最速下降法和牛頓法求最小值點的算法及結(jié)果

1、工程優(yōu)化作業(yè)姓名：龔尚映學號：11031218161.最速下降法計算：f=(x-1)2+5(x-5)212(1)程序代碼如下：#include#includedoublef1(doublex,doubley)doubler;r=(pow(x-1,2)+5*pow(y-5),2);returnr;/最速下降法求最優(yōu)解voidmain()doubleh=3,x0=3,x1,y0=6,y1,s,r0,r1;doublee0=0.000001,e1=0.000001;intk=0;s=sqrt(pow(2*x0-2,2)+pow(10*y0-50,2);printf(%dx1=%fx2=%fs=%fn

2、,k,x0,y0,s);while(se1)x1=x0;y1=y0;r0=f1(x0,y0);h=3;/一維搜索，成功失敗法while(fabs(h)e0)r1=f1(x1-h*2*(x1-1),(y1-h*10*(y1-5);if(r1e0)h=(-1)*h/4;s=sqrt(pow(2*x0-2,2)+pow(10*y0-50,2);k+;printf(%dx1=%fx2=%fs=%fn,k,x0,y0,s);共11頁第2頁共11頁第2頁姓名：龔尚映學號：1103121816printf(xl=%fx2=%f,x0,y0);(2)初始值設(shè)為x1=3,x2=6時，運行結(jié)果如下圖1-1：rxl

3、=3.000900 x2=6.0S0000s=10.7703301xl-2.2CO90Ox2-4.12EOQ0s-9.1801372xl=1.781250 x2=5.765625s=7.8140623xl=1.634766x2=5.0-17852s=l.35671?4xl=1.158691x2.868408s=1.353651xl=1.128937x2.991776s=0.270671xl=1.032234x2=5.02261?s=0.2351827xl=1.020146x2M.980210s=0.2019628xl=1.016369x2=4.998763s=0.0349979xl=1.004

4、092x2=5.003401s=0.0349851Rxl=1.003325x2=5.600213s=0.0069Bl11xl=1.009831x2=4.999415s=0.00607812xl=1.009520 x2=5.600512s=Q.0S522Q13xl=1.000422x2=5.000325=0.00090314xl-1.009106x2-4.999912s-0.00090415xl=1.009086x2=4.999995S=0.0001B016xl=1.009821x2=5.6000153=0.00015717xl=1.000013x2=4.999987$=0.00013518xl

5、=1.009011x2=4.999999S=0.00002319xl=1.009003x2=5.00002S=0.00002320 xl=1.009002x2=5.S00000S=0.00009521xl=1.009001x2=5.S00000s=0.00009422xl=1.009000 x2=5.600000S=0.00009323xl=1.QBBBBBx2=5-QQBSBBst=0_0(313001xl=1.000000 x2=5.0000S0Pressanykeytocontinue圖1-1(3)初始值設(shè)為x1=30000,x2=60000時，運行結(jié)果如下圖1-2：工程優(yōu)化作業(yè)工程優(yōu)化

6、作業(yè)共11頁姓名：龔尚映學號：1103121816第3頁兀xl=36000.000600 x2=0000.000000s=602942.5864081xl=24375.187500 x2=3754.687500s=61501.3796592xl=15234.867188x2=-3275.976563s=44774.586076axl=9522.166992x2=2875.854492s=34449.8336014xl=5951.7293701x2=-2506.997681s=27796.7256155xl=3720.205856v2=2202.997971s=23204.514144Exl=2

7、325.5036601x2=-1918.248224s=19786.4004387xl=1889.65922tj#includevmath.hdoublefl(doublex,doubley)return(pow(x-l,2)+5*pow(y-5,2);牛頓法求最優(yōu)解voidmain()doubleh=3,x0=3,xl,y0=6,yl,s,r0,rl;doublee0=0.000001,el=0.000001;共11頁第4頁共11頁第4頁姓名：龔尚映學號：1103121816intk=0;s=sqrt(pow(2*x0-2,2)+pow(10*y0-50,2);printf(%dx=%fy=

8、%fs=%fn,k,x0,y0,s);while(se1)x1=x0;y1=y0;r0=f1(x0,y0);h=3;一維搜索while(fabs(h)e0)r1=f1(x1-h*(x1-1),(y1-h*(y1-5);if(r1e0)h=(-1)*h/4;s=sqrt(pow(2*x0-2,2)+pow(10*y0-50,2);k+;printf(%dx=%fy=%fs=%fn,k,x0,y0,s);printf(x=%fy=%f,x0,y0);初始值設(shè)為x1=3,x2=6時，運行結(jié)果如下圖2-1：x=3.000000y=6.000000s=10.7703301x-l.500000y-5.25

9、00S03-2.6925622x=l.125000/=5.062500s=0.6731463x=1.031250y=5.0丄5625s=0.1682864x=l.007813y=5.003906s=0.042072bx=1.001953y=5.000977$=0.EJ1EJ518x=1.000488y=5.000244$=0.002629?x=1.000122V=5-000061s=0.000657Bx=1.000031U=5.000015s=0.0001649X=1.000008y=5.000004S=0.00004110X=1.000002y=5.000001S=0.00001011X=1

10、.000000y=5.000000S=0.00000312X=1.000000y=5.000000S=0.000001X=1.000000y=5i.000080Pressanykeytocontinue.圖2-1(3)初始值設(shè)為x1=30000,x2=60000時，運行結(jié)果如下圖2-2：兀X=38000.0080009=60冏00.000冏00s=6029475B64881y=15003s=1507352x=1875.937500y=3754.687500s=37683.9116503x=469.734375y=942.421875s=9420.977913斗x=118.183594y=239

11、.355469s=2355.2444785x=3B.295898y=63.588867s=588.811120Gx-B.323975y-19.64721?s-147.2027807x=2.830994#includevmath.hdoublef1(doublex,doublex1,doubley,doubley1)doubler;r=(pow(x-1,2)+pow(x1-1,2)+5*pow(y-5,2)+5*pow(y1-5,2);returnr;最速下降法求最優(yōu)解voidmain()x1,x2,x3,x4分別為多項式函數(shù)中四個分量的初始值x5,x6,x7,x8分別為對應(yīng)的下一個值doubl

12、ed=3,x1=3,x5,x2=4,x6,x3=8,x7,x4=6,x&s,r0,r1;doublee0=0.000001,e1=0.000001;intk=0;s=sqrt(pow(2*x1-2,2)+pow(2*x2-2,2)+pow(10*x3-50,2)+pow(10*x4-50,2);printf(%dx1=%fx2=%fx3=%fx4=%fs=%fn,k,x1,x2,x3,x4,s);while(se1)x5=x1;x6=x2;x7=x3;共11頁第6頁共11頁第6頁姓名：龔尚映學號：1103121816x8=x4;r0=f1(x1,x2,x3,x4);d=3;/一維搜索while

13、(fabs(d)e0)r1=f1(x5-d*2*(x5-1),(x6-d*2*(x6-1),(x7-d*10*(x7-5),(x8-d*10*(x8-5);if(r1e0)d=(-1)*d/4;s=sqrt(pow(2*x1-2,2)+pow(2*x2-2,2)+pow(10*x3-50,2)+pow(10*x4-50,2);k+;printf(%dx1=%fx2=%fx3=%fx4=%fs=%fn,k,x1,x2,x3,x4,s);printf(nx1=%fx2=%fx3=%fx4=%fs=%fn,x1,x2,x3,x4,s);(2)初始值設(shè)為x1=3,x2=4,x3=8,x4=6時，運行結(jié)

14、果如下圖3-1：工程優(yōu)化作業(yè)姓名：龔尚映學號：工程優(yōu)化作業(yè)姓名：龔尚映學號：1103121816共11頁第8頁工程優(yōu)化作業(yè)工程優(yōu)化作業(yè)共11頁姓名：龔尚映學號：1103121816xl=3.Q&0&QQxl=2.625600 xl=1.406250 xl=1.253906xl=1.206299xl=1.051575xl=1.041904xl=1.010476xl=1.006548xl=1.005320*2=4.000000 x2=3.4375005(2=1.609375x2=1.380859k2=1.309448x2=1.077362x2=1.0628575(2=1.015714x2=1.00

15、9821xl=3.Q&0&QQxl=2.625600 xl=1.406250 xl=1.253906xl=1.206299xl=1.051575xl=1.041904xl=1.010476xl=1.006548xl=1.005320*2=4.000000 x2=3.4375005(2=1.609375x2=1.380859k2=1.309448x2=1.077362x2=1.0628575(2=1.015714x2=1.009821k2=1.007980 x3=8.000000 x3=5.187500 x3=4.484375x3=5.451172x3=5.028198x3=4.9224S5x3

16、=4.995153x3=5.013328x3=4.988338x3=4.999271x4=6.000000 x4=5.062500 x4=4.828125x4=5.150391x4=5.009399x4=4.974152x4=4.998384x4=5.004443x4=4.996113x4=4.999757s=32.434549s=6.183395s=5.629078s=4.843081s=0.801011s=0.838283s=B.159492s=B.145479s=0.125175s=0.02066310 xl=1001330 x2=1-001995x3=5.002004x4=5.00%6

17、8s=032166611xl=l001081x2=1-001621x3=5.000125x4=5.000042s=000411412xl=100Q270 x2=1.000405x3=4.999655x4=4.999885s=000376013xl=1006169x2=1.000253x3=5.000301x4=5.000100s=000323514xl=l00S137x2=1.00Q206x3=5.000019x4=5.000006s=000053315xl=1006034x2=1.000051x3=4.999948x4=4.999983s=000056016xl=l006028x2=1.00

18、0042x3=4.999997x4=4.999999s=000010617xl=100S007x2=1.000010 x3=5.000009x4=5.000003s=000009718xl=l00S004x2=1.QQ&QQ7x3=4.999992x4=4.999997s=000008419xl=lx2=1.00S005x3=5.000000 x4=5.000000s=000001420 xl=100S001x2=1.000001x3=5.000001x4=5.000000s=000001421xl=l006001x2=1.000001x3=5.&Q&00 x4=5.000000s=00000

19、0322xl=100S000 x2=1.000000 x3=5.000000 x4=5.000000s=000000323xl=lx2=1.QQ&QQQx3=5.000000 x4=5.000000s=000000224xl=100S000 x2=1.000000 x3=5.000000 x4=5.000000s=0300000工程優(yōu)化作業(yè)姓名：龔尚映學號：工程優(yōu)化作業(yè)姓名：龔尚映學號：1103121816共11頁第8頁工程優(yōu)化作業(yè)工程優(yōu)化作業(yè)共11頁工程優(yōu)化作業(yè)姓名：龔尚映學號：工程優(yōu)化作業(yè)姓名：龔尚映學號：1103121816共11頁第8頁工程優(yōu)化作業(yè)工程優(yōu)化作業(yè)共11頁xl=1.0060

20、00 x2=1.006000 x3=5.000000 x4=s.000000S=0.000000Pressanykeytocontinue圖3-1(3)初始值設(shè)為xl=30000,x2=40000,x3=80000,x4=60000時，運行結(jié)果如下圖3-2：Hxl=30600.000000 x2=40000.000000k3=86000.000060 x4=60000.000000s=100491?.63E9571xl=24375.187500 x2=32500.187500k3=5604.687500 x4=3754.687500s=102503.150987$xl=15234.867188

21、x2=20312.992188k3=-4369.726563x4=-32?5.976563s=74625.028703xl=9522.166992x2=12695.995117x3=3832.885742x4=2B75.854492s=57417.014593xl=5951.729370 x2=7935.371948x3=-334.400024x4=-2506.997681s=46328.425833x2=4959.98248x3=2935.725821x4=2202.997971s=38t74.674145$xl=2325.503660 x2=3100.364042x3=-2559.3843

22、94x4=-1918.248224s=32977.759233卜xl=1889.659224x2=2519.233284x3=-155.274025x4=-115.203014s=6606.6468738xl=473.164806x2=630.558321x3=445.753568x4=335.558289s=5729.780283Pxl=296.103004x2=394.473951x3=-380.659372x4=-284.238502s=492B.044345xl=240.771191x2=320.697585x3=-19.103711x4=-13.077406s=854.145905x

23、l=68.942798x2=80.924396x3=71.285205x4=54.712868s=852.310945xl=49.703523x2=65.938572x3=9.142825x4=8.107854s=170.405058xl=13.175881x2=17.234643x3=-6.392770 x4=-3.544399s=148.07922914xl=S.G09925x2=11.146652x3=14.9GSG73x4=12.47634?8=127.16322815xl=7.183064x2=9.244155x3=5.623042x4=5.467272s=22.03265616xl

24、=2.545766x2=3.061039x3=3.286634x4=3.715002s=22.02800917xl=2.255935x2=2.674594x3.892915x4=4.919C88s=439525718xl=1.313984x2=l.418648x3=5.294485x4=5.220859s=382693119xl=1.196240 x2=l.261655x3M.742326x4=4.806748s=32866562&xl=1.159445x2=l.212595x3=4.983895x4=4.987922s=0568332xl=1.039861x2=1.05314?x3=5.04

25、1Z88x4=5.B33215s=B567315xl=1.032387x2=l.043183x3=5.002768x4=5.002076s=0113367S3xl-1.00809?x2-1.01079(,x3-4.992388x4-4.9942910?890304xl=1.005061x2=l.00674?x3=5.006660 x4=5.004995s=0084947xl=1,004112x2=l.005482x3=5.00B416x4=5.000312s=001466026xl=1.001028x2=l.001371x3M.998855x4=4.999141s=001471427xl=1.

26、000835x2=l.001114x3.999928x4=4.999946s=000292428xl=1.000209x2=l.000278x3=5.000197x4=5.000148s=0002556E9xl=1.000130 x2=l.000174x3M.999828x4=4.999071s=0002196xl=1.000106x2=l.000141x34.999989x4=4.999992s=000037831xl=1.000027x2=lx3=5.009030 x4=5.000022s=00003802si-1.000022x2-1.Q00O29x3-5.009002x4-5.0006

27、013-000007533xl=1.000005x2=l.000007x3.999995x4=4.999996s=0000066xl=1.0QQ0Q3x2=1-BQQQQ4x2=5-QQH0Q4x4=5.B0BQS32=00BBB5735xl=1.000003x2=l.000004x3=5.009000 x4=5.000S00s=0000010肚xl=1.000001x2=lx3M.999999x4=4.999999s=0000010a?xl=1.000001x2=l.000001x3=5.009000 x4=5.000S00s=000000238xl=1.000000 x2=l.000000

28、 x3=5.009000 x4=5.000S00s=0000002xl=1.000000 x2=lx3=5.00B000 x4=5.000000s=000000140 xl=1.000000 x2=l.000000 x3=5.009000 x4=5.000S00s=0000000圖3-2牛頓法計算：f=(x-1)2+(x-1)2+5(x-5)2+5(x-5)21234(1)程序代碼如下：#includevstdio.h#includevmath.hdoublef1(doublex,doublex1,doubley,doubley1)doubler;r=(pow(x-1,2)+pow(x1-1,

29、2)+5*pow(y-5,2)+5*pow(y1-5,2);returnr;最速下降法求最優(yōu)解voidmain()工程優(yōu)化作業(yè)工程優(yōu)化作業(yè)姓名：龔尚映學號：1103121816工程優(yōu)化作業(yè)工程優(yōu)化作業(yè)共11頁共11頁第9頁/x1,x2,x3,x4分別為多項式函數(shù)中四個分量的當前值/x5,x6,x7,x8分別為多項式函數(shù)中四個分量對應(yīng)的下一個值doubled=3,x1=3,x5,x2=4,x6,x3=8,x7,x4=6,x8,s,r0,r1;doublee0=0.000001,e1=0.000001;intk=0;s=sqrt(pow(2*x1-2,2)+pow(2*x2-2,2)+pow(10

30、*x3-50,2)+pow(10*x4-50,2);printf(%dx1=%fx2=%fx3=%fx4=%fs=%fn,k,x1,x2,x3,x4,s);while(se1)x5=x1;x6=x2;x7=x3;x8=x4;r0=f1(x1,x2,x3,x4);d=3;/一維搜索while(fabs(d)e0)r1=f1(x5-d*(x5-1),(x6-d*(x6-1),(x7-d*(x7-5),(x8-d*(x8-5);if(r1e0)d=(-1)*d/4;s=sqrt(pow(2*x1-2,2)+pow(2*x2-2,2)+pow(10*x3-50,2)+pow(10*x4-50,2);k

31、+;printf(%dx1=%fx2=%fx3=%fx4=%fs=%fn,k,x1,x2,x3,x4,s);printf(nx1=%fx2=%fx3=%fx4=%fs=%fn,x1,x2,x3,x4,s);(2)初始值設(shè)為x1=3,x2=4,x3=8,x4=6時，運行結(jié)果如下圖4-1：工程優(yōu)化作業(yè)工程優(yōu)化作業(yè)共11頁工程優(yōu)化作業(yè)工程優(yōu)化作業(yè)共11頁xl=3.0S0000 xl-1.500000 xl=1.125000 xl=1.031250 xl=1.007813xl=1.081953xl=1.060488xl=1.060122xl=1.000031xl=1.0S00881.0080021.0

32、000001.0000001.000000 xlxlxlxlxl=3.0S0000 xl-1.500000 xl=1.125000 xl=1.031250 xl=1.007813xl=1.081953xl=1.060488xl=1.060122xl=1.000031xl=1.0S00881.0080021.0000001.0000001.000000 xlxlxlxlx2=4.000900 x3=8.Q&QQ&Qx2-1.750000 x3-5.?50SS0 x2=1.187500 x3=5.187500 xZ=1.046875x3=5.B46875x2=1.011719x3=5.011719x2=1.002?30 x3=5.002930 x2=1.000732x2=1.000183x2=1.000346x2=1.000011x2=1.003003Ix2=1.009001IX2=1.009000Ix2=1.009000 x4=6.006000s=32.43454