Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties
Levenberg-Marquardt methods for constrained nonlinear equations with strong local convergence properties
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发表时间：
2004
影响因子：
2.4
通讯作者：
C. Kanzow
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文献类型：
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作者：
C. Kanzow
我们考虑寻找约束（不一定是平方）方程组的解的问题，即，我们考虑非线性方程组，并希望找到属于某个可行集的解。为此，我们提出了两个Levenberg-Marquardt型算法，它们计算搜索方向的方式不同。第一种方法在每次迭代中求解一个严格凸极小化问题，而第二种方法在每一步中只求解一个线性方程组。这两种方法收敛局部二次误差界的假设下，比标准的非奇异性条件弱得多。这两种方法都可以很容易地全球化。第二种方法的数值结果表明，该算法在实践中是相当不错的。
We consider the problem of finding a solution of a constrained (and not necessarily square) system of equations, i.e., we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set. To this end, we present two Levenberg-Marquardt-type algorithms that differ in the way they compute their search directions. The first method solves a strictly convex minimization problem at each iteration, whereas the second one solves only one system of linear equations in each step. Both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition. Both methods can be globalized in an easy way. Some numerical results for the second method indicate that the algorithm works quite well in practice.