数值最优化算法与理论第11章习题

1、? ? 2007-12-1 1 ? minf(x) s.t.gI 0,I = 1,m1, hE(x) = 0,E = m1,m. ?x(k)?minFk(x)x Rn? ? x?x(k)? x?(*)? ? ?8.2.1 ? xK-K-T? x?(1)?K-K-T ? ? 2 ? (1) ?x(k)? ( Fuk(x(k) Fuk(x(k1) Fuk1(x(k1) Fuk1(x(k) 1 ? u1 k s(xk) + u1 k1s(xk1) u 1 k s(xk1) + u1 k1s(xk) s(xk) s(xk1) uk s(xk) s(xk1) uk1 s(xk) s(xk1)(u1 k

2、u1 k1) 0 ?uk?s(xk) s(xk1). ? (2) ?Fu?x(k)? Fuk+1(x(k+1)Fuk+1(x(k) =f(x(k) + u1 k+1s(xk) f(x(k) + u1 k s(xk) =Fuk(x(k) ?Fuk(x(k)? (3) ?Fuk(x(k1) Fuk(x(k)? fx(k1)+ u1 k s(xk1) f(x(k) + u1 k s(xk) f(x(k) fx(k1)+ u1 k (s(xk1) s(xk) fx(k1) ?f(xk)? 3 ? ?f?gi,i I?x1,x2 D? 0,1 ? f(x1+ (1 )x2) f(x1) + (1 )f(

3、x2) gi(x1+ (1 )x2) gi(x1) + (1 )gi(x2),i I 2 ?G(x) = logx,?G(x) = logx? loggi(x1+ (1 )x2)loggi(x1) + (1 )gi(x2) loggi(x) + (1 )log gi(x2) ? Fu(x1+ (1 )x2)=f(x1+ (1 )x2) u1 X iE loggi(x1+ (1 )x2) f(x1) + (1 )f(x2) u1 X iE loggi(x1) + (1 )loggi(x2) =f(x1) u1 X iE loggi(x1) + (1 )f(x2) u1 X iE loggi(x2

4、) =Fu(x1) + (1 )Fu(x2) ?FuD0? ? 4 (1) ? ? minf(x) = 1 2x 2 1+ x 2 2 s.t. g1(x) = 3x1 x2+ 10 0 g2(x) = x1 1 0 g3(x) = 4 x1 0 ?x?D? Fu(x)=f(x) + 1 2us(x) =f(x) + 1 2u(3x1 + x2 10)2+ (x1 1)2+ (x1 4)2 ? Fu(x) = x1+ 3u(3x1+ x2 10) + u(x1 1) + u(x1 4) 2x1+ u(3x1+ x2 10) ! 3 ?Fu(x) = 0? xu= ( 5u 2u 5,10 15

5、u + 5 2u 5 )T ?u ?xu (5 2, 5 2) T = x (2) ? ? minf(x) = 1 2x 2 1 x1x2+ x 2 2 2x1 s.t. g1(x) = 3x1 x2+ 2 0 g2(x) = 1 x1 0 ?x?D? Fu(x)=f(x) + 1 2us(x) =f(x) + 1 2u(3x1 + x2 2)2+ (x1 1)2 ? Fu(x) = x1 x2 2 + 3u(3x1+ x2 2) + u(x1 1) x1+ 2x2+ u(3x1+ x2 2) ! ?Fu(x) = 0? xu= ( 2u 3u 1 u + 2 3u 1 u2+ 3u + 2

6、u2+ 33u 1 , u2+ 3u + 2 u2+ 33u 1 )T ?u ?xu (1,1)T= x 5 ? F(x)=f(x) 1 X iI loggi(x) = 1 2x 2 1+ x 2 2 2x1 1(log(x1 + x2 2) + logx1+ log(1 x1) 4 ?minF(x),x Rn, ? F(x) = (x12 1 (x1+ x2 2) 1 x1 + 1 (1 x1),2x2 1 (x1+ x2 2) T = 0 ? x() = ? 6 ?x(k) kK x,?x?11.8? ?x(k)? min Lk(x) ? Lk(x(k) Lk(x) = f(x) + (k

7、)ThE(x) + 1 2kkhE(x )k2 = f(x) Lk(x(k) = f(x(k) + (k)ThE(x(k) + 1 2kkhE(x (k)k2 f(x) 0 khE(x(k)k2 2(f(x) f(x(k) (k)ThE(x(k) k ?k?k?k +,?k , ?x(k) kK x,k ,? S( x) = 0 ? x?11.8?h( x) = 0. ?11.2.1?f(x) f(x(k), ?k ,?f(x) f( x), ?x(k)?11.8? 5 7 8 (1) ? Lu(x,)=L(x,) + 1 2us(x) =f(x) h(x) + 1 2uh 2(x) = 1 2x 2 1+ 1 6x 2 2 (x1+ x2 1) + 1 2u(x1 + x2 1)2 Lu(x,) = x1 + u(x1+ x2 1) 1 3x2 + u(x1+ x2 1) ! ?Lu(x,) = 0? x(k)= (k + uk 4uk+ 1 , 3(k+ uk) 4uk+ 1 )T= (k + 6 25 , 3(k+ 6) 25 )T k+1=k ukh(x(k) =k 6(4(k + 6) 25 1) = 1 25k + 6 25 = 1 25k+