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Descent line search scheme using Geršgorin circle theorem. (English) Zbl 1409.90148
Summary: In the context of the minimization of a real function, we propose a line search scheme that involves a new positive definite modification of the Hessian. In this framework, a safeguard based on Geršgorin Circle’s theorem provides an approximation of the Hessian that improves with iteration count. Convergence analysis of the scheme is validated by numerical experiments.

90C26 Nonconvex programming, global optimization
65K05 Numerical mathematical programming methods
90C53 Methods of quasi-Newton type
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[1] Andrei, Neculai, An unconstrained optimization test functions collection, Adv. Model. Optim, 10, 1, 147-161, (2008) · Zbl 1161.90486
[2] Bunch, James R.; Kaufman, Linda, Some stable methods for calculating inertia and solving symmetric linear systems, Math. Comp., 163-179, (1977) · Zbl 0355.65023
[3] Chen, Yannan; Sun, Wenyu, A dwindling filter line search method for unconstrained optimization, Math. Comp., 84, 291, 187-208, (2015) · Zbl 1307.65085
[4] Cheng, Sheung Hun; Higham, Nicholas J., A modified Cholesky algorithm based on a symmetric indefinite factorization, SIAM J. Matrix Anal. Appl., 19, 4, 1097-1110, (1998) · Zbl 0949.65022
[5] Dolan, E. D.; Moré, Jorge J., Benchmarking optimization software with performance profiles, Math. Program., 91, 2, 201-213, (2002) · Zbl 1049.90004
[6] Horn, Roger A.; Johnson, Charles R., Matrix Analysis, (2012), Cambridge University Press · Zbl 0576.15001
[7] Nocedal, J.; Wright, S. J., (Numerical Optimization, Springer Series in Operations Research and Financial Engineering, (2006), Springer, NY)
[8] Schnabel, Robert B.; Eskow, Elizabeth, A revised modified Cholesky factorization algorithm, SIAM J. Optim., 9, 4, 1135-1148, (1999) · Zbl 0958.65034
[9] Schneider, Reinhold; Uschmajew, André, Convergence results for projected line-search methods on varieties of low-rank matrices via lojasiewicz inequality, SIAM J. Optim., 25, 1, 622-646, (2015) · Zbl 1355.65079
[10] Ueda, Kenji; Yamashita, Nobuo, A regularized Newton method without line search for unconstrained optimization, Comput. Optim. Appl., 59, 1-2, 321-351, (2014) · Zbl 1302.90218
[11] Varga, Richard S., Geršgorin and his Circles, vol. 36, (2010), Springer Science & Business Media
[12] Wan, Zhong; Yang, ZhanLu; Wang, YaLin, New spectral PRP conjugate gradient method for unconstrained optimization, Appl. Math. Lett., 24, 1, 16-22, (2011) · Zbl 1208.49039
[13] Zhou, Weijun, On the convergence of the modified Levenberg-Marquardt method with a nonmonotone second order Armijo type line search, J. Comput. Appl. Math., 239, 152-161, (2013) · Zbl 1258.65050
[14] Zoutendijk, G., Nonlinear programming, computational methods, Integer Nonlinear Program., 143, 1, 37-86, (1970) · Zbl 0336.90057
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