Convergence and Complexity Analysis of a Levenberg–Marquardt Algorithm for Inverse Problems

The Levenberg–Marquardt algorithm is one of the most popular algorithms for finding the solution of nonlinear least squares problems. Across different modified variations of the basic procedure, the algorithm enjoys global convergence, a competitive worst-case iteration complexity rate, and a guaranteed rate of local convergence for both zero and nonzero small residual problems, under suitable assumptions. We introduce a novel Levenberg-Marquardt method that matches, simultaneously, the state of the art in all of these convergence properties with a single seamless algorithm. Numerical experiments confirm the theoretical behavior of our proposed algorithm

Convergence properties of a family of inexact Levenberg-Marquardt methods

We present a family of inexact Levenberg-Marquardt (LM) methods for the nonlinear equations which takes more general LM parameters and perturbation vectors. We derive an explicit formula of the convergence order of these inexact LM methods under the H$\mathrm{\ddot{o}}$derian local error bound condition and the H$\mathrm{\ddot{o}}$derian continuity of the Jacobian. Moreover, we develop a family of inexact LM methods with a nonmonotone line search and prove that it is globally convergent. Numerical results for solving the linear complementarity problem are reported

A globally convergent neurodynamics optimization model for mathematical programming with equilibrium constraints

summary:This paper introduces a neurodynamics optimization model to compute the solution of mathematical programming with equilibrium constraints (MPEC). A smoothing method based on NPC-function is used to obtain a relaxed optimization problem. The optimal solution of the global optimization problem is estimated using a new neurodynamic system, which, in finite time, is convergent with its equilibrium point. Compared to existing models, the proposed model has a simple structure, with low complexity. The new dynamical system is investigated theoretically, and it is proved that the steady state of the proposed neural network is asymptotic stable and global convergence to the optimal solution of MPEC. Numerical simulations of several examples of MPEC are presented, all of which confirm the agreement between the theoretical and numerical aspects of the problem and show the effectiveness of the proposed model. Moreover, an application to resource allocation problem shows that the new method is a simple, but efficient, and practical algorithm for the solution of real-world MPEC problems