1、? ? ( ? ? ) ? 1? 1. ( !#$% ?A8BCD(barrel) = A E 5BCD = BFGH;IJKLNM OPQRS -TJKU = NVWXAY38Z/DNJK_-a = NVWAX Y 33Z/DLNMbcdJKefNVghiHj process 1process 2process 3 input crude A315 input crude B513 output gasoline413 onput heating oil314 cost$51$11$40 k /l/m S D/Y/n/o/Lqp/i/qr/:/;/e/f/s/t/q,/-3 D /= A E

2、 5 D /= B R/S J/K4D/U = E 3Dvuw-/x/y = Lz/|vw/( /= /E/J/K/e/f /)./8/9/z/ /4/5/V R/S/ /T/7/H/;/I/,/L ,-Excel Matlab.012389L 3jH;I,-r:;efJKxr;efJKyrc;efJKz Nm S BCDYnoLN,Y f(x,y,z)= (38 4 + 33 3 51)x + (38 + 33 11)y + (38 3 + 33 4 40)z = 200 x + 60y + 206z V$Y minimize200 x + 60y + 206z subject to3x +

3、 y + 5z 8 5x + y + 3z 5 x,y,z 0. 63x = 0,y = 0.5,z = 1.5BCL 2.,- k 23H89 minimization(x1 2)2+ (x2 1)2 Subject tox2 1 x2 0, x1+ x2 2. j R -3N4N23L 3. aYnlAYn nL.0f1(x) = aTx E f2(x) = xTAxEHessianL 1 3AAjNa = (a1,an)T,A = (aij)nn,i,j = 1,2,n, f1(x) = aTx = a1x + a2x + + anx k S f1(x) = (a1,an)T= a,2f

4、1(x) = nn(n ) fx(x) = xTAx = a11x2 1+a22x22+annx2n+2a12x1x2+2a1nx1xn+2a23x2x3+ + 2a2nx2xn+ + 2an1nxn1xn k S f2 x1 = 2a11x1+ 2a12x2+ + 2a1nxn= 2(a11,a1n)x. f2 x2 = 2a21x1+ 2a22x2+ + 2a2nxn= 2(a21,a2n)x. f2 x2 = 2a21x1+ 2a22x2+ + 2a2nxn= 2(a21,a2n)x. f2 xn = 2an1x1+ 2an2x2+ + 2annxn= 2(an1,ann)x. k S

5、f2(x) = 2Ax, 2f2 xixj = 2aij k S 2f2(x) = 2A 4.cos(1/x)xTaylorcosx:x cTaylorNx = 1.0 3j(cos1 x) 0 = sin1 x( 1 x2) = 1 x2sin 1 x (cos1 x) 00 = 2 x3sin 1 x + 1 x2cos 1 x( 1 x2) = 2 x3sin 1 x 1 x4cos 1 x k S cos1 x x0TaylorYj (cos1 x) = cos 1 x0 + 1 x2sin 1 x0(x x0) 1 2( 2 x3 0 sin 1 x0 + 1 x4 0 cos 1

6、x0)(x x0) 2 k S cosx:x0cTaylorYj cosx = cosx0 sinx0(x x0) 1 2cosx0(x x0) 2 + 1 6sinx0(x x0) 3. 2 ? ? ( ? ? ) ? 2? 1. 2 2.2 !#$% minimize|x| + |y| + |z| subject tox + y 1, 2x + z = 3. =A7)BC-DE/!1FGHI#$%J minimizex+ x+ y+ y+ z+ z subject tox+ x+ y+ y+ s = 1, 2x+ 2x+ z+ z= 3, x+,x,y+,y,z+,z,s 0. 2. 2

7、2.3KLMNFGOPQRST1f(x) = Maximum(cT 1x +d1,cT2x +d2,cTpx +dp). U VWX OP7)YZ minimizef(x) subject toAx = b, x 0. .2 - !FGHI_ 3. 2 2.4a x1+ 8 3x2 4, x1+ x2 2, 2x1 3, x1 0,x2 0. b Vc #$%de bfgh K 7,(1) f x = x1+ (1 )x2 7 W x 1 S 7)(x,y) S7)*x c 1 S_)*7)x1,x2 S, + x16= x2, (0,1), x = x1+ (1 )x2.(2) ( y1=

8、b Ax1,y2= b Ax2,(3) *K(x1,y1),(x2,y2) S + (x1,y1) 6= (x2,y2)7)7,(2) (3) f (x,y) = (x1,y1) + (1 )(x2,y2) 7 W (x,y) S /KK Vc d_ 5. 2 2.7 3 x3 9 g 7*:y3= B1a3= (1/2,3,4). 8AB9:C+%XcTx + akx.n%/ N=/ T=B a1a2a3a4a5b x4013121 x1111012 rT012024 a1a2a3a4a5b x301/311/32/31/3 x112/301/31/35/3 rT01/302/32/310/

9、3 Rrj 0,de . x= (1,1,1/2,0). (b)+%/ NO0 ZO 5h x3)/ x1x2x3x4x5x6b 6301040 3300123 2110011 b x+%/= (2/3,1/3,10/3). Y deX1. 7 ? ? B( ? ? ) ? 4?5? ? 1. f(x) = 8x1+ 12x2+ x2 1 2x22! # $ % 5 f = ? 8 + 2x1 12 4x2 ? , ? 4f = 0 x= ? 4 3 ?AB # )MN ; 8OP Q(4,3) )R S ; , T U V W X Y E 2.Z minkAx bk2 )* O A - m

10、n _ b - mab 5 (a) c 7 8 d e f 5 (b) g 7 . h i 8 j k l m 5)n 0 - # $ o p l m qsr (c) . h B # qsrutv - w x r (d)y z c 7 . h 8 # | L qsru G e y H 8 5 (e) c 7 8 A b: A = 110 021 010 101 , b = 2 1 1 0 5 (a)d e b A8 : R(A)(vA8 8 ) 8 . / E b Q R(A) 8 A c Ax = c 8 x A . h f = 0, ATAx = ATb F 2f(x) = 2ATA A

11、G _ )H I f(x) A )C 3 0 A o p l m E (c) . h , # G B # E) b R(A) ( rank(A) min(m,n) Ax = b )3 Ax = b 8 A . h E (d) ATA - G 8 (A - 8) c 7.h 8 #| L x= (ATA)1ATb. (e) 0, v ki= 0,i = 0,1,l, C 3 p0,p1,pl ; i E 5. 4 4.9 Z f(x1,x2) = (x1+x2 2)2. Q 8 . f(x) = 1 2x TQx + bTx, * O Q G . h H I x1= x0+ x0p0= x0 1

12、 f0 = x0 1 (x0 x) = x. H I ). Q #? 5.2 ? / f(x1,x2) = 5x2 1+ 5x 2 2 x1x2 11x1+ 11x2+ 11. (a) # $ # j k l m 8 f(x) + f(x)T(y x), x 6= y. 0 4V3PBQx(k)6= x(k+1) Z (b)VMQ,-f . 1 HI f(0) = 1 f(1) = 1 4 . ?3Bl $3Bz Z ;Z(a)4)3jB,-.gBb f(k) f(k+1)+ g(k+1) T(x(k) x(k+1) f(k+1) f(k)+ g(k) T(x(k+1) x(k) M .0 (

13、g(k) g(k+1)T(x(k+1) x(k) 0. (b) f(x) = x2+ 7 4x 1 .0Rf0(x) = 2x + 7 4. f(0) = 1,f(1) = 1 4,f 0(0) =7 4,f 0(1) = 1 4. C D xk= 0,xk+1= 1.0RsT kyk = (1 0)(1 4 7 4) = 2 0,i = 0,1,k 1.hi2h7 8.0h() ) !3jB,-f 5.93B? (b)ABC$%kk?klkmkKknkokprqstYkOmk(s)#kukktkvr0sos= ( 1 21,1)( s1awx : s2ayx : ).zDEF = 0GH =

14、 2Is| NJ OJ$% # l= 404 24084 = 0.0495)! 0,1/2 + 202 = 0,0.0499.C$% x= (3 2, 9 4) h 0lVWnP:heop0lHIKKT+q ;r s !70(VWx ; !70 1 8.6tuvCw4+xy+!z( J Euclidean | )0 W P minimize f(x,y) = x2+ y2subject to (x 1)3= y2. (a)7 7 7Y 7=(a)7 = V W: 05101520253035 10 8 6 4 2 0 2 4 6 8 10 x y +P:eb:e y!7x= 1,y= 0. (

15、b) 3 k 2 SC:h ; y!7 2 Lagrange? y!7 2 E Lagrange? b? ? = 1 6 . ?!7(y 78= = 1 y!7 0*9M ? 0a CBTA1B ? 0. 7B=(M IRnn,C IRmm PM ? 0(C BTA1B ? 0. ? ? C BTA1B ? 0, A ? 0. y IRnn, - T1y = ? x1 x2 ? , j:x1 IRnm,x2 IRm. ; yTMy =yTTT ? A0 0C BTA1B ? T1y xT 1 x2 ? A0 0C BTA1B ? x1 x2 ? = xT 1Ax1+ xT2C BTA1Bx2

16、!?,34? ; PC BTA1B ? 0, A ? 0 (M ? 0. 6 ? ? B 9? ? 9.1! #$% (b)minimize x1+ x2subject tox2 1+ x22 1,x1+ x2= 3; (c)minimize x1x2subject tox2 1+ x22= 2. (a)()*+,-(1,1),./(a)0213(1,1). (b)+,43,./(b)5 7 4 ?0 x1x2 A 0B13-1.CD(c)x= (1,1) (1,1). 9.2EF-x(0) GH 4x | ATx = b(IJ7AKLM) NOP 0 Q +DRS3TUVW minimize

17、 1 2(x x (0)T(x x(0), subject toATx = b. XJY Z = (ATA)1(b ATx(0), Z x= x(0)+ A(ATA)1(b ATx(0). Y = A = a)*KJ_8abcd02.x(0) Gef aTx = b NOP |b aTx(0)| kak . XJY L(x,) = 1 2x Tx xT 0 x + 1 2x T 0 x0 T(Ax b) xL(x,) = x x0 AT 1 KKTgh x x0 AT= 0, Ax b= 0. i 3 A,jLjMj02k Q gjhjjlj)j*jmjnjojpjqAx Ax0 AAT =

18、0. r Ax = bsutu0wvAAT = bAx0.xzyA,uLuMu0wAAT+uu v G ! #%m i (b) x(0)= 0,(0)= 1; v G ! #! Q Lagrange L(x,) = x1+ x2 (x2 x2 1). q x(k)02c(x(k) = x(k) 2 (x(k) 1 )2, a(k)= 2x(k) 1 1 # ,g(k)= ? 1 1 ? ,W(k)= 2L(x(k),(k) = ? 2(k)0 00 ? . 6 SQP7802TUVW 3 minimize 1 2s TW(k)s + g(k)Ts subject toc(x(k) + a(k)

19、 Ts 0, = k = 0,x(k)= (0,0)?0 c(x(0) = 0,a(0)= (0,1),g(0)= (1,1),W(0)= ? 2(0)0 00 ? . rSJ_8st02v minimize0s2 1+ s1+ s2 subject tos2 0. (a)(0)= 0,:TUVW ; 5 k = (b)j(0)= 1,jjSjTjUjVjW jjvjs(0)= (0.5,0), j Lagrange (1)= 1.qx(1)= x(0)+ s(0)= (0.5,0). lTU s c(x(1) = 0.25,a(0)= (1,1),g(0)= (1,1),W(1)= ? 20

20、 00 ? . 3 rSJ_8stTUVW 02v minimizes2 1+ s1+ s2 subject to0.25 s1+ s2 0. STUVW vs(1)= (0,0.25), Lagrange (2)= 1. qx(2)= x(1)+ s(1)= (0.5,0.25). X Z k. Q -$0y8SQP s U+v G 9.14EuFuux2 1+x22= 1. u$ 6 uKu-(0,0),(0,1),(0.1,0.02),(0.1,0.02) x2 1+ x22= 1 6 x2 3c(x2) + c(x2)T(x x2) = 002 x2= 1;+vx2 1+ x22= 1 6 I-%=ABCDEF GH. +?IJ = ABFKLM H.%k = 0,1,21/0$ (d)NO J #.P7RQ y k = 1 z + 1= 0.01,x1= (0,1/33) ; 1 y k = 2 z + 2= 0.001,x2= (0,1/333) ; | qr 34Lagrange 5 * LA(x,) = 1 2(