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Ghalavand, Nasim; Khorram, Esmaile; Morovati, Vahid

An adaptive nonmonotone line search for multiobjective optimization problems. (English) Zbl 1511.90374

Comput. Oper. Res. 136, Article ID 105506, 9 p. (2021).
Summary: This paper aims to propose an adaptive nonmonotone line search for direction-based multiobjective optimization (MO) algorithms. All direction-based MO algorithms can be equipped with this line search, especially the BFGS quasi-Newton algorithm is applied in this study. To validate the proposed line search some well-known line searches, including Armijo, maximum nonmonotone and average nonmonotone line searches, were considered. In order to make a comprehensive comparison between the proposed line sereach and the aforementioned line searches two criteria are considered: the computational effort and quality of the approximated nondominated frontier. The results confirm the remarkable superiority of the proposed line search over the mentioned line searches. In addition, this line search preserves the quality of the obtained nondominated frontier. It should be noted that we established the convergency of BFGS quasi-Newton algorithm equipped with the proposed line search under some mild conditions.

Cited in 4 Documents

MSC:

90C29 Multi-objective and goal programming
65K05 Numerical mathematical programming methods
90C53 Methods of quasi-Newton type

Keywords:

multiobjective optimization; non-scalarization algorithms; nonmonotone line searches; quasi-Newton algorithm; efficient solution; weakly efficient solution
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References:

[1] Basirzadeh, H.; Morovati, V.; Sayadi, A., A quick method to calculate the super-efficient point in multi-objective assignment problems, J. Math. Comput. Sci., 10, 3, 157-162 (2014)
[2] Brito, A. S.; Neto, J. C.; Santos, P. S.M.; Souza, S. S., A relaxed projection method for solving multiobjective optimization problems, European J. Oper. Res., 256, 1, 17-23 (2017) · Zbl 1394.90510
[3] Carrizo, G. A.; Lotito, P. A.; Maciel, M. C., Trust region globalization strategy for the nonconvex unconstrained multiobjective optimization problem, Math. Program., 159, 1-2, 339-369 (2016) · Zbl 1345.90081
[4] Carrizosa, E.; Frenk, J. B.G., Dominating sets for convex functions with some applications, J. Optim. Theory Appl., 96, 2, 281-295 (1998) · Zbl 0905.90134
[5] Castro, J.; Nasini, S., On geometrical properties of preconditioners in IPMs for classes of block-angular problems, SIAM J. Optim., 27, 3, 1666-1693 (2017) · Zbl 1369.90102
[6] Castro, J.; Nasini, S., A specialized interior-point algorithm for huge minimum convex cost flows in bipartite networks, European J. Oper. Res., 290, 3, 857-869 (2021) · Zbl 1487.90641
[7] Cocchi, G.; Lapucci, M., An augmented Lagrangian algorithm for multi-objective optimization, Comput. Optim. Appl., 77, 1, 29-56 (2020) · Zbl 1447.90055
[8] Dai, Y. H., On the nonmonotone line search, J. Optim. Theory Appl., 112, 315-330 (2002) · Zbl 1049.90087
[9] Das, I.; Dennis, J. E., Normal-boundary intersection: A new method for generating Pareto optimal points in Nonlinear Multicriteria Optimization Problems, SIAM J. Optim., 8, 631-657 (1998) · Zbl 0911.90287
[10] Dolan, E. D.; More, J. J., Benchmarking optimization software with performance profiles, Math. Program., 91, 2, 201-213 (2002) · Zbl 1049.90004
[11] Drummond, L. G.; Iusem, A. N., A projected gradient method for vector optimization problems, Comput. Optim. Appl., 28, 1, 5-29 (2004) · Zbl 1056.90126
[12] Ehrgott, M., Multicriteria Optimization, Vol. 491 (2005), Springer Science & Business Media · Zbl 1132.90001
[13] El Moudden, M.; El Ghali, A., Multiple reduced gradient method for multiobjective optimization problems, Numer. Algorithms, 79, 4, 1257-1282 (2018) · Zbl 1461.65146
[14] Evans, G. W., An overview of techniques for Solving Multiobjective Mathematical Programs, Manage. Sci., 30, 11, 1268-1282 (1984) · Zbl 0551.90090
[15] Fliege, J.; Drummond, L. G.; Svaiter, B. F., Newton’s method for multiobjective optimization, SIAM J. Optim., 20, 2, 602-626 (2009) · Zbl 1195.90078
[16] Fliege, J.; Svaiter, B. F., Steepest descent methods for multicriteria optimization, Math. Methods Oper. Res., 51, 3, 479-494 (2000) · Zbl 1054.90067
[17] Fliege, J.; Vaz, A. I.F., A method for constrained multiobjective optimization based on SQP techniques, SIAM J. Optim., 26, 4, 2091-2119 (2016) · Zbl 1349.90735
[18] Fu, Y.; Diwekar, U. M., An efficient sampling approach to multiobjective optimization, Ann. Oper. Res., 132, 1-4, 109-134 (2004) · Zbl 1090.90171
[19] Goncalves, M. L.N.; Prudente, L. F., On the extension of the Hager-Zhang conjugate gradient method for vector optimization, Comput. Optim. Appl., 76, 3, 889-916 (2019) · Zbl 1446.90142
[20] Grippo, L.; Lampariello, F.; Lucidi, S., A nonmonotone line search technique for Newton’s method, SIAM J. Numer. Anal., 23, 4, 707-716 (1986) · Zbl 0616.65067
[21] Jin, Y., Olhofer, M., Sendhoff, B., 2001. Dynamic weighted aggregation for evolutionary multiobjective optimization: Why does it work and how? In: Proceedings of the Genetic and Evolutionary Computation Conference. pp. 1042-1049.
[22] Kim, I. Y.; De Weck, O. L., Adaptive weighted-sum method for bi-objective optimization: Pareto front generation, Struct. Multidiscip. Optim., 29, 2, 149-158 (2005)
[23] Laumanns, M.; Thiele, L.; Deb, K.; Zitzler, E., Combining convergence and diversity in evolutionary multiobjective optimization, Evol. Comput., 10, 3, 263-282 (2002)
[24] Mita, K.; Fukuda, E. H.; Yamashita, N., Nonmonotone line searches for unconstrained multiobjective optimization problems, J. Global Optim., 75, 1, 63-90 (2019) · Zbl 1428.90155
[25] Morovati, V.; Basirzadeh, H.; Pourkarimi, L., Quasi-Newton methods for multiobjective optimization problems, 4OR, 6, 3, 261-294 (2017) · Zbl 1407.90300
[26] Morovati, V.; Pourkarimi, L., Extension of Zoutendijk method for solving constrained multiobjective optimization problems, European J. Oper. Res., 273, 1, 44-57 (2019) · Zbl 1403.90614
[27] Morovati, V.; Pourkarimi, L.; Basirzadeh, H., Barzilai and Borwein’s method for multiobjective optimization problems, Numer. Algorithms, 72, 3, 539-604 (2016) · Zbl 1347.65108
[28] Povalej, Z., Quasi-Newton’s method for multiobjective optimization, J. Comput. Appl. Math., 255, 765-777 (2014) · Zbl 1291.90316
[29] Preuss, M.; Naujoks, B.; Rudolph, G., Pareto set and EMOA behavior for simple multimodal multiobjective functions, (Runarsson, T. P.; etal., Proceedings of the Ninth International Conference on Parallel Problem Solving from Nature (PPSN IX) (2006), Springer: Springer Berlin), 513-522
[30] Qu, S.; Ji, Y.; Jiang, J.; Zhang, Q., Nonmonotone Gradient Methods for Vector Optimization with a Portfolio Optimization Application, European J. Oper. Res., 263, 2, 356-366 (2017) · Zbl 1380.90246
[31] Sayadi Bander, A.; Morovati, V.; Basirzadeh, H., A super non-dominated point for multi-objective transportation problem, Appl. Appl. Math., 10, 1, 544-551 (2015) · Zbl 1326.90078
[32] Toint, P. L., An assessment of nonmonotone linesearch techniques for unconstrained optimization, SIAM J. Sci. Comput., 17, 3, 725-739 (1996) · Zbl 0849.90113
[33] Vieira, D. A.; Takahashi, R. H.; Saldanha, R. R., Multicriteria optimization with a multiobjective golden section line search, Math. Program., 131, 1-2, 131-161 (2010) · Zbl 1235.90141
[34] Zhang, H.; Hager, W. W., A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim., 14, 4, 1043-1056 (2004) · Zbl 1073.90024
[35] Zitzler, E.; Deb, K.; Thiele, L., Comparison of multiobjective evolutionary algorithms: Empirical results, Evol. Comput., 8, 2, 173-195 (2000)
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