Abstract
In this work an iterative method to solve the nonlinear multiobjective problem is presented. The goal is to find locally optimal points for the problem, that is, points that cannot simultaneously improve all functions when we compare the value at the point with those in their neighborhood. The algorithm uses a strategy developed in previous works by several authors but globalization is obtained through a nonmonotone technique. The construction of a new ratio between the actual descent and predicted descent plays a key role for selecting the new point and updating the trust region radius. On the other hand, we introduce a modification in the quadratic model used to determine if the point is accepted or not, which is fundamental for the convergence of the method. The combination of this strategy with a Newton-type method leads to an algorithm whose convergence properties are proved. The numerical experimentation is performed using a known set of test problems. Preliminary numerical results show that the nonmonotone method can be more efficient when it is compared to another algorithm that use the classic trust region approach.
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The authors declare that the data supporting the findings of this study are available within the article and its references. Nevertheless, the findings of this research are available at the request of the corresponding author Viviana A. Ramirez.
References
Thomann, J., Eichfelder, G.: A trust region algorithm heterogeneous multiobjective optimization. Siam J. Optim. 29, 1017–1047 (2019)
Gobbi, M., Levi, F., Mastinu, G., Previati, G.: On the analytical derivation of the Pareto-optimal set with applications to structural design. Struct. Multidiscip. Optim. 51, 645–657 (2015)
Soleimani-Damaneh, M.: An optimization modelling for string selection in molecular biology using Pareto optimality. Appl. Math. Model. 35, 3887–3892 (2011)
Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds.): Pareto Optimality, Game Theory and Equilibria. Springer (2008)
Kamal-Sayed, H., Ma, J., Tseung, H., Abdel-Rehim, A., Herman, M.G., Beltran, C.J.: Adaptive method for multicriteria optimization of intensity modulated proton therapy. Med. Phys. 45, 5643–52 (2018)
Tapia, M., and Coello, C.: Applications of multi-objective evolutionary algorithms in economics and finance: a survey. In: Proceedings of the IEEE Congress on Evolutionary Computation, IEEE Press, Piscataway, NJ, pp. 532–539 (2007)
Ignatius, J., Mustafa, A.: A multiobjective sensitivity approach to training providers evaluation and quota allocation planning. Int. J. Inf. Technol. Decis. Mak. 10(1), 147–174 (2011)
Şakar, C.T., Köksalan, M.: Effects of multiple criteria on portfolio optimization. Int. J. Inf. Technol. Decis. Mak. 13(1), 77–99 (2014)
Cesarone, F., Scozzari, A., Tardella, F.: An optimization-diversification approach to portfolio selection. J. Glob. Optim. 76, 245–265 (2020)
Andreani, R., Ramirez, V.A., Santos, S.A., Secchin, L.D.: Bilevel optimization with a multiobjective problem in the lower level. Numer. Algorithms 81(3), 915–946 (2019)
Ramirez, V.A.: Reformulação de um problemma de Programação não linear com restrições multiobjetivo. Tese de Doutorado, Instituto de Matemática, Estatística e Computação Científica, UNICAMP, Brasil (2015)
Fliege, J., Graña Drummond, L.M., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20, 602–626 (2009)
Carrizo, G., Lotito, P., Maciel, M.: Trust region globalization strategy for the nonconvex unconstrained multiobjective optimization problem. Math. Prog. Ser. A 159, 339–369 (2016)
Villacorta, K.D.V., Oliveira, P.R., Soubeyran, A.: A Trust-Region Method for Unconstrained Multiobjective Problems with Applications. J. Optim. Theory Appl. 160, 865–889 (2014)
Thomann, J., Eichfelder, G.: Numerical results for the multiobjective trust region algorithm MHT. Data Brief 25, 104103 (2019)
Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23(4), 707–716 (1986)
Deng, N., Xiao, Y., Zhou, F.: Nonmonotonic trust region algorithm. J. Optim. Theory Appl. 76, 259–285 (1993)
Sun, W.: Nonmonotone trust region method for solving optimization problems. Appl. Math. Comput. 156, 159–174 (2004)
Mo, J., Zhang, K., Wei, Z.: A nonmonotone trust region method for unconstrained optimization. Appl. Math. Comput. 171, 371–384 (2005)
Ahookhosh, M., Amini, K.: An efficient nonmonotone trust-region method for unconstrained optimization. Numer. Algorithms 59(4), 523–540 (2012)
Chen, J., Sun, W., Yang, Z.: A non-monotone retrospective trust-region method for unconstrained optimization. J. Ind. Manag. Optim. 9(4), 919–944 (2013)
Maciel, M.C., Mendonça, M.G., Verdiell, A.B.: Monotone and nonmonotone trust-region-based algorithms for large scale unconstrained optimization problems. Comput. Optim. Appl. 54, 27–43 (2013)
Nocedal, J., Wright, S.J.: Numerical Optimization, Springer Series in Operations Research. Springer, Berlin (1999)
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm, NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable Test Problems for Evolutionary Multi-Objective Optimization, Kanpur Genetic Algorithms Lab. Indian Inst. Technol., Report 2 (2001)
Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Prog. Ser. A 91, 201–213 (2002)
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The authors are grateful to the anonymous reviewers, whose comments improved the presentation of the manuscript.
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The funding was provided by Universidad Nacional del Comahue (AR) (Grant No. 04/E106).
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Ramirez, V.A., Sottosanto, G.N. Nonmonotone trust region algorithm for solving the unconstrained multiobjective optimization problems. Comput Optim Appl 81, 769–788 (2022). https://doi.org/10.1007/s10589-021-00346-8
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DOI: https://doi.org/10.1007/s10589-021-00346-8