In the present paper, we propose a modified inexact Levenberg-Marquardt method (LMM) and its global version by virtue of Armijo, Wolfe or Goldstein line-search schemes to solve nonlinear least squares problems (NLSP), especially for the underdetermined case. Under a local error bound condition, we show that a sequence generated by the modified inexact LMM converges to a solution superlinearly and even quadratically for some special parameters, which improves the corresponding results of the classical inexact LMM in Dan et al. (Optim Methods Softw 17:605-626, 2002). Furthermore, the quadratical convergence of the global version of the modified inexact LMM is also established. Finally, preliminary numerical experiments on some medium/large scale underdetermined NLSP show that our proposed algorithm outperforms the classical inexact LMM.
Abstract-The rapid development of "Internet +" technology has promoted the modernization process of education. The advanced mathematics course is an essential basic course of a university. Its teaching model has direct influence on the teaching efficiency and teaching quality. Combined with the Wechat platform and PC software, with the use of "internet + education", the paper takes the "advanced mathematics" course for example to build an interactive teaching platform of Smart Classroom. Practice has proved that the teaching platform promotes the reform of teaching model of "advanced mathematics", increases the learning enthusiasm and initiative of students, and provides a new way for promoting the modernization of education and personalized education.
In this paper, we study the initial boundary problem of fifth-order Korteweg-de Vries equation with nonlinear boundary values. First, we establish a so-called sharp boundary trace regularity associated with the linearized fifth-order Korteweg-de Vries equation. Then, aided by the sharp boundary trace regularity, we verify that initial-boundary value problem of fifth-order Korteweg-de Vries equation with nonlinear boundary conditions is locally well-posed when initial and boundary values are properly chosen.
We consider a generalized equilibrium problem involving DC functions. By using the properties of the epigraph of the conjugate functions, some sufficient and/or necessary conditions for the weak and strong duality results and optimality conditions for generalized equilibrium problems are provided.
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