Variable Metric Method for Unconstrained Multiobjective Optimization Problems

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  • 1. Department of Mathematics, Shanghai University, Shanghai, 200444, China;
    2. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China;
    3. National Center for Applied Mathematics of Chongqing, and School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

Received date: 2022-01-23

  Revised date: 2022-07-15

  Online published: 2023-09-07

Supported by

This research was supported by the Major Program of the National Natural Science Foundation of China (Nos. 11991020 and 11991024), the National Natural Science Foundation of China (Nos. 11971084 and 12171060), the Natural Science Foundation of Chongqing (No. cstc2019jcyj-zdxmX0016) and Foundation of Chongqing Normal University (Nos. 22XLB005 and 22XLB006).

Abstract

In this paper, we propose a variable metric method for unconstrained multiobjective optimization problems (MOPs). First, a sequence of points is generated using different positive definite matrices in the generic framework. It is proved that accumulation points of the sequence are Pareto critical points. Then, without convexity assumption, strong convergence is established for the proposed method. Moreover, we use a common matrix to approximate the Hessian matrices of all objective functions, along which a new nonmonotone line search technique is proposed to achieve a local superlinear convergence rate. Finally, several numerical results demonstrate the effectiveness of the proposed method.

Cite this article

Jian Chen, Gao-Xi Li, Xin-Min Yang . Variable Metric Method for Unconstrained Multiobjective Optimization Problems[J]. Journal of the Operations Research Society of China, 2023 , 11(3) : 409 -438 . DOI: 10.1007/s40305-022-00447-z

References

[1] Marler, R.T., Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidisc. Optim. 26(6), 369–395(2004)
[2] Tapia, M., Coello, C.: Applications of multi-objective evolutionary algorithms in economics and finance: a survey. In: IEEE Congress on Evolutionary Computation, pp. 532–539(2007)
[3] Handl, J., Kell, D.B., Knowles, J.: Multiobjective optimization in bioinformatics and computational biology. IEEE/ACM Trans. Comput. Biol. Bioinform. 4(2), 279–292(2007)
[4] Reed, P.M., Hadka, D., Herman, J.D., Kasprzyk, J.R., Kollat, J.B.: Evolutionary multiobjective optimization in water resources: the past, present, and future. Adv. Water Resour. 51, 438–456(2013)
[5] Gass, S., Saaty, T.: The computational algorithm for the parametric objective function. Naval. Res. Logs. 2(1), 39–45(1955)
[6] Zadeh, L.: Optimality and non-scalar-valued performance criteria. IEEE Trans. Autom. Control 8(1), 59–60(1963)
[7] Haimes, Y., Lasdon, L., Wismer, D.: On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans. Syst. Man Cybern. 1(3), 296–297(1971)
[8] Charnes, A., Cooper, W., Ferguson, R.: Optimal estimation of executive compensation by linear programming. Manag. Sci. 1(2), 138–151(1955)
[9] Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22(3), 618–630(1968)
[10] Mukai, H.: Algorithms for multicriterion optimization. IEEE Trans. Autom. Control 25(2), 177–186(1980)
[11] Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494(2000)
[12] Fliege, J., Drummond, L.M.G., Svaiter, B.F.: Newton's method for multiobjective optimization. SIAM J. Optim. 20(2), 602–626(2009)
[13] Graña Drummond, L.M., Iusem, A.N.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28(1), 5–29(2004)
[14] Povalej, Ž: Quasi-Newton's method for multiobjective optimization. J. Comput. Appl. Math. 255, 765–777(2014)
[15] Qu, S.J., Goh, M., Chan, F.T.S.: Quasi-Newton methods for solving multiobjective optimization. Oper. Res. Lett. 39(5), 397–399(2011)
[16] Lucambio Pérez, L.R., Prudente, L.F.: Nonlinear conjugate gradient methods for vector optimization. SIAM J. Optim. 20(3), 2690–2720(2018)
[17] Carrizo, G.A., Lotito, P.A., Maciel, M.C.: Trust region globalization strategy for the nonconvex unconstrained multiobjective optimization problem. Math. Program. 159, 339–369(2016)
[18] Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970(2005)
[19] Moudden, M.E., Mouatasim, A.E.: Accelerated diagonal steepest descent method for unconstrained multiobjective optimization. J. Optim. Theory Appl. 188(1), 220–242(2021)
[20] Powell, M.J.D.: Variable metric methods for constrained optimization. In: Mathematical Programming: The State of the Art, pp. 288–311(1983)
[21] Ansary, M.A.T., Panda, G.: A modified quasi-Newton method for vector optimization problem. Optimization 64(11), 2289–2306(2014)
[22] Broyden, C.G.: The convergence of a class double-rank minimization algorithms. J. Inst. Math. Appl. 6, 76–90(1970)
[23] Fletcher, R.: A new approach to variable metric algorithms. Comput. J. 13, 317–322(1970)
[24] Goldfarb, D.: A family of variable metric methods derived by variation mean. Math. Comput. 23, 23–26(1970)
[25] Shanno, D.F.: Conditioning of quasi-Newton methods for function minimization. Math. Comput. 24, 647–656(1970)
[26] Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1988)
[27] Jian, J.B.: Researches on superlinear and quadratically convergent algorithms for nonlinearly constrained optimization. Ph.D. Theies, Xi'an Jiaotong University (2000)
[28] Dennis, J.E., Moré, J.J.: Quasi-Newton methods, motivation and theory. SIAM Rev. 19, 46–89(1977)
[29] Grippo, L., Lampariellof, L., Lucidi, S.: A nonmonotone line search technique for Newton's method. SIAM J. Numer. Anal. 23(4), 707–716(1986)
[30] Zhang, H.C., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14(4), 1043–1056(2004)
[31] Wolfe, P.: Finding the nearest point in a polytope. Math. Program. 11(1), 128–149(1976)
[32] Hohenbalken, B.V.: A finite algorithm to maximize certain pseudoconcave functions on polytopes. Math. Program. 9(1), 189–206(1975)
[33] Frank, M., Wolfe, P.: An algorithm for quadratic programming. Naval Res. Log. Q. 3, 95–110(1956)
[34] Diamond, S., Boyd, S.: CVXPY: a python-embedded modeling language for convex optimization. J. Mach. Learn. Res. 17, 1–5(2016)
[35] Deb, K.: Multi-objective genetic algorithms: problem difficulties and construction of test problems. Evol. Comput. 7(3), 205–230(1999)
[36] Jin, Y., Olhofer, M., Sendhoff, B.: Dynamic weighted aggregation for evolutionary multi-objective optimization: Why does it work and how? In: Proceedings of the Genetic and Evolutionary Computation Conference, pp. 1042–1049(2001)
[37] Preuss, M., Naujoks, B., Rudolph, G.: Pareto set and EMOA behavior for simple multimodal multiobjective functions. In: Parallel Problem Solving from Nature-PPSN IX, pp. 513–522(2006)
[38] Witting, K.: Numerical algorithms for the treatment of parametric multiobjective optimization problems and applications. Ph.D. thesis, Universitätsbibliothek (2012)
[39] Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731(2007)
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