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1 ,, 2 2.,, A, PC/AT, MB, 5GB,,,, ( ) MB, GB 2,5,, 8MB, A, MB, GB 2 A,,,? x MB, y GB, A (), x + 2y () 4 (,, ) (hanba@eee.u-ryukyu.ac.jp), A, x + 2y() x y, A, MB ( ) 8 MB ( ) 5GB ( ) ( ), x x x 8 (2) y y 5 x + 25y,,, (x, y), x + 2y (2), x, y

2 HDD(GB) y A 6, A 4, A 2, 4 2 x = x = 8 : x + 25y 4 8 y =5 x (MB) () n x,...,x n, m a x a n x n b... a k x a kn x n = b k... a l x a ln x n b l... (3), x i,...,x ir, x j,...,x js, (3), () c x c n x n (4), (), ( ) x x n (),,, 947 G. B. Dantzig,,, 4,, 984 N. Karmarkar,,,, (, ),,,, ,,,,,,,, A m n, b m, c, x n, A m n A R m n, x n x R n i), A i a i, b, c i b i, c i, x w n, x w, x w, x i w i (i =,...,n) i) R 2

3 n, m n ii), : min{z = c T x : Ax = b, x }, Ax = b, x z = c T x, min minimize( ),,. a i, x + + a i,n x n b i : y i R, a i, x + + a i,n x n y i b i,y i 2. a i, x + + a i,n x n = b i : 3. a i, x + + a i,n x n b i : y i R, a i, x + + a i,n x n + y i b i,y i 4. x i : 2 x i, x i+, x i = x i+ x i, x i+, x i, a p,i x i a p,i x i+ a p,i x i, c i x i c i x i+ c i x i 5. x i : 6. x i : x i = x i, a p,i x i a p,i x i, c i x i c i x i 7. c T x ( ): c = c, (c ) T x,,3,4,, : min{z = c T x : Ax b, x } ii),,,,,,, [5] 2.3, [3],[4] : (STD) min { z = c T x : Ax = b, x } (5), A R m n, b R m, c R n, n m, A,, (STD), ( ),, A a,...,a n m, B =[a i,...,a im ] (6), A B, N =[a j,...,a jn m ] (7) B, N,, B x i,...,x im x B =[x i,...,x im ] T (8) x B, N x j,...,x jn m x N =[x j,...,x jn m ] T (9) 3

4 x N, c, c B =[c i,...,c im ], c N =[c j,...,c jn m ] (), z = c T x, x z = c T B x B + c T N x N () x B = B b, x N = (2) Ax = b Bx B + Nx N = b (3), (2) x Ax = b (2) x,, x, B b, x B, x N, x x,, x (STD) (2) x (STD),, B iii),, 2.3. B, Ax = b (3),, (3) b = B b, Ā N = B N (4) x B = b ĀNx N (5) (5) (), z B = c T B b, c T N = ct N ct BĀN (6) iii) (, ),, z = z B + c T N x N (7), ĀN i ā i, Ā N =[ā,...,ā n m ] (8) (5) (7), (SP) (BF) min{z = z B + c T Nx N : x B = b ĀNx N, x B, x N } (9) (BF) B, (2) (BF),, (2) z, (5) (7), (6) c N, s c N,s c N,s,, z c N,s <, (7) c N,s x js z, x js (9) x js θ, x B = b ā s θ b b,..., b m, ā s ā s;,...,ā s;m x B, ā s,i, θ b i /ā s;i, ā s,i =, θ, {,...,m} ā s;i >, θ = min{ b i /ā s;i :ā s;i >, i m}, θ θ, a s <=, θ, z 4

5 iv) () B B (9) ( ) { c N,,..., c N,n m } c N,s, c N,s, x B = B b, x N = (2) c N,s < 2 ( 2) N s a js, Bā s = a js (2) ā s ā s (), θ = min{ b i /ā s;i :ā s;i >, i m} r θ = b r /ā s;r, r (, ) ( 3) 2 r s B =[a i,...,a ir, a ir, a ir+,...,a im ] (22) B =[a i,...,a ir, a js, a ir+,...,a im ] (23), c N,,, c N c T B B N, B T c N = c B c N c N N (c T BB N) T = N T (B T ) c B, ĀN = B N, B, B x B = b (25) x B x N x B x N x,, x, x 3, x 5, x B = [, 2, 3], [,, 2,, 3,,...] ( ), (4) b,, B b = b (24) iv),,,, ( ),, 2, 2 min{x + x 2 : x 2, x 2 2,x,x 2 } (26) (26) 2 x + x 2,,, x =,x 2 = 5

6 x 2 3 2, 2 ( ), 4, 5 4,, 4,, B, B b 2 3 x, x x 2, 2 (26), x, x 3, x x 3 =,x 3 x 2, x 4, x + x 4 =2,x 4 x 2 x 2 x 5 =,x 5 x 2 + x 6 =2,x 6,, x =[x,x 2,x 3,x 4,x 5,x 6 ] T min{c T x : Ax = b, x }, A =, b = 2, c = 2 (27), (27) 6 4, ( ) 6 =5 v) 4 v) n n! = r r!(n r)! x B (x,x 2 ) [x,x 2,x 3,x 5 ] T (2, 2) [x,x 2,x 3,x 6 ] T (2, ) [x,x 2,x 4,x 5 ] T (, 2) [x,x 2,x 5,x 6 ] T (, ) 2 4, x B =[x,x 2,x 3,x 5 ] T, (x,x 2 )=(2, 2), 2 (x,x 2 )= (, ) x B = [x,x 2,x 3,x 5 ] T, x N = [x 4,x 6 ] T ( (x,x 2 )=(2, 2) ) B =[a, a 2, a 3, a 5 ]=, N =[a 4, a 6 ]=, c B =[,,, ] T, c N =[, ] T, z B =4, 2 [ ] b = 2, ĀN =, c N = 6

7 ( -) c N, 2, s = ( -2) ā =[,,, ] T, 3 b /ā ; =2 b 3 /ā ;3 =, b 3 /ā ;3, r =3 ( -3) B =[a i, a i2, a i3, a i4 ] B =[a i, a i2, a j, a i4 ] x B = [x,x 2,x 3,x 5 ] T, x B = [x 4,x 6 ] T, i =,i 2 =2, i 3 =3,i 4 =5,j =4,j 2 =6, B =[a, a 2, a 4, a 5 ] x B = [x,x 2,x 4,x 5 ] T 2, x B =[x,x 2,x 4,x 5 ] T, x N =[x 3,x 6 ] T, (x,x 2 )=(, 2), z B =3, [ ] b = 2, Ā N =, c N = ( 2-) c N 2, s =2 ( 2-2) ā =[,,, ] T, 2 4 b 2 /ā 2;2 =2, b 4 /ā 2;2 =, b 4 /ā 2;4, r =4 ( 2-3) B =[a i, a i2, a i3, a i4 ] B =[a i, a i2, a i3, a j2 ] x B = [x,x 2,x 4,x 5 ] T, x B = [x 3,x 6 ] T, i =,i 2 =2, i 3 =4,i 4 =5,j =3,j 2 =6, B =[a, a 2, a 4, a 6 ] x B = [x,x 2,x 4,x 6 ] T 3 2, x B =[x,x 2,x 4,x 6 ] T, x N =[x 3,x 5 ] T, (x,x 2 )=(, ),, z B =2, [ ] b =, ĀN =, c N = (28) ( 3-) c N,, x B =[x,x 2,x 4,x 6 ] T x B = b, x N =, (28), x =,x 2 =,x 3 =,x 4 =,x 5 =, x 6 =, x =,x 2 =,, (x,x 2 )=(2, 2), (x,x 2 )=(, 2) (x,x 2 )=(, ) M 2.3. B,,

8 , 2 M, M (STD) M (STD) b, b i b i, A α T i, b i b i, α T i α T i, b, ξ =[ξ,...,ξ m ] T (29), c 2 =[c T,M T m] T, A 2 =[A, I m ] (3) ( m m, M, I m m ), (5) { [ ] [ ] } min c T x x 2 : A 2 = b, x, ξ (3) ξ ξ b, x =, ξ = b (3),, (3) ξ,,,,,, M, x ξ x ξ, ξ = x = x (5), ξ (5), M, ξ, M, x , - 2 [5, 6],, x = n i= x2 i Hadamard, xs =[x s,...,x n s n ] T,, x, /x = [/x,...,/x n ] T, x 2 = [x 2,...,x 2 n] T 2.4. (P) min { z = c T x : Ax b, x } (32) (32) (32),, { } (D) max w = b T y : A T y c, y (33), max maximize,, x R n, y R m vi) -,, (P) (D),, (P) (D) (SP),, x y x >, y >, p =/x, t =/y b, c, β b = t + b Ax c = p c + A T y β = b T y + c T x (34) vi) (STD), (D n ) max w = b T y : A T o y + s = c, s,,,,, 8

9 , M q m m A b b M = A T n n c c b T c T β b T c T β (35) m q = n n + m +2, (SP) min { q T ξ : Mξ+ q, ξ } (36), ξ =[y T, x T,κ,θ] T n + m +2 (SP), M vii), q, ξ, (SP) Mξ + q s(ξ),, S, Ξ s(ξ) =Mξ + q (37) S = diag[s,...,s m+n+2 ], Ξ = diag[ξ,...,ξ m+n+2 ] (38), diag[x,...,x k ] x,...,x k, 2, Dikin (Dikin ) () ε, ξ := [(y ) T, (x ) T,, ] T, s := s(ξ), α <α () while ξ T s ε do begin ξ := ξ + α ξ s := s(ξ) end vii) M, M T = M, M, ξ (S + ΞM) ξ = ξ2 s 2 ξs (39), α <α 2 n + m +2 viii), α =/(2 n + m +2) (4), 2(n + m + ) log n + m +2 ε (4) [5], (4), (36) ξ, q T ξ = ξ =[y, x,κ,θ ] T κ, x = x κ, ȳ = y κ (42) κ =,,,, 2.4.2,, ( ) () ε, ξ := ξ, s := s(ξ), σ =.4/ n + m + 2 (43) viii) n + m +2 ο 2, ο dim ο, α (4) <α 2 dim ο 9

10 () while ξ T s ε do begin µ := s T ξ/(n + m +2) ξ := ξ + α ξ s := s(ξ) end, ξ (S + ΞM) ξ = sξ + σµ (44) (44) Dikin, Newton Dikin Dikin,,,,, Mehrotra., 2.4.2, M, Dikin 2.4.4, Newton, (SP), 2.4 M, (P) (D) (Ax b, x ) P, (A T y c, y ) D P D,, x P y D, b T y (x T A T )y = x T (A T y) x T c, b T y c T x,, x P y D b T y = c T x, x, y,, x y, b T y = c T x, b T y = c T x b T y c T x (45) b T y c T x, (45),,, Ax b A T y + c b T y c T x, A b y A T c x, x, y b T c T (46) (46), (46) κ A b y y A T c x, x (47) b T c T κ κ (47) (y T, x T,κ), κ>, (y T /κ, x T /κ, ) (46), κ (46), κ (47) x =, y =, κ =, (47), (46) κ, (46) κ ( ), κ (46)

11 , (47) (35), (35) M (47)?, ( ),,,, (47) (35) (47), x >, y > ( ), p =/x, t =/y b, c, β (34), M q (35), (36) (SP), y ξ = x (48) (36) ξ >, Mξ + q >, (34), Ax b + b Mξ + q = A T y + c + c b T y c T x + β b T y c T x β +(n + m +) t = p (49) Dikin, ξ (39) s(ξ) =Mξ + q >, ξ > ξ, s(ξ) s, M, ξ T Mξ = ix), q T ξ q T ξ = ξ T s x), ξ ξ + ξ,, s = s(ξ + ξ) s(ξ) (ξ + ξ) T (s + s) ξ T s = ξ T s + ξ T s xi),, ξ T s + ξ T s ξ,, ξ, ξ ξ + s s (5), (5) ( Dikin ) ξ T s + ξ T s ξ (39) (5), ξ s, ξ s, ξ s, ξ s,,, Dikin, 2, (P) (D), (P) (D) 4, 2 Dikin 3 2 [ ] T [ ] ξ min : ξ 2 [ ][ ] [ ] [ ] [ ξ +, ξ 2 ξ ξ 2 ] [ ]} ix) ο T Mο M T = M, ο T Mο =(ο T Mο) T = ο T M T ο = ο T M T ο, (5) 2ο T Mο = x) ο T Mο =, ο T s = ο T (Mο + q) = ο T q xi) ( ο) T s =( ο) T M ο =

12 (5), ξ, ξ 2 ξ, ξ = 3 (5) Dikin ξ ξ 32 Dikin, Dikin Newton, Newton 2.4.2,, (SP) min { q T ξ : Mξ + q, ξ }, M, q, (SP) M, q T ξ = s T ξ xii), (SP) ξ =, 2.4.2, ξ (P) (D),, ξ xii), s = Mο+ q (s T x = ), ξ ξ, s, ξ s, ξ i = s i >, ξ i > s i =,, s + ξ > (52), (SP) ( ) ,, µ, sξ = µ (53), (SP) ( ), (53) (SP) s = Mξ + q, (54) sξ = µ ξ, µ ξ(µ), ξ(µ) µ µ, ξ(µ), ξ(µ)( ),, , 2, (SP) x s [ ] Mξ s + q F (ξ, s) = (55) sξ 2

13 ξ s = Mξ+q, ξ = ξ + ξ a s = s + s a F (ξ, s )= ξ a s a Tailor, ξ a s a [ ][ ] [ ] M I ξ a = (56) S Ξ s a sξ, α, ξ = ξ + α ξ a s = s + α s a, M(ξ )+q = s + αm ξ a = s, α, s ξ s = Mξ + q (56) ( ξ T a, st a )T,,,, µ, µ, / dim ξ, µ =(/ dim ξ)s T ξ,, [ ] Mξ s + q G(ξ, s) = (57) sξ µ ξ = ξ + ξ c, s = s + s c, (57) Tailor, ξ c s c [ ][ ] [ ] M I ξ c = (58) S Ξ s c sξ + µ (58) ( ξ T c, st c )T, Newton , Newton, σ, [ ] [ ] [ ] ξ ξ =( σ) a ξ + σ c (59) s s a s c, ( ξ T, s T ) T [ ][ ] [ ] M I ξ = S Ξ s sξ + σµ (6), (6) ξ s, (6) s s = M ξ, (6) ( ) S + ΞM ξ = sξ + σµ (6) ξ. (6) ( ξ T, s T ) T, (Newton ) () ε, ξ := ξ, s := s(ξ) () while ξ T s ε do begin σ µ := s T ξ/(n) end (6) ξ, s α ξ := ξ + α ξ s := s + α s 2.4., α =,σ =.4/ dim ξ, σ,, α σ, [2, 5, 6] 4 3,, 3 (5), ξ 2 =/2 3

14 , (ξ,ξ 2 )=(, /2) 4,, µ =(/2)s T ξ, ξ ξ (a) ξ ξ (b) 4 4

15 3 Scilab /home/b/teacher/hanba/info-eng Scilab sample.sci functions.sci 4 4. ( M ) 2 (Dikin ) 3 ( ), A b /home/b/teacher/hanba/info-eng/ practice.dat problem.dat 2 practice.dat, problem.dat problem.dat, practice.dat A A a ij, b b i, A b a a 2 a n b a 2 a 22 a 2n b 2 (62), A, 2,..., n b c [] Scilab,, L A TEX, Scilab /home/b/teacher/hanba/info-eng/ practice.dat problem.dat sample.tex L A TEX f.eps PostScript sample.sci Scilab functions.sci practice.dat practice.dat problem.dat, Scilab ~harada/lab/scilab-intro.pdf sample.sci Dikin 2,, 3 5

16 , 2, (44), functions.sci sample.sci, functions.sci, L A TEX PostScript sample.tex,,,, ( ),,,,, ( ) 5 Dikin Dikin ( 3 ) /home/b/teacher/hanba/info-eng,.dat 2.dat, ( 3 ) [] S. -C. Fang, S. Puthenpura: Linear Optimization and Extensions: Theoory and Algorithms, Prentice Hall (993) [2],,, :, (2) [3] :, (987) [4],, :, (994) [5] C. Roos, T. Terlaky and J. -Ph. Vial: Theory and Algorithms for Linear Optimization, Wiley (997) [6] S. J. Wright: Primal-Dual Interior-Point Mehtods, SIAM (997) 6

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