由(7.4)，(7.6)得 l(JJ)-1S-[J(x*)TJ(x*】-1S(x)川 ≤I(JgJ)-1IS-S(x)川 +I(JJ)-1-[J(a)FJ(x】-S(ax*)川 ≤2al[J(z*)YJ(x*】-1lxk-x‖+yS(x*)-x* =O(xk-x*)川. 即 (JRJ)-1Sk‖≤[J(a)TJ(x*-1S(x*)川 +I(JgJ)-1Ss-[J(c*)TJ(x*】-1S(x*)川 = Il[J(*)TJ(c*】-1S(*)‖+O(lzk-x.(7.8) 将(7.8)代入(7.7)即得本定立的结论.证毕. Back Close
11/33 JJ II J I Back Close d (7.4), (7.6) k(J T k Jk) −1Sk − [J(x ∗ ) T J(x ∗ )]−1S(x ∗ )k ≤ k(J T k Jk) −1 kkSk − S(x ∗ )k + k(J T k Jk) −1 − [J(x ∗ ) T J(x ∗ )]−1 kkS(x ∗ )k ≤ 2αk[J(x ∗ ) T J(x ∗ )]−1 kkxk − x ∗ k + γkS(x ∗ )kkxk − x ∗ k = O(kxk − x ∗ )k. = k(J T k Jk) −1Skk ≤ k[J(x ∗ ) T J(x ∗ )]−1S(x ∗ )k + k(J T k Jk) −1Sk − [J(x ∗ ) T J(x ∗ )]−1S(x ∗ )k = k[J(x ∗ ) T J(x ∗ )]−1S(x ∗ )k + O(kxk − x ∗ )k. (7.8) Ú (7.8) ì\ (7.7) = ½·(ÿ. y.
注7.1若问题(7.1)满足定理7.2的条件且最优解x*使得目标 函数值取零，则S(x*)=0,上面的结论表明迭代点列二阶收敛到x*. 但当F(x)在最优解点的函数值不为0时，由于Vf(x)略去了不容 忽视的项S(x),因而难于期待Gauss-Newton算法会有好的数值效果 §7.2 Levenberg-Marquardt方法 Gauss-Newton算法在迭代过程中要求矩阵J(xk)列满秩，而这一 条件限制了它的应用.为克服这个困难，Levenberg-Marquardt方法通 过求解下述优化模型来获取搜索方向 dk arg min ‖Jd+F2+d2, d∈Rm 其中k>0.由最优性条件知d满足 V(IIJkd+Fkl2+pld)=2[(+uI)d+JF]=0. Back Close
12/33 JJ II J I Back Close 5 7.1 eØK (7.1) ˜v½n 7.2 ^áÖÅ`) x ∗ ¶8I ºÍä", K S(x ∗ ) = 0, ˛°(ÿL²Sì: ¬Ò x ∗ . F(x) 3Å`):ºÍäÿè 0 û, du ∇2 f(x) — ÿN ¿ë S(x), œ Juœñ Gauss-Newton é{¨k–ÍäJ. §7.2 Levenberg-Marquardt ê{ Gauss-Newton é{3SìLߕᶛ J(xk) ˜ù, ˘ò ^áÅõ ßA^. èé—˘á(J, Levenberg-Marquardt ê{œ L¶)e„`z .5º|¢êï dk = arg min d∈Rn kJkd + Fkk 2 + µkkdk 2 , Ÿ• µk > 0. dÅ`5^á dk ˜v ∇(kJkd + Fkk 2 + µkkdk 2 ) = 2[(J T k Jk + µkI)d + J T k Fk] = 0.