Abstract
In this work we consider the solution of large scale (possibly nonconvex) unconstrained optimization problems. We focus on Truncated Newton methods which represent one of the commonest methods to tackle such problems. In particular, we follow the approach detailed in Caliciotti et al. (Comput Optim Appl 77:627–651, 2020), where a modified version of the Bunch and Kaufman decomposition (Bunch and Kaufman, Math Comput 31:163–179, 1977) is proposed for solving the Newton equation. Such decomposition is used within SYMMBK routine as proposed by Chandra (Conjugate gradient methods for partial differential equations, Ph.D. thesis, Yale University, New Haven, 1978; see also Conn et al., Trust–Region Methods, MPS–SIAM Series on Optimization, Philadelphia, PA, 2000; HSL: A collection of Fortran codes for large scale scientific computation, https://www.hsl.rl.ac.uk/; Marcia, Appl Numer Math 58(4):449–458, 2008) for iteratively solving symmetric possibly indefinite linear systems. The proposal in Caliciotti et al. (Comput Optim Appl 77:627–651, 2020) enabled to overcome a relevant drawback of nonconvex problems, namely the computed search direction might not be gradient-related. Here we propose further extensions of such approach, aiming at improving the pivoting strategy of the Bunch and Kaufman decomposition and enhancing its flexibility.
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References
Bunch, J., Kaufman, L.: Some stable methods for calculating inertia and solving symmetric linear equations. Math. Comput. 31, 163–179 (1977)
Caliciotti, A., Fasano, G., Nash, S.G., Roma, M.: An adaptive truncation criterion, for linesearch-based truncated Newton methods in large scale nonconvex optimization. Oper. Res. Lett. 46, 7–12 (2018)
Caliciotti, A., Fasano, G., Nash, S.G., Roma, M.: Data and performance profiles applying an adaptive truncation criterion, within linesearch-based truncated Newton methods, in large scale nonconvex optimization. Data in Brief 17, 246–255 (2018)
Caliciotti, A., Fasano, G., Roma, M.: Preconditioned nonlinear conjugate gradient methods based on a modified secant equation. Appl. Math. Comput. 318, 196–214 (2018)
Caliciotti, A., Fasano, G., Potra, F., Roma, M.: Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation. Comput. Optim. Appl. 77, 627–651 (2020)
Chandra, R.: Conjugate gradient methods for partial differential equations. Ph.D. thesis, Yale University, New Haven (1978). Research Report 129
Conn, A.R., Gould, N.I.M., Toint, P.L.: Trust–Region Methods. MPS–SIAM Series on Optimization, Philadelphia, PA (2000)
Dembo, R., Steihaug, T.: Truncated-Newton algorithms for large-scale unconstrained optimization. Math. Program. 26, 190–212 (1983)
Dembo, R., Eisenstat, S., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)
Fasano, G.: Planar–conjugate gradient algorithm for large–scale unconstrained optimization, Part 1: theory. J. Optim. Theory Appl. 125, 523–541 (2005)
Fasano, G.: Planar–conjugate gradient algorithm for large–scale unconstrained optimization, Part 2: application. J. Optim. Theory Appl. 125, 543–558 (2005)
Fasano, G.: Lanczos-conjugate gradient method and pseudoinverse computation, in unconstrained optimization. J. Optim. Theory Appl. 132, 267–285 (2006)
Fasano, G., Roma, M.: Iterative computation of negative curvature directions in large scale optimization. Comput. Optim. Appl. 38, 81–104 (2007)
Fasano, G., Roma, M.: Preconditioning Newton–Krylov methods in nonconvex large scale optimization. Comput. Optim. Appl. 56, 253–290 (2013)
Grippo, L., Lampariello, F., Lucidi, S.: A truncated Newton method with nonmonotone linesearch for unconstrained optimization. J. Optim. Theory Appl. 60, 401–419 (1989)
HSL: A collection of Fortran codes for large scale scientific computation. https://www.hsl.rl.ac.uk/
Marcia, R.: On solving sparse symmetric linear systems whose definiteness is unknown. Appl. Numer. Math. 58(4), 449–458 (2008)
Nash, S.: A survey of truncated-Newton methods. J. Comput. Appl. Math. 124, 45–59 (2000)
Nash, S., Sofer, A.: Assessing a search direction within a truncated-Newton method. Oper. Res. Lett. 9, 219–221 (1990)
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Fasano, G., Roma, M. (2022). An Improvement of the Pivoting Strategy in the Bunch and Kaufman Decomposition, Within Truncated Newton Methods. In: Amorosi, L., Dell’Olmo, P., Lari, I. (eds) Optimization in Artificial Intelligence and Data Sciences. AIRO Springer Series, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-030-95380-5_5
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