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An outcome space algorithm for minimizing the product of two convex functions over a convex set
On the LevenbergMarquardt methods for convex constrained nonlinear equations
1.  Department of Mathematics, and MOELSC, Shanghai Jiao Tong University, Shanghai 200240, China 
References:
[1] 
S. Bellavia, M. Macconi and B. Morini, An affine scaling trustregion approach to boundconstrained nonlinear systems,, Appl. Numer. Math., 44 (2003), 257. doi: 10.1016/S01689274(02)001708. 
[2] 
S. Bellavia and B. Morini, An interior global method for nonlinear systems with simple bounds,, Optim. Methods Software, 20 (2005), 1. 
[3] 
H. Dan, N. Yamashita and M. Fukushima, Convergence properties of the inexact LevenbergMarquardt method under local error bound conditions,, Optim. Meth. Software, 17 (2002), 605. 
[4] 
J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations,", PrenticeHall, (1983). 
[5] 
S. P. Dirkse, M. C. Ferris, MCPLIB: a collection of nonlinear mixed complementarity problems,, Optim. Meth. Software, 5 (1995), 319. doi: 10.1080/10556789508805619. 
[6] 
M. E. ElHawary, "Optimal Power Flow: Solutions Techniques, Requirements, and Challenges,", IEEE Service Center, (1996). 
[7] 
J. Y. Fan and J. Y. Pan, Convergence properties of a selfadaptive LevenbergMarquardt algorithm under local error bound condition,, Computational Optimization and Applications, 34 (2006), 47. 
[8] 
J. Y. Fan and Y. X. Yuan, On the quadratic convergence of the LevenbergMarquardt method without nonsingularity assumption,, Computing, 74 (2005), 23. doi: 10.1007/s0060700400831. 
[9] 
C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer and C. A. Schweiger, "Handbook of Test Problems in Local and Global Optimization, Nonconvex Optimization and its Applications,", Vol. 33, (1999). 
[10] 
W. Hock and K. Schittkowski, "Test Examples for Nonlinear Programming Codes,", Lecture Notes in Economics and Mathematical Systems, (1981). 
[11] 
C. Kanzow, An active settype Newton method for constrained nonlinear systems,, in, (2001), 179. 
[12] 
C. Kanzow, N. Yamashita and M. Fukushima, LevenbergMarquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints,, Journal of Computational and Applied Mathematics, 173 (2005), 321. 
[13] 
C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", SIAM, (1995). 
[14] 
D. N. Kozakevich, J. M. Martinez and S. A. Santos, Solving nonlinear systems of equations with simple bounds,, Comput. Appl. Math., 16 (1997), 215. 
[15] 
K. Levenberg, A method for the solution of certain nonlinear problems in least squares,, Quart. Appl. Math., 2 (1944), 164. 
[16] 
D. W. Marquardt, An algorithm for leastsquares estimation of nonlinear inequalities,, SIAM J. Appl. Math., 11 (1963), 431. 
[17] 
K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems,, Appl. Math. Comput., 22 (1987), 333. 
[18] 
K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems,, ACM Trans. Math. Software, 16 (1990), 143. 
[19] 
R. D. C. Monteiro and J. S. Pang, A potential reduction Newton method for constrained equations,, SIAM J. Optim., 9 (1999), 729. doi: 10.1137/S1052623497318980. 
[20] 
J. J. Moré, The LM algorithm: implementation and theory,, in, (1978), 105. 
[21] 
J. M. Ortega and W. C. Rheinboldt, "Iterative solution of Nonlinear Equations in Several Variables,", Academic Press, (1970). 
[22] 
L. Qi, X. J. Tong and D. H. Li, An activeset projected trust region algorithm for box constrained nonsmooth equations,, Journal of Optimization Theories and Applications, 120 (2004), 601. 
[23] 
M. Ulbrich, Nonmonotone trustregion methods for boundconstrained semismooth equations with applications to nonlinear mixed complementarity problems,, SIAM J. Optim., 11 (2001), 889. 
[24] 
T. Wang, R. D. C. Monteiro and J. S. Pang, An interior point potential reduction method for constrained equations,, Math. Programming, 74 (1996), 159. 
[25] 
A. J. Wood and B. F. Wollenberg, "Power Generation, Operation, and Control,", John Wiley and Sons, (1996). 
[26] 
N. Yamashita and M. Fukushima, On the rate of convergence of the LM method,, Computing (Supplement 15), (2001), 237. 
[27] 
Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least sqaures,, Numerical Algebra, 1 (2011), 15. 
show all references
References:
[1] 
S. Bellavia, M. Macconi and B. Morini, An affine scaling trustregion approach to boundconstrained nonlinear systems,, Appl. Numer. Math., 44 (2003), 257. doi: 10.1016/S01689274(02)001708. 
[2] 
S. Bellavia and B. Morini, An interior global method for nonlinear systems with simple bounds,, Optim. Methods Software, 20 (2005), 1. 
[3] 
H. Dan, N. Yamashita and M. Fukushima, Convergence properties of the inexact LevenbergMarquardt method under local error bound conditions,, Optim. Meth. Software, 17 (2002), 605. 
[4] 
J. E. Dennis and R. B. Schnabel, "Numerical Methods for Unconstrained Optimization and Nonlinear Equations,", PrenticeHall, (1983). 
[5] 
S. P. Dirkse, M. C. Ferris, MCPLIB: a collection of nonlinear mixed complementarity problems,, Optim. Meth. Software, 5 (1995), 319. doi: 10.1080/10556789508805619. 
[6] 
M. E. ElHawary, "Optimal Power Flow: Solutions Techniques, Requirements, and Challenges,", IEEE Service Center, (1996). 
[7] 
J. Y. Fan and J. Y. Pan, Convergence properties of a selfadaptive LevenbergMarquardt algorithm under local error bound condition,, Computational Optimization and Applications, 34 (2006), 47. 
[8] 
J. Y. Fan and Y. X. Yuan, On the quadratic convergence of the LevenbergMarquardt method without nonsingularity assumption,, Computing, 74 (2005), 23. doi: 10.1007/s0060700400831. 
[9] 
C. A. Floudas, P. M. Pardalos, C. S. Adjiman, W. R. Esposito, Z. H. Gumus, S. T. Harding, J. L. Klepeis, C. A. Meyer and C. A. Schweiger, "Handbook of Test Problems in Local and Global Optimization, Nonconvex Optimization and its Applications,", Vol. 33, (1999). 
[10] 
W. Hock and K. Schittkowski, "Test Examples for Nonlinear Programming Codes,", Lecture Notes in Economics and Mathematical Systems, (1981). 
[11] 
C. Kanzow, An active settype Newton method for constrained nonlinear systems,, in, (2001), 179. 
[12] 
C. Kanzow, N. Yamashita and M. Fukushima, LevenbergMarquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints,, Journal of Computational and Applied Mathematics, 173 (2005), 321. 
[13] 
C. T. Kelley, "Iterative Methods for Linear and Nonlinear Equations,", SIAM, (1995). 
[14] 
D. N. Kozakevich, J. M. Martinez and S. A. Santos, Solving nonlinear systems of equations with simple bounds,, Comput. Appl. Math., 16 (1997), 215. 
[15] 
K. Levenberg, A method for the solution of certain nonlinear problems in least squares,, Quart. Appl. Math., 2 (1944), 164. 
[16] 
D. W. Marquardt, An algorithm for leastsquares estimation of nonlinear inequalities,, SIAM J. Appl. Math., 11 (1963), 431. 
[17] 
K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium systems,, Appl. Math. Comput., 22 (1987), 333. 
[18] 
K. Meintjes and A. P. Morgan, Chemical equilibrium systems as numerical test problems,, ACM Trans. Math. Software, 16 (1990), 143. 
[19] 
R. D. C. Monteiro and J. S. Pang, A potential reduction Newton method for constrained equations,, SIAM J. Optim., 9 (1999), 729. doi: 10.1137/S1052623497318980. 
[20] 
J. J. Moré, The LM algorithm: implementation and theory,, in, (1978), 105. 
[21] 
J. M. Ortega and W. C. Rheinboldt, "Iterative solution of Nonlinear Equations in Several Variables,", Academic Press, (1970). 
[22] 
L. Qi, X. J. Tong and D. H. Li, An activeset projected trust region algorithm for box constrained nonsmooth equations,, Journal of Optimization Theories and Applications, 120 (2004), 601. 
[23] 
M. Ulbrich, Nonmonotone trustregion methods for boundconstrained semismooth equations with applications to nonlinear mixed complementarity problems,, SIAM J. Optim., 11 (2001), 889. 
[24] 
T. Wang, R. D. C. Monteiro and J. S. Pang, An interior point potential reduction method for constrained equations,, Math. Programming, 74 (1996), 159. 
[25] 
A. J. Wood and B. F. Wollenberg, "Power Generation, Operation, and Control,", John Wiley and Sons, (1996). 
[26] 
N. Yamashita and M. Fukushima, On the rate of convergence of the LM method,, Computing (Supplement 15), (2001), 237. 
[27] 
Y. X. Yuan, Recent advances in numerical methods for nonlinear equations and nonlinear least sqaures,, Numerical Algebra, 1 (2011), 15. 
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