. .-. . . . ()

. .-. . . . ()

. . . . . ()

. .-. . . . ()

. .-. . . . ()

. .-. . . . ()

. .-. . . . ()

. . ()

. . ()

. . . . . ()

. .-. . . . ()

. .-. . . . ()

. .-. . . . ()

. . . ()

. .-. . . . ()

. .-. . . . ()

. .-. . . . ()

. .-. . . . ()

. .-. . . . ()

..-.. . . ()

. ..

, 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:abramov@iacp.dvo.ru

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: antipin@ccas.ru

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: astnn@imm.uran.ru

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: beresnev@math.nsc.ru

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: vasilev@math.nsc.ru

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:gimadi@math.nsc.ru

30

. . , . .

(), , -

. (-,

, ) -

. - -

, . -

,

.

. ,

. -

, -

.

, -

. ,

, -

.

, , ,

. -

-

. -

MATLAB.

(

) ( ).

P-IV 2.6 -

. .

(, CPLEX) -

MATLAB

-

. .

8

30 .

06-01-00547

-2240.2006.1.

, .. ,

. 40, , 119991, , . (495) 135-00-20. E-mail: evt@ccas.ru

, .. ,

. 40, , 119991, , . (495) 135-61-61. E-mail: gol@ccas.ru

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: adil@math.nsc.ru

, . ..

, . , 4, 630090, , , . (8-383) 333-21-89,

(8-383) 333-25-98, E-mail: slava@math.nsc.ru

, . .. ,

. , 4, 630090, , , . (383) 333-37-88, (8-383) 333-

25-98, E-mail: orlab@math.nsc.ru

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:zork@isem.sei.irk.ru

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: iljev@math.omsu.omskreg.ru

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-, -

. -

,

.

06-01-00058 07-07-00022.

1. Kel'manov A.V., Jeon B. A Posteriori Joint Detection and Discrimination of Pulses in

a Quasiperiodic Pulse Train // IEEE Trans. on Signal Proc. 2004. Vol. 52, No. 3 P. 1-12.

2. .. -

:

// XII -

(-12). , 2005. . 125-128.

3. .. -

-

// 3-

. , 2006. . 37-41.

4. .., ..

//

. 2001. .41, 5 . 807-820.

5. Kel'manov A.V., Khamidullin S.A., Okol'nishnikova L.V. A Posteriori Detection of

Identical Subsequences in a Quasiperiodic Sequence // Pattern Recognition and Image

Analysis. 2002. Vol. 12, No. 4. P. 438-447.

6. .., .., .., .. -

// . . . . 2006. .9, 1(25).

. 55-75.

,

. .. ,

. 4, , 630090, ,

.: (8383) 333-3291, : (8383) 333-2598, e-mail: kelm@math.nsc.ru

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],

.

-

.

1. .. : -

// . "

", , 1998. . 410.

2. .., .., .. -

. : , 1978. 160 .

3. .. -

. : - , 2005. 408 .

4. .., .., .. -

. , 1978. 333 .

5. .., .., .., .. -

// XII . -, , 2001. . 1. . 132137. http://math.nsc.ru/AP/benchmarks/

6. .., .., .. -

. . : ,

1997. 26 .

7. .., .. -

// . 2004. 3.

. 4854.

8. ., .., .. -

// . 2004. 2.

. 7992.

9. .. -

// . 1994. 2. . 1839.

10. .., .., ..

55

//

. 2005. 2(31). . 7680.

11. .., ..

: -. . : , 2004. 40 .

12. .., .., ..

// . 2006.

4(38). . 62-67.

13. .., ..

// XII -

" ". , , 2001. . 1.

. 207-210.

14. ..

p-. . : , 2006. 23 .15. .., .., .. , -

// III "

". , 2006. . 105.

16. .., .. -

p- // . 2004. 3. . 8088.17. . : 2- . .:

, 1991. 702 .

18. Avella P., Sassano A., Vasil'ev I. Computational study of large scale p-median problems//Mathematical Programming. 2006. Vol. 109. 1. P. 89114.

19. Beasley J.E. A note on solving large p-median problems //European Journal of Opera-tional Research. 1985. 21. P. 270273.

20. Benders J. F. Partitioning procedures for solving mixed-variables programming problems

//Numerische Mathematik. 1962. Vol. 4. .3. P. 238252.

21. Devyaterikova M.V., Kolokolov A.A. On the stability of some integer programming

algorithms //Operations Research Letters. 2006. 34(2). P. 149154.

22. Discrete Location Theory (Ed. by Pitu B. Mirchandani and Richard L. Franscis), by

John Wiley and Sons, Inc., 1990. 576 p.

23. Facility Location. Applications and Theory (Ed. by Zvi Drezner and Horst W. Hamacher),

Springer, 1990. 457 p.

24. Kolokolov A.A., Kosarev N.A. Analysis of decomposition algorithms with Benders

cuts for p-median problem //Journal of Mathematical Modelling and Algorithms. 2006. 5. P. 189199.

25. Kosarev N.A., Kolokolov A.A. On stability of some benders decomposition algorithms

for p-median problem //Proceedings of European Chapter on Combinatorial OptimizationXVIII. Minsk, 2005. P. 26.

26. Reinelt G. TSPLIB: A traveling salesman problem library //ORSA Journal on Compu-

ting. 1991. 3. P. 376384.

http://www.iwr.uni-heidelberg.de/groups/comopt/software/TSPLIB95/.

, ,

. .. ,

. , 13, , 644099, , . (3812) 23-67-39, (3812) 23-45-84.

E-mail: kolo@om.oscsbras.ru, nkosarev@mail.ru

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].

, ,

on-line, , [14], -

= O(N1/2 logN

).

on-line . , -

( )

. -

: = (N2/3

). 05-01-00798.

1. Garey M.R., Johnson D.S., Computers and Intractability, A Guide to the Theory of

NP-Completeness, Freeman, San Francisco, 1979.

2. Baker B.S., Coman E.J., Rivest R.L. Orthogonal packings in two dimensions// SIAM

J. Computing, 1980, V. 9, P. 846-855.

3. Baker B.S., Schwarz J.S. Shelf algorithms for two dimensional packing problems//

SIAM J. Computing, 1983, V. 12, P. 508-525.

4. Csirik J., Woeginger G.J., Shelf Algorithms for On-Line Strip Packing// Inf. Process.

Lett., 1997, V. 63, N 4, P. 171-175.

5. Van Vliet A. An improved lower bound for on-line bin packing algorithms// Inf. Process.

Lett., 1992, V. 43, P. 277284.

6. Graham R. L. Bounds for certain multiprocessing anomalies // Bell System Technical

Journal, 1966, V. 45, P. 15631581.

7. Albers S. Better bounds for online scheduling// Proc. of the 29th ACM Symp. on

Theory of Computing, 1997, P. 130-139.

8. Bartal Y., Fiat A., Karlo H., Vohra R. New algorithms for an ancient scheduling

problem // J. Comput. Syst. Sci., 1995, V. 51, N 3, P. 359-366.

9. Faigle U., Kern W., Turan G. On the performance of on-line algorithms for partition

problems //Acta Cybernetica, 1989, V. 9, P. 107-119.

10. Karger D. R., Phillips S. J., Torng E. A better algorithm for an ancient scheduling

problem // Proc. of the 5th Annu. ACM-SIAM Symp. on Discrete Algorithms, 1994, P.

59

132140.

11. Bartal Y., Karlo H., Rabani Y. A better lower bound for on-line scheduling // Inf.

Proces. Letters, 1994, V. 50, P. 113116.

12. .. -

// , 2006, . 18, N 1 C. 91105.

13. E.G. Coman, Jr., C. Courcoubetis, M.R. Garey, D.S. Johnson, P.W. Shor, R.R.

Weber, M. Yannakakis, Perfect packing theorems and the average-case bahavior of optimal

and online bin packing // SIAM Review, 2002, V. 44, N 1, P. 95108.

14. R.M. Karp, M. Luby, A. Marchetti-Spaccamela, A probabilistic analysis of multidimensional

bin packing problems // Proc. Annu. ACM Symp. on Theorty of Computing, 1984, P.

289298.

15. P. W. Shor, The average-case analysis of some on-line algorithms for bin packing//

Combinatorica, 1986, V. 6, P. 179-200.

16. P. W. Shor, How to pack better than Best Fit: Tight bounds for average-case on-line

bin packing // Proc. 32nd Annual Symposium on Foundations of Computer Science, 1991,

P. 752-759.

17. E. G. Coman, Jr., D. S. Johnson, P. W. Shor and G. S. Lueker. Probabilistic analysis

of packing and related partitioning problems // Statistical Science, 1993, V. 8, P. 40-47.

18. E. G. Coman, Jr., P. W. Shor. Packings in two dimensions: Asymptotic average-case

analysis of algorithms// Algorithmica, 1993, V. 9, P. 253-277.

19. .., ..,

// , 2006, . 18, N 1, C.

7690.

20. .., .., -

// , 2006, . 12, C.

1726.

21. Han X., Iwama K., Ye D., Zhang G. Strip packing vs. bin packing// Electronic

Colloquium on Cpmputational Complexity, 2006, Report N 112.

,

, . , 25, , 109004, , . (8-495) 912-56-59,

(8-495) 912-15-24, e-mail: nnkuz@ispras.ru

60

..

. -

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 - .

4. DTime(2O(n)) 6= NTime(2O(n)), -

.

, -

, -

P-

. .

. -

.

,

[4]. -

66

, -

, P-

. -

P- .

-

. ,

"" NP

[11]. 2,

, -

[13].

, -

, -

. , -

.

. ..

,

.

, -

, ,

.

, , ,

.

L [16]. -

a2t, a t , a < 2L. [16], NP- -

, L ,

L . - , .

06-01-00075.

1. . , . . . .:

, 1982.

2. M. Atallah. Algorithms and theory of computation handbook. Boca Raton; FL: CRC

Press, 1999.

3. G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, M. Protasi.

Complexity and approximation: combinatorial optimization problems and their approxi-

mability properties. Berlin: Springer, 1999.

4. C. Blum, A. Roli. Metaheuristics in combinatorial optimization: overview and conceptual

comparison.// ACM Computing Surveys. 2003. Vol. 35, No. 3. P. 268-308.

5. B. Bollobas. Random Graphs. Academic Press, 1985.

6. S. Ben-David, B. Chor, O. Goldreich, M. Luby. On the theory of average case complexity.//

J. Comput. System Sci. 1992. V. 44. P. 193-219.

7. E. Eberbach. Toward a theory of evolutionary computation.// BioSystems. 2005. V.

67

82. P. 1-19.

8. Y. Gurevich. Average case completeness.// J. Comput. System Sci. 1991. V. 42. P.346-

398.

9. W. E. Hart, R. K. Belew. Optimizing an arbitrary function is hard for the genetic

algorithm.//Proceedings of the Fourth International Conference on Genetic Algorithms.

1991. P. 190-195

10. R. Impagliazzo. A personal view of average-case complexity.// In Proceedings of the

10th Conference on Structure in Complexity Theory. 1995. P. 134-147.

11. R. Impagliazzo, L. A. Levin. No better ways to generate hard NP instances than

picking uniformly at random.// In Proc. of the 31st IEEE Symp. on Foundation of

Computer Science. 1990. P. 812-821.

12. L. Levin. Average case complete problems.// SIAM Journal on Computing. 1986. V.

15, N. 1. P. 285-286.

13. N. Livne. All natural NPC problems have average-case complete versions.// Electronic

Colloquium on Computational Complexity, Revision 1 of Report No. 122 (2006). P. 1-17.

14. P. Moscato. Memetic algorithms: a short introduction// D. Corne, F. Glover, M.

Dorigo (eds.) New Ideas in Optimization. McGraw-Hill. 1999.

15. E. Nagel, J. R. Newman, D. R. Hofstadter. Godel's proof. New York University Press,

2002.

16. J. Orlin, A. Schulz, S. Sengupta. -optimization schemes and L-bit precision: alternativeperspectives in combinatorial optimization.// Proceedings of the Thirty-Second Annual

ACM Symposium on Theory of Computing. 2000. P. 565-572.

17. C. H. Papadimitriou. Computational complexity. Addison Wesley. 1994.

18. A. A. Razborov, S. Rudich. Natural proofs.// J. Comput. System Sci. 1997. V. 55, N.

1. P. 24-35.

19. J. Wang. Average-case computational complexity theory. //L. Hemaspaandra, A.

Selman (eds.) Complexity Theory Retrospective II. Springer Verlag. 1997. P. 295-328.

20. J. Wang. Average case intractable NP-problems.// D.-Z. Du, K. Ko (eds.) Advances

in Complexity and Algorithms. Kluwer Academic Publishers. 1997. P. 313-378.

21. D. Whitley, J. P. Watson. Complexity theory and the no free lunch theorem.//

E. K. Burke, G. Kendall (eds.) Search Methodologies: Introductory Tutorials in Optimization

and Decision Support Techniques. New York: Springer. 2005.

, . ..

, . 4, , 630090, , . (383) 333-20-86,

(383) 333-25-98. E-mail:apljas@math.nsc.ru

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: popkov@sscc.ru.

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:popld@imm.uran.ru

,

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: strekal@icc.ru

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: khamisov@isem.sei.irk.ru

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:mkhachay@imm.uran.ru

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:shev@uic.nnov.ru

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: Pierre.Hansen@gerad.ca

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: Nenad.Mladenovic@brunel.ac.uk.

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:

astrakov@yandex.ru

, . .. , . -

, 4, 630090, , , . (8-383) 333-37-88, (8-383) 333-25-98.

E-mail: adil@math.nsc.ru

, , . -

, 2, 630090, , . E-mail: takhonov@gmail.com

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

n+ 1

)n+1

2

] 1 lnn

n.

2.

|Sk||Sk1|

(k

k 1)k1

(

k

k + 1

)k< 1 Sk, k (1 k n) Sk, . , - , n , :

|Sk||Sk1|

1

n

[ nk2

(k

k 1)k1

(

k

k + 1

)k+1

2

] 1 0.79

n.

3. xk Sk {Sk} -, :

|Sk||Sk1|

(1 n

n+ 1

)< 1.

, -

.

06-01-00465

95

1. .., .. -

. // . : , 1982.

2. .., ..

// . 1984. . 27, 3. . 348385.

, ,

. .. 664033, . , .

, 130 . 8(3952)42-84-40, e-mail: elv@isem.sei.irk.ru

,

, 664004, . ,

. , 14

96

. .

() M Rn : Rn Rn M , x / M |(x)M | < |xM |, |xM | =infyM

|x y|. ,

[1,2]. , ,

, .

1 - , M Rn, x0 Rn , xn = (xn1). {xn} - M . -, {xn} M , M ., M [3], - r > 0 : M + rB Unp(M), Unp(M) - x Rn - piM(x) M , B - . C2- [3] . 2 - , - M Rn, x0 Rn, xn+1 = (xn) n=0

[xn, xn+1] M = . (*) N, > 0 : x {xn} , n > N cos ((x) x, piM(x) x) > . xn M. M Rn - , M - (*).

, -5595.2006.1.

1. .., .. -

. . : , 2005.

2. .., .. -

, .: "", 1979.

3. Federer H., Curvature Measures // Trans. Amer. Math. Soc. 93, 3 (1959), 418-493.

, , -

4, , 620083, , . 8-902-253-39-40, e-mail: senyag@hotmail.com

97

,

. . , . .

, -

( ), -

.

.. , (

10.1 [1]). -

,

. ,

. ,

.

.

,

. , -

, -

.

, ,

, -

. -