj. , dij , - MAD .
MAD , -
dij . q, .
, . . Qs. s - Qs. i I = E G j J = Ss=1Qs . aij = 1, i j, aij = 0 . xj j xj = 1, j , xj = 0 . :
iI
(max
{0,jJ
aijxj qi})2 min
xj{0,1};
jQsxj = 1, s = 1, . . . , S.(2)
33
qi, i I. ,
(2). -
.
f(y) =
iI(max{0, yi qi})2, yi =
jJ aijxj. 0. , , xj = 1/|Qs|, j Qs, s = 1, . . . , S. yi =
jJ aijxj, i I, L .
1. gi = max{0, yi qi}, i I f(y) = iI(gi)2. - f(y),
f(y) f(y) +f(y)(y y) = f(y) +iI
2gi(yi yi),
y . ,
iIgiyi =
iI
gijJ
aijxj =jJ
(iI
giaij)xj min
x.
s.
minx
jJ
(iI
giaij)xj =
Ss=1
minjQs
(iI
giaij).
GY =
iI giyi;GAj =
iI giaij, js = argminjQs GAj GZ =S
s=1GAjsGY . f(y) f(y) + 2GZ. L = max{L, f(y) + 2GZ}. f(y) L max{1, L}, > 0 , .
= (j)
j =
{1 xj, j = js;xj, j 6= js; j Qs, s = 1, . . . , S;
z = (zi) zi =
jJ aijj, i I. , x x = (xj), xj = 1, j {j1, . . . , jS}, xj = 0 . t (0, 1] h(t) =
f(y+ zt) =
iI(max{0, yi qi+ zit})2, r(t) =
iI(gi+ zit)2. , h(0) = r(0), h(0) = r(0) h(t) r(t) t [0, 1]. t = argmin{r(t)|t [0, 1]} = min{1,GZ/ZZ}, GZ = iI gizi ZZ =
iI(zi)2. y , h(0) < 0 , ,
r(0) < 0. t > 0. h(0) > h(t) h(0) =r(0) > r(t) h(t), . . t . , xj = xj + j t, j J ; yi = yi + zit, i I, 1.
34
. -
O(|I| |J |), O(1 ln 1).
-
, 1900 3500 .
2 20. G 4100 7400. .
-
05-01-00395.
1. J. Hu, S. S. Sapatnekar. A Survey on Multi-net Global Routing for Integrated Circuits.
// Integration, the VLSI Journal. 2002. V. 31. P. 149.
2. J. Rubinstein, P. Peneld, M. A. Horowitz. Signal Delay in RC Tree Networks. // IEEE
Trans. on CAD. 1983. V. 2. P. 201211.
3. M.R. Kramer, J. van Leenwen. Wire-Routing is NP-Complete. // Technical Report
RUU-CS-84-4, Department of Computer Science, Rijksuniversiteit Utrecht. 1982.
, . .. ,
. , 4, 630090, , , . (8-383) 333-37-88, (8-383)
333-25-98, E-mail: [email protected]
, . ..
, . , 4, 630090, , , . (8-383) 333-21-89,
(8-383) 333-25-98, E-mail: [email protected]
, . .. ,
. , 4, 630090, , , . (383) 333-37-88, (8-383) 333-
25-98, E-mail: [email protected]
35
..
,
. -
[1]. ,
,
, -
[2].
-
.
(
) . -
.
j = 1, . . . , n n 2. Rn+, R
n++ -
.
. -
Q(P, v, U) = argmax
U(Q) : Q Rn+,nj=1
PjQj = v
, (1) P Rn++, v 0 - U . U Rn+ , - Q(P, v, U) (1) v., Rn+ , [3].
U , P Rn++, v 0, 0
Q(P, v, U) = Q(P, v, U).
U, U , P Rn++, v 0
Q(P, v, U) = Q(P, v, U).
U i i = 0, . . . , k k 2. i = 1, . . . , k , i = 0 . , P Rn++, vi 0, i = 1, . . . , k
Q(P,ki=1
vi, U i) =ki=1
(Q(P, vi, U i).
, . ..
, , 130, , 664033, , . (8-3952) 42-88-27,
(8-3952) 42-67-96, E-mail:[email protected]
36
, -
, . [4, 5]
[6]
1. -
, -
.
C -
-
. ,
-
. ,
. ,
- .
. -
. Itp , Itq
j = 1, . . . , n, t , t > . - ,
[7-10]. ,
Rn++:
Itp = f(P , Q , P t, Qt), Itq = (Q
, P , Qt, P t).
P , Q , P t, Qt , - , t, . Rn++ R
1+.
-
, -
. f , [8], .
1. ( ) -
Itp Itq =nj=1
P tjQtj/
nj=1
P j Qj . (2)
2. () -
()
maxj=1,...,nPtj /P
j Itp minj=1,...,nP tj /P j , (3)
maxj=1,...,nQtj/Q
j Itq minj=1,...,nQtj/Qj . (4)3. , t, l
Itp I tlp = Ilp , (5)
37
Itq I tlq = Ilq . (6) [11, 12]
2. (2) (6) . f , . -
( ) -
. -
. [13] , .
[14].
P (s), Q(s) - s Rn++, - j = 1, . . . , n. - , t
Dtp = exp t
nj=1
Qj(s)
V (s)dPj(s), D
tq = exp
t
nj=1
Pj(s)
V (s)dQj(s),
V (s) =nj=1
Pj(s)Qj(s)
t.
Dtp Dtq = V (t)/V ()
. -
. , -
[15],
P (t) = P (), Q(t) = Q()
Dtp > 1, Dtq < 1.
, ,
[, t] 1. "" - , ,
. [, t] , .
[10, 15, 16] , -
, , ,
(1). , [15], ,
, -
.
38
U , P (s) - Rn++, v(s) s [, t]. s [, t] -
Q(s) = argmin
U(Q) :nj=1
Pj(s)Qj = v(s), Q Rn+ . (7) P (s), Q(s) s [, t] - U , Q(s) (7) v(s) > 0 s [, t]. U - , ,
, P (s), Q(s) P (s), Q(s), s [, t] , ..
P () = P (), Q() = Q(), P (t) = P (t), Q(t) = Q(t),
Dtp = Dtp , D
tq = D
tq .
: Dtp , Dtq P (s), Q(s); D
tp , D
tq
P (s), Q(s).C
3. U ,
.
06-02-00266 .
1. . . . , . .: , 1948.
2. . , . . . .: , 1971.
3. . . . .: , 1983.
4. .. . : -
? .: ,
1997.
5. .. .
// -
. . .: , 1994.
6. .. . . .:
, 2000.
7. .. . . .: , 1963.
8. . . . .: , 1928.
9... . ( -
). .: , 1992.
10. . . . .: , 1980.
11. .. . . .: ,
39
1991.
12. .. . // -
, 1993. 2.
13. Devisia. Economic rationelec. Paris, 1928.
14. .. . //
, 1929. 9-10.
15. .. . : ,
1996.
16. .. . . .: ,
2004.
40
. .
1.
U A 2U , : A A, A A A A. S = (U,A) U . A , - U . D. , D : D D, D D D D. A, D S, S = (U,A) S = (U,D) , .
S -, .
:
max{f(X) : X B}, (1)min{f(X) : X C}, (2) B , C S = (U,A) = (U,D), f : 2U R+ .
-
,
, ,
p-, , k- - . ,
NP-.
, (1)
(2), , -
. , f : 2U R+ , - X, Y U f(XY )+f(XY ) f(X)+f(Y ), , . -
f . , f() = 0 , . (1)
GA ( ).
0. X0 , 1. i (i 1). xi / Xi1, f(Xi1 {xi}) = max
x/Xi1,Xi1{x}A
f(Xi1 {x}).
Xi Xi1 {xi}, i+ 1. xi / Xi1 , GA Xi1..
(2)
.
41
GR.
0. X0 U , 1. i (i 1). xi Xi1, f(Xi1 \ {xi}) = min
xXi1,Xi1\{x}D
f(Xi1 \ {x}).
Xi Xi1 \ {xi}, i+ 1. xi Xi1 , GR Xi1..
, GA , GR -
, . . GA (1), GR (2).
2.
.
S = (U,A) = (U,D) W U . W , W . W , W .
cA(S) = minWU,W /A
rmin(W )
rmax(W ), cD(S) = max
WU,W /D
gmax(W ) |W |gmin(W ) |W | ,
rmin(W ) rmax(W ) W , gmin(W ) gmax(W ) W , . , cA(S) 1 cD(S) n 1 S. S , cA(S) = 1, -, cD(S) = 1. rmax(U) , gmax() - . p- S = (U,A) p- S = (U,D), A = {A U : |A| p}, D = {D U :|D| p}, p , p < |U |. .
1 (-) [5, 12]. S = (U,A) - U . GA (1) S f : U R+ , S . -
. (2) , -.
2. S = (U,D) . GR - (2) S , S . [6].
1, ,
(1)
. [8, 9] GA
:
f(GA)
f(OPT ) cA(S), (3)
42
OPT (1). [6] GR -
(2) :
f(GR)
f(OPT ) cD(S). (4)
-
, (3) (4), GA GR
(1) (2) .
, [2] , (2) -
,
-
, (2).
3.
(1) , -
.
,
. -
(1) ,
, -.
U , f : 2U R+. V U |V | 3 k |V | 2. X = {x1, . . . , xk} V GA- V ,
f({x1}) f({x}) x V ,f({x1, x2}) f({x1, x}) x V \ x1,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .f(X) f({x1, . . . , xk1, x}) x V \ {x1, . . . , xk1}. f : 2U R+ GA-, V U |V | 3 GA- X ={x1, . . . , xk} f(V \ xk) f(V \ x) x V \X., GA-. - (1).
3. S = (U,A) . GA - (1) S GA-- f : 2U R+ , S . -
(1) .
4. f : 2U R+ . - GA (1) f S = (U,A) , f - GA-. (2): -
(2) , GR ( GR- ), 2 (2)
.
43
4.
(1), B p, f : 2U R+ , f() = 0. (1) p- f(X) =
jJ
maxiX
cij, (cij) n m U J . [4] GA p- :
f(GA)
f(OPT ) 1
(p 1p
)p e 1
e 0, 63. (5)
[10] (5) -
p- . [3] :
f(GA)
f(OPT ) 1
c
(1
(p cp
)p),
c [0, 1] f , - . c :
c = maxxU,
f({x})>f()
f({x}) f() (f(U) f(U \ {x}))f({x}) f() ,
c = 0 , f ( f() = 0). (2), C - p, f : 2U R+ ,f(U) = 0. p- - f(X) =
jJ
miniX
cij, (cij) nm U J . , f() = max
X,YU,XY=
{f(X) + f(Y ) f(X Y )}
p- . , GR
. , ,
, p- , , ,
P = NP [11]., (2)
GR.
f
s = maxxU,
f({x})
[1] s < 1 - GR
p- :
f(GR)
f(OPT ) 1
t
((q + t
q
)q 1
) e
t 1t
,
q = n p, t = s/(1 s). [7] (2) -
p. GR
p- (cij).
1. .. .
// . . . C. 1. 1998. . 5,
N 4. C. 45-60.
2. .. , .. . // .
. . C. 1. 2003. . 10, N 3. C. 54-66.
3. M. Conforti, G. Cornuejols. Submodular set functions, matroids and the greedy al-
gorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds the-
orem // Discrete Appl. Math. 1984. V. 7, N 3. P. 251-274.
4. G. Cornuejols, M.L. Fisher, G.L. Nemhauser. Location of bank accounts to optimize
oat: An analytic study of exact and approximate algorithms // Management Science.
1977. V. 23. P. 789-810.
5. J. Edmonds. Matroids and the greedy algorithm // Math. Programming. 1971, V. 1,
N 2. P. 127-136.
6. V. Il'ev. Hereditary systems and greedy-type algorithms // Discrete Appl. Math. 2003.
V. 132, N 1-3. P. 137-148.
7. V. Il'ev, N. Linker. Performance guarantees of a greedy algorithm for minimizing a
supermodular set function // European J. Oper. Res. 2006. V. 171, N 2. P. 648-660.
8. Th.A. Jenkyns. The ecacy of the greedy algorithm // Proc. 7th S-E Conf. Combi-
natorics, Graph Theory and Computing. 1976. P 341-350.
9. B. Korte, D. Hausmann. An analysis of the greedy heuristic for independence systems //
Annals of Discrete Mathematics. 1978. V. 2. P. 65-74.
10. G.L. Nemhauser, L.A. Wolsey, M.L. Fisher. An analysis of approximations for maxi-
mizing submodular set functions I // Math. Programming. 1978. V. 14. P. 265-294.
11. G.L. Nemhauser, L.A. Wolsey. Integer and combinatorial optimization. New York.:
John Wiley & Sons, Inc., 1988.12. R. Rado. Note on independence functions // Proc. London. Math. Soc. 1957. V. 7,
N 3. P. 300-320.
,
, . , 55, , 644077, , .
(3812) 22-56-96. E-mail: [email protected]
45
NP-
. .
, -
, -
-
.
,
( ) (o-line) -
. -
, ()
, -
(); . [1-3] .
,
q, - :
xn =mM
unnm(m), n = 0, . . . , N 1,
unnm(m) = 0, n nm 6= 0, . . . , q 1; (u0(m), . . . , uq1(m)) Rq,0 < (u0(m), . . . , uq1(m))
X X(n1, . . . , nM , U1, . . . , UM), -
.
() -
( X() Y X(),2I), . -
Y X()2 .
Yn = (yn, . . . , yn+q1), n = 0, . . . , N q + 1.1. . -
, Um = U = (u0, . . . , uq1), m M, .. ,
:
xn =mM
unnm , n = 0, . . . , N 1.
X = X(n1, . . . , nM , U) .1.1. ; .
: Y RN , U Rq M . : (n1, . . . , nM) M ,
mM(Ynm , U) max .
. , -
O[M(Tmax Tmin + q)(N q + 1)] = O(MN2), [4].1.2. ; .
: Y RN U Rq. : (n1, . . . , nM) , mM
{2(Ynm , U) U2} max .
, O[(TmaxTmin + q)(N q + 1)] = O(N2), [5].1.3. ; .
: Y RN , M q. : (n1, . . . , nM) M,
mM
Ynm max .
NP-. ,
O[M(Tmax Tmin + q)(N q + 1)] = O(MN2), [6].1.4. ; .
: Y RN , q. : (n1, . . . , nM) ,
1
MmM
Ynm2 max .
47
. , NP-.
2. -
-. -
:
xn =mL
unnm +
mM\Lunnm(m), n = 0, . . . , N 1,
L M, |L| = L. U , Um,m M\L, . X = X(n1, . . . , nM , U,L, {Um,m M\L}) .
2.1. .
2.1.1. . -
, Um {U : U Rq, 0 < U < },m M \ L. .2.1.1.1. .
: Y RN , U Rq, M L. : (n1, . . . , nM) M L ,
mL2(Ynm , U) +
mM\L
Ynm2 max .
2.1.1.2. , .
: Y RN , U Rq, L. : (n1, . . . , nM) L - , EYnm 6= 0 ( E ), m M \ L.2.1.1.3. , .
: Y RN , U Rq, M . : (n1, . . . , nM) M L ,
mL{2(Ynm , U) U2}+
mM\L
Ynm2 max .
2.1.1.4. .
: Y RN U Rq. : (n1, . . . , nM) L, , , EYnm 6= 0, m M \ L.2.1.2. . -
, U A Um A, m M \ L, A {U : U Rq, 0 < U < }, |A| = K, .. - () A . - .
2.1.2.1. .
: Y RN , A K, M L. : (n1, . . . , nM) M , L {Um A \ {U},m M \ L},
mL2(Ynm , U) +
mM\L
{2(Ynm , Um) Um2} max .
48
2.1.2.2. , .
: Y RN , A K, L. : (n1, . . . , nM) , L {Um A \ {U},m M \ L}, .
2.1.2.3. , .
: Y RN , A K, M . : (n1, . . . , nM) M , L {Um A \ {U},m M \ L},
mL{2(Ynm , U) U2}+
mM\L
{2(Ynm , Um) Um2} max .
2.1.2.4. .
: Y RN , A K. : (n1, . . . , nM) , L {Um A \ {U},m M \ L} , .
2.2. .
2.2.1. . -
1
LmL
Ynm2 +
mM\L{2(Ynm , Um) Um2} max,
. NP-
( NP- 1.3).
2.2.1.1. .
: Y RN , A K, M L. : (n1, . . . , nM) M , L {Um A,m M \ L}.2.2.1.2. , .
: Y RN , A K, L. : (n1, . . . , nM) , L {Um A,m M \ L}. . ,
NP- ( 1.4).
2.2.1.3. , .
: Y RN , A K, M . : (n1, . . . , nM) M , L {Um A,m M \ L}.2.2.1.4. .
: Y RN , A K. : (n1, . . . , nM) , L {Um A,m M \ L}.2.2.2. . -
1
LmL
Ynm2 +
mM\LYnm2 max,
.
2.2.2.1. .
: Y RN , M L. : (n1, . . . , nM) M L.
49
2.2.2.2. , .
: Y RN , L. : (n1, . . . , nM) L , EYnm 6= 0, m M \ L. NP- ( NP- -
1.3). .
, NP- ( 1.4).
2.2.2.3. , .
: Y RN , M . : (n1, . . . , nM) M L.2.2.2.4. .
: Y RN . : (n1, . . . , nM) L , EYnm 6= 0, m M \ L.
-
. -
, NP-, -
. -
,
.
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.: (8383) 333-3291, : (8383) 333-2598, e-mail: [email protected]
50
. . , . .
-
, , -
, [2, 4, 22, 23].-
NP- . -
.
, ,
. -
, ,
. -
(), p- ( ), ,
, [2, 4, 6, 1113, 16, 18, 24].
-
(). ,
p- , : . ,
,
[2, 10, 11, 13, 14, 20].
-
. ,
"" , ,
.
1.
. p- ( Pmin) . - I = {1, . . . ,m} , , J = {1, . . . , n}. . -
i- j- cij, i I, j J . p, ,
, -
.
: zi = 1, i- - , zi = 0, i I;xij = 1, j- i- , xij = 0 , i I, j J .
51
:
f(z, x) =iI
jJ
cijxij min (1)
iIzi = p, (2)
iIxij = 1, j J, (3)
xij zi, i I, j J, (4)xij, zi {0, 1}, i I, j J. (5) z = (z1, . . . , zm) (1)-(5), (2) (5).
cij dij i- j- . p- (Pmax): p , .
Pmin , , .. (2),
c0i , i I, :
iIc0i zi +
iI
jJ
cijxij min .
-
. , -
,
[11].
. -
-
, , c0i i- , cij j- , i- , i I, j J. j i, dij .
.
[6].
, -
, [3].
[15].
, NP-.
2.
Pmin. -
.
52
, F (k) k- , F (0) =.
D (1) = . k (k 1) 1. z(k) (k). - , : ,
F (k1), . 2. T (z(k)) - z(k):
iIjJ
cijxij min
iIxij = 1, j J,
0 xij z(k)i , i I, j J.,
, x(k), - xij . f(z
(k))
T (z(k)). F (k) = min{f(z(k)), F (k1)} - .
3. ( ):
(k)1 z1 + . . .+
(k)m zm (k)0 . (6)
(k+1) (k), - (6). .
(6) , :
a) z(k);b) z (k), f(z) < F (k). "a" D, "b" .
"a", , ,
.
, "a" "b", -
iIk0
zi 1, (7)
Ik0 = {i I : z(k)i = 0}. - [9]. , (7)
(k) z(k). , D Cpm.
53
(6) [17], -
. 1 z(k) - T (z(k)).
jJuj
iIjJ
wijz(k)i max
uj wij cij, wij 0, i I, j J.(8)
, (-
) u(k)j , w
(k)ij , i I, j J . , z(k) , :
iI
jJ
w(k)ij zi >
jJ
u(k)j F (k). (9)
(8) ,
z(k) . , .. -
, (k). [14] p- - .
.
3.
Pmin - , , , 1, M (M 2). C. , (9) -
1.
, D - . -
(7).
, -
, z , - [10].
p-, D . Pmin (m m)- , - 0, (,
m p+ 1m p ) > 0. , Pmin Cpm D -. Pmin , NP-. Pmax [10,24].
,
C,
54
c0i = 1, i I. z(k), , 1 [10].
[8, 21, 25]. , [7, 21]
L- - . , (
) . [25] , D (
), -
,
z(1), . . . , z(K) . [11, 13, 14] , -
, p- . -
L- , , - .
OR-Library [19], TSPLIB [26], [5],
.
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56
. .
-
. ,
, . -
. -
T = {T1, . . . , Tn}, h(Tj) w(Tj) Tj, C = {C1, . . . , Cm} , wi i- . -
T ( , - ), ,
-
.
, NP-,
m [1]. .
HO(T,C) HA(T,C) , A. , 1.
A
RA = supT,C{HA(T,C)/HO(T,C)},
RA = limk
supT,C{HA(T,C)/HO(T,C) | HO(T,C) k}.
. -
, T , - , -
T C - - .
-
. [2].
[3] -
, 1.7. [4]
,
, 1.69103, ,
. [5] ,
-
, 1.5401.
[21] A RA 1.5889.
57
-
m , , m 1
1. [6] , -
.
2 1/m, m . [8] - 1.986
m > 70. 1.945, 1.923 [10] [7]. [11] ,
, 1.837
m. [7] 1.852.
. [12] , -
(
),
10.
e . - ...
1. A : RA e. r , 0 < r < 1.2. Ar , R
Ar 2er .,
[13-21].
-
. [4] -
. r (0, 1). , . -
R, k, rk+1 < h(R) rk rk. - ,
rk. - . , ,
( k), - , .
A(E) ,
E.
U([0, 1]) [0, 1]. - , hi wi [0, 1]. , wi, hi . - , ,
. ,
N (N ). E
E
(L
Ni=1
wi
)= O (f(N)) ,
58
L - , E {wi}.. [20] E f(N) = N log N, 0 < < 1, 0. A(E) :
= O
(N
(log N
1+ N1
)).
= N (1)/(2) log(/(2))N ,
= O(N1/(2) log(/(2))N
).
, , -
, FF (rst t): f(N) = N2/3, BF (best t): f(N) = N1/2(logN)3/4, (best on-line) BO: f(N) = (N logN)1/2 [16]. , A(FF) =O(N3/4), A(BF) = O(N2/3(logN)1/2) [19], A(BO) = O(N2/3(logN)1/3). [17].
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..
. -
NP-
[4]. -
(
theoretical computer science), ,
,
,
.
,
, . , ,
, ,
. -
. " P 6= NP, ..." - [1-3, 6]. , -
,
[9]
[14],
[2]. -
. , -
[4], .
, , .
, -
P 6= NP . -
, .
,
.
. , -
.
"" [7], -
NP- .
, -
. , -
. -
, ..
(. [2, 3, 17]). ,
-
61
"No free lunch"[21].
,
, .
, -
, .
, , -
, , . -
-
[2, 3, 17].
[2].
1 ,
. , -
.
1.
P 6= NP - [1, 17].
, () -
, . P 6= NP NP, -
[17].
[15]. , ,
. , -
, .
[18] . , -
,
P = NP. , .. . ,
P 6= NP . , -
. , P = NP ,
. , -
, NP.
P = NP. - , -
P = NP, , ,
NP- . -
[2], ,
-
. ,
, , P 6= NP. -
62
, , P = NP. [10] . , , -
?
, ..
.
, -
. ,
.
[10] ,
-
.
.
, , .
. , -
, .
,
. , , -
,
P 6= NP. , NP - . , -
.
-
, , -
, .
. -
1/2. pM(x) , M x. x . A [2]:1. , pM(x) > 1/2, x A, pM(x) = 0, x 6 A.2. , pM(x) > 1/2+ , x A, pM(x) < 1/2 0, x 6 A, > 0. RP , -
. BPP
,
. -
P. , P RP BPP . , RP =NP BPP = NP, P = NP, [2]. , RP 6= NP BPP 6= NP. , NP - .
1. P 6= NP, - .
, P 6= NP,
63
.
2.
NP-, ,
. -
, NP-, -
. -
.
k- [19], -
[19].
[5].
.
[12], P NP .
L, . (L, ) [6,8,12]. L. [6,8,12,13,19,20]. B = {0, 1}. B , .
1. , :B [0, 1], (x) 0 x B
xB(x) = 1.
2. f : B N , k, c > 0 ,
xB
f 1/k(x)
|x| (x) c.
3. L , ( ). (L, ) .
4. (L, ) AvgP , L . AvgP P - .
5. P-, - M , x B k 1
M(x, 1k) (x) 2k.
6. (L, ) DistNP, L NP P-.
64
DistNP
NP. P 6= NP DistNP 6 AvgP. .
7. (L, ) (L, ), - f , x L f(x) L q , y B,
(y) 1q(|y|)
xf1(y)
(x).
, , (L, ) AvgP, (L, ) . f , , x B
(x) (x)q(|x|) .
, L L [1].
8. (L, ) DistNP-, DistNP
.
[6,8,12,20], DistNP- .
6, [6]. -
DistNP-
(K,K) [8,12,20]:: (i, x, 1n), x B, i, n N: Mi x n ? K : i, x n
K(i, x, 1n) =
2|i|
|i|2 2|x|
|x|2 1
n2.
[20]. [13]
, . -
NP- ,
DistNP-.
NP- [17],
P- NP-
, DistNP-.
9 [13]. L -, q S : 1B 7 B , 1. S .2. x n S(1n, x) L x L.3. x n , n > |x|
|S(1n, x)| = q(n).
65
q S. S - , .
10 [13]. L -, E : B B 7 B - D : N B 7 B , 1. S D .2. x, p B E(p, x) L x L.3. |x1| = |x2|, |p1| = |p2| p1 p2,
E(p1, x1) E(p2, x2).4. |x1| = |x2| |p1| = |p2|, |E(p1, x1)| = |E(p2, x2)| |x1| < |x2|
|p1| < |p2|, |E(p1, x1)| < |E(p2, x2)|.5. x, p B D(|p|, E(p, x)) = p D(k, w) , x p , |p| = k E(p, x) = w.
D , .
2 [13]. L NP- , - - -. P- -
, L DistNP- .
DistNP 6 AvgP? , , ,
(K,K) - K .
3. DistNP 6 AvgP, - .
3.
, DTime(2O(n)) 6= NTime(2O(n)) P6= NP, RP 6= NP DistNP 6 AvgP [2,6]. 1 3 - .
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P-
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68
. . ,
, - . ( s, t ), NP. , - .
1. -
. , - [1].
1.1.
AS = (X, V, R; P, F, W), - :
X = (x1, x2,..., xn) ; V = (v1, v2, ..., vg) ; R = (r1, r2, ..., rm) ; P : V 2X , vV
P(v) X . PS = (X,V; P); F : R VPS2 , rR F(r)
V . F FS = (V, R; F). - VPS2 , PS, . , .
rR W : r 2P(F(r)) , rR - W(r) P(F(r)) , P(F(r)) PS, - F(r) V. , W WS = (X, R; W).
PS AS, WS - AS.
, AS = (X, V, R), , . :
P1(x) = {v: xP(v)}; F1(v) = {r: vF(r)}; W1(x) = {r: xW(r)}; ,
. PS = (X, V; P) WS0 = (X, R0, W0), WS1 = (Y1, R1; W1), ... ,
WSk = (Yk, Rk; Wk), PSWSWSWS kk kki 1111 ... :}{
69
k AS = (PS, WS1, ..., WSk; 1, ..., k), Yk Yk1 ... Y1 X i ( , )1 k ri Ri {ri1} Ri1, Wi(ri) Wi1({ri1}) &{ri1} - Wi1. , - WSi. k (X,V, R1, ..., Rk; F1, F2, ..., Fk) (X, V, R1, ..., Rk).
, P, F, W PS, FS, WS , , - AS.
- . , , , . , - xX rR, vV, xP(v) vF(r), xP(F(r)). , . .
, . , - , - , . - - , . , x1 x2 X V, vV x1P(v) x2P(v), r, rR x1W(r) x2W(r). AS.
AS , - PS, FS WS.
1.2.
S = (X, V, R; P, F, W) , vV P(v)= 2, rRW(r)= 2, rR F(r) V PS = (X, V). , PS WS S ,
F WS = (X, R) PS = (X, V). F(r) , F -
W. , F(r) - r, S (X, V, R; P, F).
S = (X, V, R) S = (X, V, R ) , , X X, V V, R R P F, , - P W. S S , X X, R R, W. - , , PS PS . , X X, V V, P.
70
2. 2.1. .
, - , :
PS = (X, V) AS. () ( - ), () S(Vi) L(Vi) PS, .
WS = (X, R) - PS = (X, V) ,
+)},({}{1
min )()(yxRir
i
jrFivi crlvS
,
(1)
{(x, y)} x, y WS. - i ri , l(ri) . , - WS, (1) .
, NP- - .
2.2 .
- S = (X, V, R), s(ri) s(vi) -,
)(:)(
irFivirirs . ( )
P, < .
}1,0{,
1)(:
1min)()(
=
=+
idicn
kkrFivi
iii
n
ii vsdrsc
U.
- . , , , .
2.3. .
, , [35]. , .
. . AS = (X, V, R; P, F, W), X =
(x1,..xn), V = (v1, v2,, vm), R = (r1, , rn), V(ri) ri. R R , :
71
x X, I(x) R, URrRir
i xrIrv
= .)( min,)( I(x) ( -
) x, I(r) ( ) r. , . , , . - , .
, 18 . , . . - ( ) - , .. .
. X V- (R-),
X V- (R-). - . , 18-. NP-.
2.4. .
AS = (X, V, R; P, F, W), Z(xi) xiX, l(vj) vj V, c(vj) vj.
=
)()()(
krFjVjk vlrl ri .
xi :
+=})({1
)()(),()},(),{(irFjv
jXjx
jjijii vcxZxxxxx ,
{zi} , mi xi xj, = )(),( kji zlxx , zk(xi, xj), (xi, xj) xi xj.
(x0) x0 - {(xi) | (xi, xj)} (x0, xj) ,
((xi), (xi, xj)) min (2) , x0
. , |,|)()(||)( PSixiWSi
PSocmiWS TxxTx ++
)( iWS x WS =(X, R), PSocmT PS = (X,V), || PSocmT . ||
PSixT ( ) PS
|| WSixT PS. WS
PS , |||| PSixWSix TT = .
.
72
AS = (X, V, R, P, F, W), - x0 AS , (x0)-min , WS , .
x0 AS, (2) - AS.
2.5. .
, , PS = (X, V), WS = (Y, R).
(vij) PS(vij) vij V; (rij) WS(rij) rij R; (vij) () vij V; (rij) rij R; (S) S.
S = (PS, WS; ), (S) k
vV )()( xrr
(S) S. - (S) .
1. .. . 6. ,1981. . 2648. 2. .. . . . 2006. 3. . . . .: , 1978. 4. ., . , . .: , 1978. 5. .. . .: , 1987. , , . , 6, , 630090, , . (383) 330-96-43. E-mail: [email protected].
73
. . , . .
, ,
. -
F (x) =12 ||(Axb)+||2, ( ) - Ax b, Amn , b = (b1, . . . , bm) x = (x1, . . . , xn) . x+ = max(0, x) x p+ = (p+1 , . . . , p
+n )
p = (p1, . . . , pn). -
[19]. , , (
-
), ( -
), -
(
), (
- -
, ) .
,
.
F (x) , - . , . . -
H(x) = (2F/xixj)nn , - .
. . [10, 11]
F (x, x0) =1
2(Ax b)TD0(Ax b) +
2||x x0||2,
x0 , , > 0 , D0 mm- Ax b x0, . . , i- 1, i- x0, 0 . - F (x) xk+1 = xk + kxk
xk = (ATDkA+ E
)1F (xk) = xk xk, E mm-, Dk mm- - Ax b xk, xk F (, xk)
74
, k ( 1), - (0, 1) ,
(, F (xk) = 1F (xk, xk))., -
H(xk) = ATDkA+E. - H(xk)x = F (xk), xk, , , , -
. -
H(xk), . , , . . , ,
.
-
, -
,
-
H = ATDA+ E (., , [1215]). , -
H. - -
A =
A11 A1,n
A22 A2,n.
.
.
.
.
.
An1,n1An1,n
. , H - (, )
H =
H11 H1,n
H22 H2,n.
.
.
.
.
.
Hn1,n1 Hn1,nHn,1 Hn,2 . . . Hn,n1 Hn,n
, . , -
[15],
.
.
- -
A =
A11
A22.
.
.
An1,n1An,1 An,2 . . . An,n1
.
75
H = ATDA+E, , - , . . . -
, H = ADTAT + E, , .
( ) -
H = ATDA+ E.,
.
, -
. , ,
, ,
.
. -
. -
, , -
-
. , --
,
, , -
H = ADAT , H = ATDA . -
. -
,
. -
.
.
-
-
-
min (c, x) : Bixi = bi (i = 1, . . . , r), Ax = b0, x = [x1, . . . , xr] 0.
c = [c1, . . . , cr] A = [A1, . . . , Ar], ci Rni ,Bi Ai mi ni- m0 ni- , m =
ri=0mi, n =
ri=1 ni. Ax = b0 .
(x) =1
2
ri=1
(||Bixi bi||2 + ||xi ||2)
{xk Arg min{(x) : Ax = b0, (c, x) = k },k+1 = k + 2
1k (x
k), k = 0, 1, 2, . . . .
76
-
-
, -
k (k > 0). 0. , , , . .
K xK . (x), - xs+1 = xs + sx
s (s = 0, 1, . . . ), s = 1 ( -
),
xs = xs xs. xs
(x, xs) =1
2
ri=1
(||Bixi bi||2 + xTi D(i)s xi)+ 2 ||x xs||2 - Ax = b0, (c, x) = k -
(BTi Bi +D(i)s + Enini) x
si + A
Ti y + sc = B
Ti bi + x
si (i = 1, . . . , r),
A1 xs1 + . . . + Ar x
sr = b0,
(c1, xs1) + + (cr, xsr) = k, D
(i)s = diag( sign((xsi )+1 ), . . . , sign((xsi )ni+) ), E - , s , > 0 - ( ).
-
. ,
. -
y = H10[ ri=1
AiH1i
(pi sci
) b0
], xsi = H
1i (pi ATi y sci), i = 1, . . . , r,
H0 =ri=1
AiH1i A
Ti , Hi = B
Ti Bi +D
(i)s + Enini , pi = B
Ti bi + x
si , i = 1, . . . , r,
s =
k + (ri=1
cTi H1i A
Ti )H
10 (
ri=1
AiH1i pi b0)
ri=1
cTi H1i pi
(ri=1
cTi H1i A
Ti )H
10 (
ri=1
AiH1i ci)
ri=1
cTi H1i ci
.
, r Hi nini H0 m0m0, Hi, i = 1, . . . , r, . [16].
, 07-01-00399.
77
1. Eremin I.I. Theory of Linear Optimization. Inverse and Ill-Posed Problems Series. VSP.
Utrecht, Boston, Keln, Tokyo, 2002.
2. .., .. -
. M.: , 1979.
3. .. -
. .: , 1982.
4. .. . .: . .
. .-.., 1988.
5. .., .. . .: ,
2003.
6. .. ,
.- .: (), 1979.
7. .. // , 1974,
.14, 4, 10521058.
8. ..
// . 1977, N 1, .5-15.
9. .., .. -
// . . 1972.
.8. .5. .740751.
10. Mangasarian O.L. A nite Newton method for classication // Optimizat. Meth.
Software. 2002. Vol.17, p.913930.
11. Kanzow C., Qi H., Qi L. On the minimum norm solution of linear program // J.
Optimizat. Theory and Appl. 2003. Vol.116. p. 333345.
12. .., .., .
// . -
. . . , 2004, 44, 9, . 15641573.
13. ., . : . . .: , 1999.
14. .
/ . . .: , 1991.
15. Gondzio J., Sarkissian R. Parallel interior-point solver for structured linear programs
// Math. Progam., Ser. A. 2003, Vol. 96, p. 561584.
16. .. -
// . . -
. . . 2007. 47, N2, .206-221.
, ,
. . , 16, , 6200219, ,
. (8-343-3)75-34-23, (8-343-3)74-25-81, e-mail:[email protected]
,
78
. .
[1], -
. -
, -
. -
. , , [2],
c, x+ d, y max(x,y)
,
x > 0, Ax+By 6 b,y Y(x) , Sol(2),
(1)c1, x+ d1, y max
y,
y > 0, A1x+B1y 6 b1,
}(2)
- (2) -
( )
d1 + vB1, y v, b1 A1xB1y = 0,x, y, v > 0.
, , d.c.
, .. .
, -
, x,
x > 0, Mx+ q > 0,x,Mx+ q = 0.
}(3)
, ( )
-
, , , -
, ..
, P : X IRm, X IRn, x X,
P (x), x x > 0 x X. (4) (3) (4) P (x) = Mx + q, X = IRm+ . (3) (4).
, (
) -
( ).
, - ,
, ,
, , .
79
, , -
f0 min, fi(x) = 0, i = 1, . . . ,m, (5)
fi(x), i = 1, . . . ,m, (..
0f0(x) +mi=1
ifi(x) = 0; (6)
i IR, i = 1, . . . ,m, 0 > 0,mi=1
|i| + 0 > 0), . , , (1)-(2), (3) (4)
, (6)
.
,
()
(x) = f0(x) + (x) minx(7)
(x) = 0(f1(x), . . . , fm(x)) , , :
1(x) =mi=1
i|fi(x)|, 2(x) =mi=1
i[fi(x)]2, p(x) =
mi=1
i[fi(x)]p, 1 < p < +.
, fi(x), i = 1, . . . ,m - (x), . (7) (
) ( -
) . ,
(5) ,
d. c. -
:
(P) f(x) min, x S,F (x) , g(x) h(x) = 0,
}(8)
f, g, h , S IRn. , (P)(8) d. c. :
f(x) min, x S,F (x) , g(x) h(x) > 0.
}(9)
(9) [6], -
[4],[5],
[3],[5]. , -
(9) .
(9) -
(8).
80
. , (P)(8) :
v S : F (v) < 0, (10)
(H) y S : F (y) = 0 . . g(y) = h(y), p = p(y) S :h(p) h(y) < g(y), p y.
}(11)
[3],[4] -
().
1 (e )[3],[4]. -
(10) (H)(11). ,
(E) (y, ) : g(y) = , y S,
h(y) 6 6 sup(h, S),h(x) > g(y), x y,x S, f(x) 6 f(z);
(12) z (P)(8). # (E)(12) (P)(8) - , (9). -
, , (10):
v S : f(v) < f(z), F (v) < 0. (13) 2 ( ). (P)-(8) (13).
, z Sol(P),
(E1)(y, ) : g(y) = , y S,h(y) > g(y), x yx S, f(x) 6 f(z).
(14). (E1)-(14) ,
(y, ) : g(y) = , y S, u S, f(u) 6 f(z),h(u) < g(y), u y. g()
0 < h(u) + g(u) g(y) = F (u),
u S, f(u) 6 f(z), F (u) > 0. (13), ]0, 1[: F (x()) = 0,
x() = u+ (1 )v S. , f(),
f(x()) 6 f(u) + (1 )f(v) < f(z),
81
z. # (E)-(12) (E1)-(14) (1)-(2) (3) - ,
.
, (1)-
(2),
.
1. .. . .: , 1975.
2. Dempe S. Foundations of bilievel programming. Dordrecht/ Boston/London: Kluwer
Academic Publishers, 2002.
3. .. , : ,
2003.
4. .. d.c. .//
. . . , 2001, .41, 12, . 1833-1843.
5. .. d.c. -
.// . . . , 2005, .45, 3, . 435-447.
6. .., .. -
// . . . . 2007, .47, 3, c. 397-
413.
,
. 134, , 644033, , . (3952)511398.
E-mail: [email protected]
82
. .
, , .
F. X Rn. x X , X = . , F. -
, , X
X =
D, (1)
= {x Rn : < xj xj xj
, .. -
, , ,
, MAPLE -
[3].
[6].
. g(x, y) = sin(xy)+0.1(x 1)2+0.2y2, X = = {x R2 : 3 xj 3, j = 1, 2} y = (0, 0). y -
(x, y) = min
(0,(x y)
2
2
)+ 0.1 0.2x.
(2)-(3) .
D
= B = {x Rn : 0 xj 1, j = 1, . . . , n},D = DB = {x Rn : hB(x) = xT (x e) = 0}, e = (1, . . . , 1)T . XB = B DB B. D
D = DI = {x Rn : hI(x) =nj=1
| sin(pixj)| = 0},
x DI x Zn, Zn n- , .. DI = Zn. , x
xj Zj = {zj1, . . . , zjkj}, j = 1, . . . , n. (4) (4).
j(xj) =
kjs=1
(x zjs).
(4) x DZ ,
DZ = {x Rn : hZ(x) =nj=1
|j(xj)| = 0}.
hB(x) , hI(x) hZ(x) - .
D (3)
D = {x Rn : F (x) 0},F (x) = max{g1(x), . . . , gm(x), |h1(x)|, . . . , |hl(x)|}.
84
F (x) - F (x, y). x0 - . F (x0) 0, x0 X .
x0 6 X (5) F (x0) > 0. F (x
0, x0) = F (x0),
F (x0, x0) > 0. (6)
, F (x, x0) F (x) x X.
F (x, x0) 0 x X. (7) (5)-(7) , F (x, x
0) = 0 x0 X. , . -
,
2
F (x, y). , x0 . -
U(x0) = {x : F (x, x0) > 0}, X. C0 - x0, . C0 F (x, x
0) = 0
(p0)Tx = s0. (8)
U(x0) (8) , ..
(p0)Tx0 > s0 (p0)Tx s0 x X.
P 0 = {x : (p0)Tx s0}., x0 6 P 0 P 0 X. () x1 P 0 x1, P 1 ,
P 0 P 1 X.
P 0 P 1 . . . P k X xk, k = 0, 1, . . . , k. , P k = k, - X = , limk xk = x X. ,
,
2
.. , .
85
. Ck
Cki ,
Ck =i
Cki int(Ckj)
int(Cki) = i 6= j.
, -
, ,
. -
. -
X. -
, -, ,
(1)-(3). . -
, [5].
.
06-01-00465-
1. .. , .. . . // -
. 2. 2004. . 44. N9. . 45-68.
2. .. . -
. // XIII - " -
", .1 " ", ,
. 2005. . 621-626.
3. .. . , -
. // .: , , . 1998. . 73-100.
4. .. . .
// . 2004. . 44. N9.
. 1552-1563.
5. V.P. Bulatov, O.V. Khamisov. The branch and bound method with cuts in En+1
for solving concave programming problem. // Lecture Notes in Control and Computer
Sciences, 180, Springer-Verlag. 1991. P. 273-282.
6. G.P. McCormik. Nonlinear programming: Theory, Algorithms and Applications. //
John Wiley and Sons. New York. 1988. 267 p.
, , . 130, , 664033,
, . (3952) 42-84-39, (3952) 42-67-96, E-mail: [email protected]
86
,
. .
[1,2] MASC
A,B Qn. , , NP - Apx( P 6= NP ). - MASC.
, ,
. , -
, (..
).
PC
, -
[3]. , [4], -
-
.
1. Q = (f1, . . . , fq), fi(x) =Ti x i , A,B Rn,
|{i Nq | fi(a) > 0}| > q2
(a A),
|{i Nq | fi(b) < 0}| > q2
(b B).
q () Q.
(MASC).
A = {a1, . . . , am1} B = {b1, . . . , bm2}, A,B Qn. Q , - A B.
1 ([2]). MASC NP-.
MASC NP -
A B {z {0, 1, 2}n : |z| 2}.
2 ([2]). MASC Apx ( P 6= NP ).
87
MASC.
3.
NP 6 TIME(2poly(logn)), MASC -
O(log log logm).
, -
. [5], n = 1 .
, n = 2 (, , n > 1) NP -.
2. L = {l1, . . . , ls}, lj = {x R2 | cTj x = dj}, P = {p1, . . . , pk} R2, p P l = l(p) L , p l. , , ,
P,A B, , , - . ,
,
.
. 1: PC PASC
(PC).
P = {p1, . . . , pk} Z2 s N. L P s?
88
(PASC).
A = {a1, . . . , am1} B = {b1, . . . , bm2}, A,B Q2, t N. Q, A B t ?
[3], PC NP -. PASC ASC [2], .
, PASC ( ASC) NP. PC PASC,
NP - . PC P = {p1, . . . , pk} Z2 s N. = max{|pi| : i Nk} = 16(2+1)+1 . - , || = 1 , {i, j} Nk [pi , pi + ] [pj, pj+] . PC PASC : A = P, B = (P ) (P + ) t = 2s+ 1 (. 1). ,
, PC.
, -
PC PASC
. , P - s , A B , 2s+ 1.
4. P = {p1, . . . , pk} Z2 s , A = P B = (P ) (P +) 2s+ 1 .
. 2:
89
,
,
P , A B, . 2.
1. PASC NP-. ASC
n > 1 NP-.
2. MASC n > 1 NP-.
, -6768.2006.1 -
5595.2006.1 , 07-07-00168.
1. ...
// , 2006, 406, 6, . 742745.
2. ... -
. //
. 2006, 1, . 3443.
3. N.Megiddo, A.Tamir. On the complexity of locating linear facilities in the plane //
Operations research letters. 1982, vol. 1, no. 5, p. 194197.
4. N.Megiddo. On the complexity of polyhedral separability // Discrete and Computational
Geometry. 1988, 3, p. 325337.
5. ... // -
. 1971. 3. . 140146.
, ,
. . , 16, , 620219, ,
. (343) 375-35-05, (343) 374-25-81, E-mail:[email protected]
90
f.
. .
A = {a1, ..., an} aj Rd, ( [A]) d- . - A TA = {S1, S2, ..., St} Si A, [Si] d- , |Si| = d + 1,
ti=1
[Si] = [A] i 6= k[Si Sk] = [Si] [Sk]. j = 0, 1, 2, ..., d (j + 1)- F A j- TA, i , F Si. fj(TA) j- f(, TA) =
d+1j=0
fj1(TA)j, f1(TA) = 1.
TA A - f(, TA) F (d, n). ( -
, ) f() ( F (d, n)).
, . . . -
, . , . 23, . , 603950, , . (8-8312) 65-78-81,
E-mail:[email protected]
91
CONTINUOUS COVERING PROBLEMS
P. Hansen
Covering problems are frequently encountered in Operations Research, Location Theory,
Telecommunications and Geometry. The most studied are the discrete ones, such as the
p-center problem. However, continuous problems are of interest also. They are of two
types:
(i) discrete-continuous ones, in which a discrete set of demand points is given, together
with a continuous set wherein facilities are to be located, the objective being to minimize
the maximum distance from a demand point to its closest facility;
(ii) fully continuous ones which dier from the former only in that the set of demand
points is continuous; this last category comprizes well-known geometric problems such as
covering disks, squares or tringles by a minimum number of disks of given radius (or with
a given number n of disks with minimum radius).
We review work on these problems and provide new heuristic and exact algorithms for
both of them.
Pierre Hansen,
GERAD and Department of Quantitative Methods in Management, HECMontreal, Canada,
phone: (1-514) 340-6052, fax: (1-514) 340-5665. E-mail: [email protected]
92
THE VARIABLE NEIGHBORHOOD SIMPLEX SEARCH
FOR CONTINUOUS OPTIMIZATION
Q. Zhao, D. Urosevic and N. Mladenovic
We rst suggest a modied version of the well-known Nelder-Mead (or simplex) method,
originally designed for solving continuous convex minimization problems. Then we propose
a natural and simple extension that allows us to solve non convex nor concave problems
as well. It ts into the variable neighborhood search scheme. Extensive computational
analysis shows the capability of our method. It appears that, in solving convex problems,
our modied simplex outperforms in average the original version as well as some other
recent modications. In solving unconstrained global optimization, it is comparable with
the state-of-the-art heuristics, but easier to implement and more user-friendly.
Nenad Mladenovic
School of Mathematics, University of Birmingham Edgbaston, Birmingham B15 2TT,
United Kingdom, e-mail: [email protected].
93
. . , . . , . .
n- , - . , -
.
.
i j cij(xij, xji) ( xij, xji , - i j (i, j) (j, i)) xij . , , -
, , ,
.
[1] -
cij(xij, xji) = axij + bxji, .
:
1. cij(xij, xji) = aixij + ajxji;
2. cij(xij, xji) = aij(xij + xji);
3. cij(xij, xji) = aixij + bijxji.
1 2 -
O(n3).
1. .., .. //
. 2005. . 8, 3(23). . 5868.
, () -
- , . , .55-, .130,
650055, , , . (8-3842) 25-33-34, (8-3842) 25-07-21. E-mail:
, . .. , . -
, 4, 630090, , , . (8-383) 333-37-88, (8-383) 333-25-98.
E-mail: [email protected]
, , . -
, 2, 630090, , . E-mail: [email protected]
94
,
. . , . . , . .
:
x = argmin{(x) : x R}, (1)
(x) x En, R En . : R , - x (1), {Sk} , |Sk| 0 x Sk k. Sk+1 xk Sk, -. .
[1, 2], :
1.
(Sk)
(Sk1)(
n1n1 1
)n11(
n
n+ 1
)n< 1 (2) Sk, n1 . , R - Ax b A. , -
1 n1 n, (2) :|Sk||Sk1|
1
n
[ nn12
(n1
n1 1)n11
(
n
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3. Federer H., Curvature Measures // Trans. Amer. Math. Soc. 93, 3 (1959), 418-493.
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