数值最优化算法与理论第3章习题答案.docx

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1、BrYQ7;X2007-10-131(1)

2、MZ*!∇f(x(k))=2x1(k)!(k)=−2x1(k)!4x2(k)d−4x2(k)αk=−2x1(k)d1(k)+4x2(k)d2(k)(k)2(k)2d1+4d2iÆH.T4(k)−t2(k)x(k+1)=x(k)+αkd(k)=(x1,x2)Tt2+8t2+8((2)n,(2)n)Tk=2nn=1,2,···x(k)27272n41Tk=2n−1n=1,2,···(27)(9,−9)(2)i==!Z*!Z*"k→∞Fx(k)→x∗=0αk=1SC/.x(k)→x∗=0.1u:2gV∇f(x)=Qx

3、+q,d=−(Qx+q)α=−(xTQ+qT)dgVdTQd&x(k+1)=x(k)+αkd(k)Vminf(x),p∇f(x(k+1))=0Qx(k+1)+q=0q-Q(x(k)+αkd(k))+q=0NeQx(k)+αkQd(k)=0&Qd(k)=−Qx(k)+q=d(k)=α−1d(k)αkαkkQd(k)=α−1d(k),d(k)=60k(k)VQOs4OswV−1.udλ=αk3u:gV(1)?∇f(x)=Qx+qd(k)=−∇f(xk)=−(Qxk+q)2lgVpk∇f(x∗)=0d∗=0,Qx∗+q=0=2αk∇f(xk)TQ(xk−x∗

4、)−α2k∇f(xk)TQ∇f(xk)=−2αkd(k)T(Qxk−Qx∗)−α2kd(k)TQd(k)=−2αkd(k)T(−d(k)−q−(d∗−q))−α2kd(k)TQd(k)}(k)T(k)−2(k)TQd(k)=2αkddαkd=(xk−x∗)TQ(xk−x∗)−(xk+1−x∗)TQ(xk−x∗)=xkTQxk−xk+1TQxk+1+2(xk−xk+1T)q=uk(k)T=xkTQxk−(x(k)(k)TQ(x(k)(k)q+αkd)+αkd)−2αkd=xTQxk−[xkTQxk(k)TQxk2(k)TQd(k)(k)Tq+2αkd+αd]

5、−2αkdk(k)TQxk2(k)T(k)(k)T(k)k)=−2αd−αdQd−2αkd(−d−Qxk()T(k)−2(k)TQd(k)=2αddαdki(2)}(1)=2αk∇f(xk)TQ(xk−x∗)−α2k∇f(xk)TQ∇f(xk)d(k)Td(k)(k)Tk∗(k)Td(k)(k)T(k)(k)T(k)=−2Q(x−x)−ddd(k)T(k)dd(k)TQd(k)(k)T(k)dQddQddQd=d(k)Td(k)[−2d(k)TQ(xk−x∗)−d(k)Td(k)]d(k)TQd(k)3=ukQ(k)T(k)(k)T(k)=ddddd(k)

6、T(k)Qd=∇f(xk)T∇f(xk)∇f(xk)T∇f(xk)∇f(xk)TQ∇f(xk)kf(x)k4=kf(xk)k2(3)aGI1(2)_uk∇f(xk)k4kxk−x∗k2=k∇f(xk)k4k2k2k2b&uk∇f(x)kk∇f(x)k−1k∇f(x)kQQQkxk−x∗k2&=1k∇f(xk)k2−1Qkkxk−x∗k2=∇f(xk)k2−1Q=∇f(xk)TQ−1∇f(xk)=d(k)TQ−1d(k)=(Qxk+q)TQ−1(Qxk+q)=xkTQQ−1Qxk+2xkTq+qTQ−1q=}xkTQxk+2xkTq+qTQ−1q=(xk−x

7、∗)TQ(xk−x∗)=xkTQxk−2xkTQx∗+x∗TQx∗=xkTQxk+2xkTq−x∗Tq4=xkTQxk+2xkTq+(Q−1q)Tq=xTQxk+2xkTq+qTQ−1qu=645u:6F(d)=∇f(xk)Tdpx_ukdkGk∇F(d(k))=0&.∇F(d)==)11∇f(xk)(dT∇2f(xk)d)2−∇f(xk)Td(dT∇2f(xk)d)2∇2f(xk)ddT∇2f(xk)d∇f(xk)dT∇2f(xk)d−∇f(xk)Td∇2f(xk)d3(dT∇2f(xk)d)2d(k)=∇2f(xk)−1∇f(xk)CIu∇F(d(k

8、))=∇f(xk)[∇f(xk)T∇2f(xk)−1∇f(xk)]−[∇f(xk)T∇2f(xk)−1∇f(xk)]∇f(xk)=03(dT∇2f(xk)d)25u:7gV(1)pk∇2f(xk)d(k)+∇f(xk)k≤ηkk∇f(xk)k,k=0,1,···&k∇f(xk)k−k∇2f(xk)d(k)k≤k∇2f(xk)d(k)+∇f(xk)k≤ηkk∇f(xk)klgV(1−ηk)k∇f(xk)k≤k∇2f(xk)d(k)k≤k∇2f(xk)kkd(k)kNeηk→0,(k→∞)&limk∇f(xk)k≤lim1k∇2f(xk)k=limk∇2f(x

9、k)k≤Mkd(k)kk→∞k→∞1−ηkk→∞k∇f(xk)kN