A GLOBAL CONVERGENCE THEORY FOR ARBITRARY NORM TRUST REGION METHODS FOR NONLINEAR EQUATIONS

[摘要] In this research we extend the Levenberg-Marquardt algorithm for approximating zeros of the nonlinear system F(x) = 0, where F is continuously differentiable from ${m I!R}sp{n}$ to ${m I!R}sp{n}.$ Instead of the $lsb 2$-norm, arbitrary norms can be used in the objective function and in the trust region constraint. The algorithm is shown to be globally convergent. This research was motivated by the recent work of Duff, Nocedal and Reid. A key point in our analysis is that the tools from nonsmooth analysis, namely locally Lipschitz analysis, allow us to establish essentially the same properties for our algorithm that have been established for the Levenberg-Marquardt algorithm using the tools from smooth optimization. In our analysis, the sequence generated by the algorithm is the couple $(xsb{k},deltasb{k})$ where $xsb{k}$ is the iterate and $deltasb{k}$ the trust region radius. Since the successor $(xsb{k+1},deltasb{k+1})$ of $(xsb{k},deltasb{k})$ is not unique we model our algorithm by a point-to-set map and then apply Zangwill;;s theorem of convergence to our case. It is shown that our algorithm reduces locally to Newton;;s method.

[发布机构] Rice University