常用的一维搜索方法

Tel:[phone]E-mail:[e-mail](x(k)+λd(k))=()s.t.SS3-+0+[a,b],x(-+),min()axbfx≤≤min()axbfx≤≤(),(),fxaxbFxothers≤≤=+∞(),(),fxaxbFxothers≤≤=+∞min()min()axbxfxFx≤≤−∞≤≤+∞=min()min()axbxfxFx≤≤−∞≤≤+∞=z“—”z0.618zzNewtonz1“—”“—”1xR,h0023,454x:=x+h,521().fxϕ=1().fxϕ=2().fxhϕ=+2().fxhϕ=+21,ϕϕ21,ϕϕ12:,:2hhϕϕ==12:,:2hhϕϕ==?hε≤?hε≤,hε≤,hε≤*xx=*xx=,4hh=−,4hh=−1ah,[c,d]1)f(a),f(a+h),f(a)f(a+h),,f(a+3h).f(a+h)=f(a+3h),c=a,d=a+3h;.2)f(a)f(a+h),1/4-h/4,f(a)f(a-h/4),c=a-h/4,d=a+h;:1.h.(,)2.f(x),()3.()20.6180.618.,1f(x)[a,b]1∃x*[a,b][a,b]2x1,x2,ax1x2b,1ºx2x*f(x1)f(x2);2ºx1x*f(x1)f(x2).f(x)[a,b]0tfx*0tfx*1fRR[a,b]ax1x2b1f(x1)f(x2)x*[x1,b],2f(x1)f(x2)x*[a,x2],x1x2bx1x2bProof.1x*[a,b]zx*[a,x1]f(x1)f(x2)2x*[x2,b]f(x1)f(x2),x1x2,f(x1)f(x2),[a,x1][x2,b].1x1–a=b-x2……“”“”2t=()x1x2t:[a,x2]([x2,b]),[,x1],x1x2b212(2)xxtbxαααα−−==−−x2=a+t(b-)x1=a+t(x2-)t2+t–1=0t0.618:t2=1-t,x1=a+(1-t)(b-)x2=a+t(b-))(251±−=t0.618[a,b],02/)15(−=tx1=+1-t(b-)x2=+t(b-)b-?STOP;x*=(+b)/2yesfx1-fx20?No=x1,x1=x2x2=+t(b–)yesb=x2,x2=x1x1=+1-t(b-)Nox1x2bx2bx1x2bx10.6183f(x)[a,b]x=(a+b)/2,f(x)=0,x,x=x*;f(x)0,x,x*x[x,b];f(x)0,x,x*x[a,x].()xbtg0f(x)xbtg0f(x)[a,b],[a,b][a,b][a,b][a,b][a,b]*x()*'0fx=*xx()fx()*'0fx*xx()'0fx()()'0,'0fafb()fx*x()*'0fx=*x02abx+=()0'0fx[]0,ax[]0,ax()0'fx*22abbax+−−[a,b],,(,,,[a,b].0x()0'0fx0x10xxx=+∆x∆()1'0fx01,axbx==()0'0fx21xxx=+∆x∆2xxx=+∆()2'0fx[]12,xx()0'0fx0x0x∆4Newton4.1Newtonf(x)xkf(x)=f(xk)+f(xk)(x-xk)+(1/2)f(xk)(x-xk)2+o||(x-xk)2||g(x)=f(xk)+f(xk)(x-xk)+(1/2)f(xk)(x-xk)2g(x)f(x)f(xk)0g(x)=f(xk)+f(xk)(x-xk)=0xk+1=xk–f(xk)/f(xk).xk+1NewtonNewtonx1,1,20k=1f'(xk)1?xkyNf(xk)0?NYxk+1=xk-f(xk)f(xk)|xk+1-xk|2YNk=k+11minf(x)=arctantdtf(x)=arctanx,f(x)=1(1+x2)xk+1=xk-(1+xk2)arctanxkx1=1kxkf(xk)1f(xk)110.785422-0.5708-0.51871.325830.1169-0.11641.01374-0.001095-0.001095x4x*=0x1=20x∫kxkf(xk)1f(xk)121.107152-3.5357-1.295213.50313.954.2f(x)2323f(x)322243332332x1x2x3f(x1),f(x2),f(x3)(“-”)ax2+bx+c=g(x)ax12+bx1+c=g(x1)ax22+bx2+c=g(x2)ax32+bx3+c=g(x3):a,b123231312122331()()()()()()()()()xxgxxxgxxxgxaxxxxxx−+−+−=−−−−222222123231312122331()()()()()()()()()xxgxxxgxxxgxbxxxxxx−+−+−=−−−2bxa−=−1.x1x2x3f(x1)f(x2),f(x2)f(x3)2.a0,g(x)3.f(x)x2x1x3x2x[]13,xxx∈2xxε−*xx=123,,,xxxx1.2.1.33.“-”4.5minf(x)x(k)d(k)x(k+1)=x(k)+kd(k):1f(x(k)+kd(k))f(x(k));2k0,,,1Goldstein(1965)f:RnRxd,fT(x)d0(d)1f(x+d)f(x)+fT(x)d2f(x+d)f(x)+(1-)fT(x)d,(0,12)=0.11f(x+d)y=f(x)+fT(x)d2f(x+d)y=f(x)+(1-)fT(x)dfT(x)d(1-)fT(x)df(x(k))=11NoYes2Yesk=No=(32)=(12)2Armijo(1966)Goldstein1f(x+d)f(x)+fT(x)d2f(x+d)f(x)+µfT(x)d012µ5101967Goldstein1f(x+d)f(x)+fT(x)d2f(x+d)f(x)+σfT(x)d012σ13Wolfe(1969)-Powell(1976)Goldstein23f(x+d)f(x)+fT(x)d4fT(x+d)dσfT(x)d012σ1=0.1,σ=0.7f(x(k))β=σfT(x)dββ=13NoYes4Yesk=No=(32)=(12)123244|fT(x+d)d|-fT(x)dfT(x)d|fT(x+d)d|-fT(x)d=0=0.1k