Levenberg–Marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints

By Christian Kanzowa, Nobuo Yamashita, Masao Fukushimab. 2004

We consider the problem of 5nding a solution of a constrained (and not necessarily square) system of equations, i.e., we consider systems of nonlinear equations and want to 5nd a solution that belongs to a certain feasible set. To this end, we present two Levenberg–Marquardt-type algorithms that di8er in the way they compute their search directions. The 5rst 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.

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Citations in the corpus

• Scene Flow to Action Map: A New Representation for RGB-D Based Action Recognition with Convolutional Neural Networks (2017, methodology): “A better estimation of H is achieved by minimising (for example, using Levenberg-Marquardt algorithm [15]) the following objective function with more matching points:”
• Trust-region quadratic methods for nonlinear systems of mixed equalities and inequalities (2009, methodology): “[24,25] proposed global projected Levenberg–Marquardt methods.”
• On the inexactness level of robust Levenberg–Marquardt methods (2010, background): “Such problems arise in different fields such as optimization, variational inequalities, and game theory, for examples see 5 , 7 .”
• A machine learning framework for sport result prediction (2019, methodology): “Five different training algorithms were applied to the data: gradient-descent BP (BP), gradient-descent with a momentum parameter (BPM), Levenberg-Marquadt (LM), and conjugate gradient descent (CGD) [37,39,20,46].”
• Backtracking Search Optimization Algorithm for numerical optimization problems (2013, background): “Since the size of the search space increases with the dimension of the optimization problem, it is difficult to find the global optimum for such problems with classical techniques [9].”
• A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints (2015, methodology): “There are numerous kinds of methods have been proposed to solve the constrained nonlinear equations, such as the trust-regionmethod [2] and the Levenberg–Marquardt method [10].”
• Lightning search algorithm (2015, background): “However, finding the ptimum solution for such problems using classical techniques ecomes complicated as the size of the search space increases with he dimension of the optimization problem [2].”
• A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications (2019, methodology): “Several algorithms were developed to solve (1), among them, are the trust-region [10] and the Levenberg-Marquardt method [11].”