JIMO
On the convergence rate of the inexact Levenberg-Marquardt method
Jinyan Fan Jianyu Pan
Journal of Industrial & Management Optimization 2011, 7(1): 199-210 doi: 10.3934/jimo.2011.7.199
In this paper we study the convergence rate of the inexact Levenberg-Marquardt method for nonlinear equations. Under the local error bound condition which is weaker than nonsingularity, we derive an explicit formula of the convergence order of the inexact LM method, which is a continuous function with respect to not only the LM parameter but also the perturbation vector. The new formula includes many convergence rate results in the literature as its special cases.
keywords: local error bound condition convergence rate. Nonlinear equations inexact Levenberg-Marquardt method
DCDS-B
Inexact Levenberg-Marquardt method for nonlinear equations
Jinyan Fan Jianyu Pan
Discrete & Continuous Dynamical Systems - B 2004, 4(4): 1223-1232 doi: 10.3934/dcdsb.2004.4.1223
In this paper, we present an inexact Levenberg-Marquardt (LM) method for singular system of nonlinear equations, where the LM parameter is chosen as the norm of the function and the trial step is computed approximately. Under the local error bound condition which is weaker than the non- singularity, we show that the new inexact LM method preserves the quadratic convergence of the traditional LM method where the parameter is chosen to be larger than a positive constant and the Jacobi at the solution is nonsingular.
keywords: local error bound condition Levenberg-Marquardt method quadratic convergence. Nonlinear equations
JIMO
On the Levenberg-Marquardt methods for convex constrained nonlinear equations
Jinyan Fan
Journal of Industrial & Management Optimization 2013, 9(1): 227-241 doi: 10.3934/jimo.2013.9.227
In this paper, both the constrained Levenberg-Marquardt method and the projected Levenberg-Marquardt method are presented for nonlinear equations $F(x)=0$ subject to $x\in X$, where $X$ is a nonempty closed convex set. The Levenberg-Marquardt parameter is taken as $\| F(x_k) \|_2^\delta$ with $\delta\in (0, 2]$. Under the local error bound condition which is weaker than nonsingularity, the methods are shown to have the same convergence rate, which includes not only the convergence results obtained in [12] for $\delta=2$ but also the results given in [7] for unconstrained nonlinear equations.
keywords: Convex constrained nonlinear equations quadratic convergence. local error bound constrained Levenberg-Marquardt method projected Levenberg-Marquardt method
JIMO
A semidefinite relaxation algorithm for checking completely positive separable matrices
Anwa Zhou Jinyan Fan
Journal of Industrial & Management Optimization 2018, 13(5): 1-16 doi: 10.3934/jimo.2018076

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $V$ such that $A = VV ^T$. A real symmetric matrix is called completely positive separable (CPS) if it can be written as a sum of rank-1 Kronecker products of completely positive matrices. This paper studies the CPS problem. A criterion is given to determine whether a given matrix is CPS, and a specific CPS decomposition is constructed if the matrix is CPS.

keywords: Completely positive matrices completely positive separability moment problems semidefinite algorithm
JIMO
On subspace properties of the quadratically constrained quadratic program
Xin Zhao Jinyan Fan
Journal of Industrial & Management Optimization 2017, 13(4): 1625-1640 doi: 10.3934/jimo.2017010

In this paper, we study subspace properties of the quadratically constrained quadratic program (QCQP). We prove that, if an appropriate subspace is chosen to satisfy subspace properties, then the solution of the QCQP lies in that subspace. So, we can solve the QCQP in that subspace rather than solve it in the original space. The computational cost could be reduced significantly if the dimension of the subspace is much smaller. We also show how to construct such subspaces and investigate their dimensions.

keywords: QCQP subspace properties subspace choices

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