×

On the employment of inexact restoration for the minimization of functions whose evaluation is subject to errors. (English) Zbl 1383.65060

Summary: Inexact Restoration is a well established technique for continuous minimization problems with constraints. Recently, it has been used by Krejić and Martínez for optimization of functions whose evaluation is necessarily inexact and comes from an iterative process. This technique will be generalized in the present paper and it will be applied to stochastic optimization and related problems. New convergence results will be given and numerical results will be presented.

MSC:

65K05 Numerical mathematical programming methods
90C30 Nonlinear programming
90C15 Stochastic programming

Software:

ALGENCAN; AMLET
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andreani, R.; Castro, S. L. C.; Chela, J. L.; Friedlander, A.; Santos, S. A., An inexact-restoration method for nonlinear bilevel programming problems, Comput. Optim. Appl., 43, 3, 307-328 (2009) · Zbl 1170.90484 · doi:10.1007/s10589-007-9147-4
[2] Banihashemi, Nahid; Yal{\c{c}}{\i}n Kaya, C., Inexact restoration for Euler discretization of box-constrained optimal control problems, J. Optim. Theory Appl., 156, 3, 726-760 (2013) · Zbl 1263.49029 · doi:10.1007/s10957-012-0140-4
[3] bastin1 F. Bastin, Trust-region algorithms for nonlinear stochastic programming and mixed logit models, Ph.D. Thesis, University of Namur, Namur, Belgium, 2004.
[4] Bastin, Fabian; Cirillo, Cinzia; Toint, Philippe L., An adaptive Monte Carlo algorithm for computing mixed logit estimators, Comput. Manag. Sci., 3, 1, 55-79 (2006) · Zbl 1136.62086 · doi:10.1007/s10287-005-0044-y
[5] Birgin, E. G.; Bueno, L. F.; Mart{\'{\i}}nez, J. M., Assessing the reliability of general-purpose inexact restoration methods, J. Comput. Appl. Math., 282, 1-16 (2015) · Zbl 1310.90088 · doi:10.1016/j.cam.2014.12.031
[6] Birgin, E. G.; Mart{\'{\i}}nez, J. M., Local convergence of an inexact-restoration method and numerical experiments, J. Optim. Theory Appl., 127, 2, 229-247 (2005) · Zbl 1116.90094 · doi:10.1007/s10957-005-6537-6
[7] Birgin, E. G.; Mart{\'{\i}}nez, J. M., On the application of an augmented Lagrangian algorithm to some portfolio problems, EURO J. Comput. Optim., 4, 1, 79-92 (2016) · Zbl 1338.91126 · doi:10.1007/s13675-015-0052-9
[8] bmr5survey E. G. Birgin, J. M. Mart\'nez, and M. Raydan, Spectral projected gradient methods: Review and perspectives, Journal of Statistical Software 60 (2014), no. 3 (DOI: 10.18637/jss.v060.i03). http://www.jstatsoft.org/v60/i03.
[9] Bueno, L. F.; Friedlander, A.; Mart{\'{\i}}nez, J. M.; Sobral, F. N. C., Inexact restoration method for derivative-free optimization with smooth constraints, SIAM J. Optim., 23, 2, 1189-1213 (2013) · Zbl 1280.65050 · doi:10.1137/110856253
[10] Escalante, Ren{\'e}; Raydan, Marcos, Dykstra’s algorithm for a constrained least-squares matrix problem, Numer. Linear Algebra Appl., 3, 6, 459-471 (1996) · Zbl 0908.90207 · doi:10.1002/(SICI)1099-1506(199611/12)3:\(6\langle459\)::AID-NLA
[11] Fischer, Andreas; Friedlander, Ana, A new line search inexact restoration approach for nonlinear programming, Comput. Optim. Appl., 46, 2, 333-346 (2010) · Zbl 1220.90122 · doi:10.1007/s10589-009-9267-0
[12] Gomes-Ruggiero, M. A.; Mart{\'{\i}}nez, J. M.; Santos, S. A., Spectral projected gradient method with inexact restoration for minimization with nonconvex constraints, SIAM J. Sci. Comput., 31, 3, 1628-1652 (2009) · Zbl 1220.90123 · doi:10.1137/070707828
[13] Higham, Nicholas J., Computing a nearest symmetric positive semidefinite matrix, Linear Algebra Appl., 103, 103-118 (1988) · Zbl 0649.65026 · doi:10.1016/0024-3795(88)90223-6
[14] tito T. Homem-de-Mello, Variable-sample methods for stochastic optimization, ACM Transactions on Modeling and Computer Simulation 13 (2003), 108-133. · Zbl 1390.65003
[15] Kaya, C. Y.; Mart{\'{\i}}nez, J. M., Euler discretization and inexact restoration for optimal control, J. Optim. Theory Appl., 134, 2, 191-206 (2007) · Zbl 1135.49019 · doi:10.1007/s10957-007-9217-x
[16] Kreji{\'c}, Nata{\v{s}}a; Krklec, Nata{\v{s}}a, Line search methods with variable sample size for unconstrained optimization, J. Comput. Appl. Math., 245, 213-231 (2013) · Zbl 1262.65066 · doi:10.1016/j.cam.2012.12.020
[17] Kreji{\'c}, Nata{\v{s}}a; Krklec Jerinki{\'c}, Nata{\v{s}}a, Nonmonotone line search methods with variable sample size, Numer. Algorithms, 68, 4, 711-739 (2015) · Zbl 1316.65062 · doi:10.1007/s11075-014-9869-1
[18] Kreji{\'c}, Nata{\v{s}}a; Mart{\'{\i}}nez, J. M., Inexact restoration approach for minimization with inexact evaluation of the objective function, Math. Comp., 85, 300, 1775-1791 (2016) · Zbl 1335.65053 · doi:10.1090/mcom/3025
[19] Martinez, J. M., Inexact-restoration method with Lagrangian tangent decrease and new merit function for nonlinear programming, J. Optim. Theory Appl., 111, 1, 39-58 (2001) · Zbl 1052.90089 · doi:10.1023/A:1017567113614
[20] Mart{\'{\i}}nez, J. M.; Pilotta, E. A., Inexact-restoration algorithm for constrained optimization, J. Optim. Theory Appl., 104, 1, 135-163 (2000) · Zbl 0969.90094 · doi:10.1023/A:1004632923654
[21] pasu2010 R. Pasupathy, On choosing parameters in Retrospective Approximation algorithms for stochastic root finding and simulation optimization, Operations Research 58 (2010), 889-901. · Zbl 1228.90069
[22] Royset, Johannes O., On sample size control in sample average approximations for solving smooth stochastic programs, Comput. Optim. Appl., 55, 2, 265-309 (2013) · Zbl 1288.90057 · doi:10.1007/s10589-012-9528-1
[23] Rubinstein, Reuven Y.; Shapiro, Alexander, Discrete Event Systems, xvi+334 pp. (1993), John Wiley & Sons, Ltd., Chichester · Zbl 0805.93002
[24] Stochastic Programming, Handbooks in Operations Research and Management Science 10, x+688 pp. (2003), Elsevier Science B.V., Amsterdam · Zbl 1115.90001
[25] Shapiro, Alexander; Dentcheva, Darinka; Ruszczy{\'n}ski, Andrzej, Lectures on Stochastic Programming, MPS/SIAM Series on Optimization 9, xvi+436 pp. (2009), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Programming Society (MPS), Philadelphia, PA · Zbl 1183.90005 · doi:10.1137/1.9780898718751
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.