Levenberg-Marquardt Algorithms for Nonlinear Equations, Multi-objective Optimization, and Complementarity Problems

The Levenberg-Marquardt algorithm is a classical method for solving nonlinear systems of equations that can come from various applications in engineering and economics. Recently, Levenberg-Marquardt methods turned out to be a valuable principle for obtaining fast convergence to a solution of the nonlinear system if the classical nonsingularity assumption is replaced by a weaker error bound condition. In this way also problems with nonisolated solutions can be treated successfully. Such problems increasingly arise in engineering applications and in mathematical programming. In this thesis we use Levenberg-Marquardt algorithms to deal with nonlinear equations, multi-objective optimization and complementarity problems. We develop new algorithms for solving these problems and investigate their convergence properties. For sufficiently smooth nonlinear equations we provide convergence results for inexact Levenberg-Marquardt type algorithms. In particular, a sharp bound on the maximal level of inexactness that is sufficient for a quadratic (or a superlinear) rate of convergence is derived. Moreover, the theory developed is used to show quadratic convergence of a robust projected Levenberg-Marquardt algorithm. The use of Levenberg-Marquardt type algorithms for unconstrained multi-objective optimization problems is investigated in detail. In particular, two globally and locally quadratically convergent algorithms for these problems are developed. Moreover, assumptions under which the error bound condition for a Pareto-critical system is fulfilled are derived. We also treat nonsmooth equations arising from reformulating complementarity problems by means of NCP functions. For these reformulations, we show that existing smoothness conditions are not satisfied at degenerate solutions. Moreover, we derive new results for positively homogeneous functions. The latter results are used to show that appropriate weaker smoothness conditions (enabling a local Q-quadratic rate of convergence) hold for certain reformulations.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Technische Universität Dresden

Year

2010

Author dc:creator

• Shukla, Pradyumn Kumar

Contributors dc:contributor

• Fischer, Andreas
• Ehrgott, Matthias

• Levenberg-Marquardt
• Multi-objective Optimization