×

On \(q\)-Newton’s method for unconstrained multiobjective optimization problems. (English) Zbl 1475.90095

Summary: In this paper, we present a method of so-called \(q\)-Newton’s type descent direction for solving unconstrained multiobjective optimization problems. The algorithm presented in this paper is implemented by applying an independent parameter \(q\) (quantum) in an Armijo-like rule to compute the step length which guarantees that the value of the objective function decreases at every iteration. The search processes gradually shift from global in the beginning to local as the algorithm converges due to \(q\)-gradient. The algorithm is experimented on 41 benchmark/test functions which are unimodal and multi-modal with 1, 2, 3, 4, 5, 10 and 50 different dimensions. The performance of the proposed method is confirmed by comparing with three existing schemes.

MSC:

90C29 Multi-objective and goal programming
65K10 Numerical optimization and variational techniques
90C52 Methods of reduced gradient type
90C53 Methods of quasi-Newton type
05A30 \(q\)-calculus and related topics

Software:

ASMO
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Luc, DT, Theory of Vector Optimization (1989), New York: Springer, New York
[2] Eschenauer, H.; Koski, J.; Osyczka, A., Multicriteria Design Optimization (1990), Berlin: Springer, Berlin · Zbl 0743.90116
[3] Kasperska, R.; Ostwald, M.; Rodak, M., Bi-criteria optimization of open cross section of the thin-walled beams with flat flanges, Proc. Appl. Math. Mech., 4, 614-615 (2004) · Zbl 1354.74206 · doi:10.1002/pamm.200410288
[4] Jüschke, A.; Jahn, J.; Kirsch, A., A bicriterial optimization problem of antenna design, Comput. Optim. Appl., 7, 261-276 (1997) · Zbl 0872.90112 · doi:10.1023/A:1008611827855
[5] Fu, Y.; Diwekar, UM, An efficient sampling approach to multiobjective optimization, Ann. Oper. Res., 132, 109-134 (2004) · Zbl 1090.90171 · doi:10.1023/B:ANOR.0000045279.46948.dd
[6] Shan, S.; Wang, GG, An efficient pareto set identification approach for multiobjective optimization on black-box functions, J. Mech. Design, 127, 866-874 (2005) · doi:10.1115/1.1904639
[7] Carrizosa, E.; Conde, E.; Mum̃oz-Màrquez, M.; Puerto, J., Planar point-objective location problems with nonconvex constraints: a geometrical construction, J. Glob. Optim., 6, 77-86 (1995) · Zbl 0834.90076 · doi:10.1007/BF01106606
[8] Carrizosa, E.; Frenk, JBG, Dominating sets for convex functions with some applications, J. Optim. Theory Appl., 96, 281-295 (1998) · Zbl 0905.90134 · doi:10.1023/A:1022614029984
[9] Kiran, KL; Lakshminarayanan, S., Treatment planning of cancer dendritic cell therapy using multi-objective optimization, IFAC Proc. Vol., 42, 109-116 (2009) · doi:10.3182/20090712-4-TR-2008.00016
[10] Petrovski, A., McCall, J., Sudha, B.: Multi-objective optimization of cancer chemotherapy using swarm intelligence. In: Taylor, N.K. (ed.) AISB Symposium on Adaptive and Emergent Behaviour and Complex Systems. UK Society for AI (2009)
[11] Kiran, KL; Jayachandran, D.; Lakshminarayanan, S.; Lim, CT; Goh, JCH, Multi-objective optimization of cancer immuno-chemotherapy, 13th Int. Conf. Biomed. Eng., 1337-1340 (2009), Singapore: Springer, Singapore
[12] Baesler, F.F., Sepúlveda, J.A.: Multi-objective simulation optimization for a cancer treatment center. In: Proceeding of Winter Simulation Conference, pp. 1401-1404. IEEE, USA (2001)
[13] Qu, SJ; Liu, C.; Goh, M., Nonsmooth multiobjective programming with quasi-Newton methods, Eur. J. Oper. Res., 235, 503-510 (2014) · Zbl 1305.90374 · doi:10.1016/j.ejor.2014.01.022
[14] Qu, SJ; Goh, M.; Wu, SY, Multiobjective DC programs with infinite convex constraints, J. Glob. Optim., 59, 41-58 (2014) · Zbl 1326.90067 · doi:10.1007/s10898-013-0091-9
[15] Qu, SJ; Zhou, YY; Zhang, YL; Wahab, MIM; Zhang, G.; Ye, YY, Optimal strategy for a green supply chain considering shipping policy and default risk, Comput. Ind. Eng., 131, 172-186 (2019) · doi:10.1016/j.cie.2019.03.042
[16] Huang, R.; Qu, S.; Yang, X.; Liu, Z., Multi-stage distributionally robust optimization with risk aversion, J. Ind. Manag. Optim. (2019) · Zbl 1474.90309 · doi:10.3934/jimo.2019109
[17] Geoffrion, AM, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22, 618-630 (1968) · Zbl 0181.22806 · doi:10.1016/0022-247X(68)90201-1
[18] Jahn, J., Scalarization in vector optimization, Math. Program, 29, 203-218 (1984) · Zbl 0539.90093 · doi:10.1007/BF02592221
[19] Luc, DT, Scalarization of vector optimization problems, J. Optim. Theory Appl., 55, 85-102 (1987) · Zbl 0622.90083 · doi:10.1007/BF00939046
[20] Eichfelder, G., Scalarizations for adaptively solving multi-objective optimization problems, J. Comput. Optim. Appl., 44, 249-273 (2009) · Zbl 1184.90152 · doi:10.1007/s10589-007-9155-4
[21] Miettinen, K., Nonlinear Multiobjective Optimization (1999), Boston: Kluwer Academic, Boston · Zbl 0949.90082
[22] Drummond, LMG; Maculan, N.; Svaiter, B., On the choice of parameters for the weighting method in vector optimization, Math. Program, 111, 201-216 (2008) · Zbl 1163.90021 · doi:10.1007/s10107-006-0071-7
[23] Jackson, FH, On \(q\)-functions and a certain difference operator, Trans. Roy. Soc. Edinburg., 46, 253-281 (1909) · doi:10.1017/S0080456800002751
[24] Ernst, T.: The history of \(q\)-calculus and a new method (Licentiate Thesis). U.U.D.M, Report (2000). http://www.math.uu.se/thomas/Lics.pdf
[25] Jackson, FH, On \(q\)-definite integrals, Quart. J. Pure Appl. Math., 41, 193-203 (1910) · JFM 41.0317.04
[26] Ernst, T., A method for \(q\)-calculus, J. Nonlinear Math. Phys., 10, 487-525 (2003) · Zbl 1041.33013 · doi:10.2991/jnmp.2003.10.4.5
[27] Rajković, PM; Marinković, SD; Stanković, MS, On \(q\)-Newton-Kantorovich method for solving systems of equations, Appl. Math. Comput., 168, 1432-1448 (2005) · Zbl 1081.65045
[28] Sterroni, AC; Galski, RL; Ramos, FM; Hu, B.; Morasch, K.; Pickl, S.; Siegle, M., The \(q\)-gradient vector for unconstrained continuous optimization problems, Oper. Res. Proc., 365-370 (2010), Heidelberg: Springer, Heidelberg · Zbl 1421.90165
[29] Gouvêa, EJC; Regis, RG; Soterroni, AC; Scarabello, MC; Ramos, FM, Global optimization using \(q\)-gradients, Eur. J. Oper. Res., 251, 727-738 (2016) · Zbl 1346.90679 · doi:10.1016/j.ejor.2016.01.001
[30] Al-Saggaf, UM; Moinuddin, M.; Arif, M.; Zerguine, A., The \(q\)-least mean squares algorithm, Signal Process., 111, 50-60 (2015) · doi:10.1016/j.sigpro.2014.11.016
[31] Chakraborty, S.K., Panda, G.: Newton like line search method using \(q\)-calculus. International Conference on Mathematics and Computing. In: Giri, D., Mohapatra, R.N., Begehr, H., Obaidat, M. (eds.) Communications in Computer and Information Science 655, pp. 196-208. Springer, Singapore (2017) · Zbl 1442.65109
[32] Fliege, J.; Drummond, LMG; Svaiter, BF, Newton’s method for multiobjective optimization, SIAM J. Optim., 20, 602-626 (2009) · Zbl 1195.90078 · doi:10.1137/08071692X
[33] Drummond, LMG; Svaiter, BF, A steepest descent method for vector optimization, J. Comput. Appl. Math., 175, 395-414 (2005) · Zbl 1058.90060 · doi:10.1016/j.cam.2004.06.018
[34] Fliege, J.; Svaiter, BF, Steepest descent methods for multicriteria optimization, Math. Methods Oper. Res., 51, 479-494 (2000) · Zbl 1054.90067 · doi:10.1007/s001860000043
[35] Ansary, MAT; Panda, G., A modified quasi-Newton method for vector optimization problem, Optimization, 64, 2289-2306 (2015) · Zbl 1327.90275 · doi:10.1080/02331934.2014.947500
[36] Schnabel, RB; Eskow, E., A revised modified cholesky factorization algorithm, SIAM J. Optim., 9, 1135-1148 (1999) · Zbl 0958.65034 · doi:10.1137/S105262349833266X
[37] Huband, S.; Hingston, P.; Barone, L.; While, L., A review of multiobjective test problems and a scalable test problem toolkit, IEEE T. Evolut. Comput., 10, 477-50 (2006) · doi:10.1109/TEVC.2005.861417
[38] Qu, S.; Goh, M.; Chan, FTS, Quasi-Newton methods for solving multiobjective optimization, Oper. Res. Lett., 39, 397-399 (2011) · Zbl 1235.90139 · doi:10.1016/j.orl.2011.07.008
[39] Povalej, Ž., Quasi-Newton’s method for multiobjective optimization, J. Comput. Appl. Math., 255, 765-777 (2014) · Zbl 1291.90316 · doi:10.1016/j.cam.2013.06.045
[40] Das, I.; Dennis, JE, 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 · doi:10.1137/S1052623496307510
[41] Eichfelder, G., An adaptive scalarization method in multiobjective optimization, SIAM J. Optim., 19, 1694-1718 (2009) · Zbl 1187.90252 · doi:10.1137/060672029
[42] Hillermeier, C., Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach (2001), Basel: Springer, Basel · Zbl 1064.90041
[43] Jin, Y., Olhofer, M., Sendhoff, B.: Dynamic weighted aggregation for evolutionary multi-objective optimization: Why does it work and how?. In: Spector, L. (ed.) Proceedings of the Genetic and Evolutionary Computation Conference. pp. 1042-1049, Morgan Kaufmann Publishers, United States (2001)
[44] Kim, IY; de Weck, OL, Adaptive weighted sum method for bi-objective optimization, Struct. Multidiscip. O., 29, 149-158 (2005) · doi:10.1007/s00158-004-0465-1
[45] Lovison, A.: A synthetic approach to multiobjective optimization (2010). arXiv:1002.0093
[46] Preuss, M.; Naujoks, B.; Rudolph, G.; Runarsson, TP; Beyer, HG; Burke, E.; Guervós, JJM; Whitley, LD; Yao, X., Pareto set and EMOA behavior for simple multimodal multiobjective functions, Parallel Problem Solving from Nature-PPSN IX, 513-522 (2006), Berlin: Springer, Berlin
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.