On the Rate of Convergence of the Levenberg-Marquardt Method

Yamashita, Nobuo; Fukushima, Masao

doi:10.1007/978-3-7091-6217-0_18

Cited by 251 publications

(223 citation statements)

References 9 publications

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“…As pointed out in [12,Theorem 5], the assumptions required for convergence of such methods are the strict complementarity and the error bound. Hence, combining the corresponding results in [16,18,38] with Theorem 1, we immediately conclude that these methods possess local quadratic convergence to some point inŪ if the strict complementarity condition holds, and the matrix in (25) has the full row rank. Observe that in Example 1, these assumptions are satisfied at any solution corresponding to t ∈ (1/2, 1).…”

Section: Newton-type Methodssupporting

confidence: 55%

See 1 more Smart Citation

On error bounds and Newton-type methods for generalized Nash equilibrium problems

Izmailov

Solodov

2013

Comput Optim Appl

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Error bounds (estimates for the distance to the solution set of a given problem) are key to analyzing convergence rates of computational methods for solving the problem in question, or sometimes even to justifying convergence itself. That said, for the generalized Nash equilibrium problems (GNEP), the theory of error bounds had not been developed in depth comparable to the fields of optimization and variational problems. In this paper, we provide a systematic approach which should be useful for verifying error bounds for both specific instances of GNEPs and for classes of GNEPs. These error bounds for GNEPs are based on more general results for constraints that involve complementarity relations and cover those (few) GNEP error bounds that existed previously, and go beyond. In addition, they readily imply a Lipschitzian stability result for solutions of GNEPs, a subject where again very little had been known. As a specific application of error bounds, we discuss Newtonian methods for solving GNEPs. While we do not propose any significantly new methods in this respect, some new insights into applicability to GNEPs of various approaches and into their convergence properties are presented.

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Section: Newton-type Methodssupporting

confidence: 55%

“…However, the local convergence results in [16,18,38] for variants of the Levenberg-Marquardt method can be applicable; see [12,Theorem 5]. Starting at a point u k , the method computes the next iterate u k+1 as the solution of the equation…”

Section: Newton-type Methodsmentioning

confidence: 99%

On error bounds and Newton-type methods for generalized Nash equilibrium problems

Izmailov

Solodov

2013

Comput Optim Appl

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“…It is worth pointing out that the local error bound condition is a weaker condition than the isolated solution. Using this condition, Yamashita and Fukushima [25], Kanzow, Yamashita and Fukushima [13], Zhou and Toh [28], Wang and Wang [24] considered the local convergence rate of the Levenberg Marquardt method and Newton method for nonlinear equations problems. Zhang, Wu and Zhang [26] considered the trust region method for unconstrained optimization.…”

Section: Then the Secant Equation Becomesmentioning

confidence: 99%

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

Yu¹,

Gan²

2016

J. Nonlinear Sci. Appl.

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In this paper, we propose a spectral projected gradient method for the convex optimization problem with singular solution. By solving the equivalent equation of the gradient function, this method combines the perturbed spectral gradient direction with the projection direction to generate the next iteration point. Under some mild conditions, we establish the global convergence and the local R-linear convergence rate under the local error bound condition. Preliminary numerical tests are given to show that the proposed method works well.

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“…We follow the choice of [36] and use µ k = G T k g k 2 (with the same approximation as above), which has all of our desirable features. If x k is far away from the optimal solution, which is the case when the mesh quality is low, µ k is large and thereby κ(…”

Section: Input Meshmentioning

confidence: 99%

Quality encoding for tetrahedral mesh optimization

Cheng

Wang

et al. 2009

Computers & Graphics

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We define quality differential coordinates (QDC) for per-vertex encoding of the quality of a tetrahedral mesh. QDC measures the deviation of a mesh vertex from a position which maximizes the combined quality of the set of tetrahedra incident at that vertex. Our formulation allows the incorporation of different choices of element quality metrics into QDC construction to penalize badly shaped and inverted tetrahedra. We develop an algorithm for tetrahedral mesh optimization through energy minimization driven by QDC. The variational problem is solved efficiently and robustly using gradient flow based on a stable semi-implicit integration scheme. To ensure quality boundary of the resulting tetrahedral mesh, we propose a harmonic-guided optimization scheme which leads to consistent handling of both the interior and boundary tetrahedra.

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On the Rate of Convergence of the Levenberg-Marquardt Method

Cited by 251 publications

References 9 publications

On error bounds and Newton-type methods for generalized Nash equilibrium problems

On error bounds and Newton-type methods for generalized Nash equilibrium problems

Global and local R-linear convergence of a spectral projected gradient method for convex optimization with singular solution

Quality encoding for tetrahedral mesh optimization

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