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Optimization
A Journal of Mathematical Programming and Operations Research
Volume 67, 2018 - Issue 1
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Original Articles

Multiple subgradient descent bundle method for convex nonsmooth multiobjective optimization

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Pages 139-158 | Received 02 Feb 2017, Accepted 16 Sep 2017, Published online: 12 Oct 2017

References

  • Mistakidis ES, Stavroulakis GE. Nonconvex optimization in mechanics. Smooth and nonsmooth algorithms, heuristics and engineering applications by the F.E.M. Dordrecht: Kluwer Academic Publisher; 1998.
  • Outrata J, Ko\^{c}vara M, Zowe J. Nonsmooth approach to optimization problems with equilibrium constraints. Theory, applications and numerical results. Dordrecht: Kluwer Academic Publishers; 1998.
  • Moreau JJ, Panagiotopoulos PD, Strang G, editors. Topics in nonsmooth mechanics. Basel: Birkhäuser; 1988.
  • Ehrgott M. Multicriteria optimization. 2nd ed. Berlin: Springer; 2005.
  • Miettinen K. Nonlinear multiobjective optimization. Boston (MA): Kluwer Academic Publishers; 1999.
  • Désidéri J-A. Multiple-gradient descent algorithm (MGDA) for multiobjective optimization. Compte rendus de l’Académie des sciences Ser I. 2012;350:313–318.
  • Fliege J, Graña Drummond LM, Svaiter BF. Newton’s method for multiobjective optimization. SIAM J Optim. 2009;20(2):602–626.
  • Fliege J, Svaiter BF. Steepest descent methods for multicriteria optimization. Math Methods Oper Res. 2000;51(3):479–494.
  • Fukuda EH, Graña Drummond LM. Inexact projected gradient method for vector optimization. Comput Optim Appl. 2013;54(3):473–493.
  • Graña Drummond LM, Svaiter BF. A steepest descent method for vector optimization. J Comput Appl Math. 2005;175:395–414.
  • Harada K, Sakuma J, Kobayashi S. Local search for multiobjective function optimization: Pareto descent method. In: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, GECCO ’06. New York (NY): ACM; 2006. p. 659–666.
  • Mukai H. Algorithms for multicriterion optimization. IEEE Trans Autom Control. 1979;ac-25(2):177–186.
  • Povalej Ž. Quasi-Newton’s method for multiobjective optimization. J Comput Appl Math. 2014;255:765–777.
  • Bagirov A, Karmitsa N, Mäkelä MM. Introduction to nonsmooth optimization: theory, practice and software. Cham Heidelberg: Springer; 2014.
  • Mäkelä MM. Survey of bundle methods for nonsmooth optimization. Optim Methods Softw. 2002;17(1):1–29.
  • Lemaréchal C. Nondifferentiable optimization. In: Nemhauser GG, Rinnooy Kan AHG, Todd MJ, editors. Optimization. Amsterdam: Elsevier North-Holland; 1989. p. 529–572.
  • Haslinger J, Neittaanmäki P. Finite element approximation for optimal shape, material and topology design. Chichester: Wiley; 1996.
  • Hiriart-Urruty JB, Lemaréchal C. Convex analysis and minimization algorithms, I and II. Berlin: Springer Verlag; 1993.
  • Kiwiel KC. A descent method for nonsmooth convex multiobjective minimization. Large Scale Syst. 1985;8(2):119–129.
  • Kiwiel KC. Methods of descent for nondifferentiable optimization. Berlin: Springer-Verlag; 1985.
  • Kiwiel KC. Proximity control in bundle methods for convex nondifferentiable optimization. Math Program. 1990;46:105–122.
  • Lemaréchal C, Strodiot JS, Bihain A. On a bundle algorithm for nonsmooth optimization. In: Mangasarian OL, Meyer GL, Robinson SM, editors. Nonlinear programming, computational methods in applied sciences. Vol. 4. New York (NY): Academic Press; 1981. p. 245–282.
  • Mäkelä MM, Neittaanmäki P. Nonsmooth optimization: analysis and algorithms with applications to optimal control. Singapore: World Scientific Publishing Co.; 1992.
  • Mifflin R. An algorithm for constrained optimization with semismooth functions. Math Oper Res. 1977;2(2):191–207.
  • Schramm H, Zowe J. A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results. SIAM J Optim. 1992;2(1):121–152.
  • Clarke FH. Optimization and nonsmooth analysis. New York (NY): Wiley; 1983.
  • Bonnel H, Iusem AN, Svaiter BF. Proximal methods in vector optimization. SIAM J Optim. 2005;15(4):953–970.
  • Bello Cruz JY, Iusem AN. A strongly convergent method for nonsmooth convex minimization in Hilbert spaces. Numer Funct Anal Optim. 2011;32(10):1009–1018.
  • Mäkelä MM, Karmitsa N, Wilppu O. Proximal bundle method for nonsmooth and nonconvex multiobjective optimization. In: Tuovinen T, Repin S, Neittaanm{\"a}ki P, editors. Mathematical modeling and optimization of complex structures. Vol. 40, Computational methods in applied sciences. Cham: Springer; 2016. p. 191–204.
  • Qu S, Liu C, Goh M, Li Y, Ji Y. Nonsmooth multiobjective programming with quasi-Newton methods. Eur J Oper Res. 2014;235(3):503–510.
  • Wang S. Algorithms for multiobjective and nonsmooth optimization. In: Kleinschmidt P, Radermacher FJ, Sweitzer W, Wildermann H, editors. Methods of operations research. Vol. 58. Frankfurt: Athenaum Verlag; 1989. p. 131–142.
  • Miettinen K, Mäkelä MM. Interactive bundle-based method for nondifferentiable multiobjective optimization: NIMBUS. Optimization. 1995;34:231–246.
  • Bello Cruz JY. A subgradient method for vector optimization problems. SIAM J Optim. 2013;23(4):2169–2182.
  • Da Cruz Neto JX, Da Silva GJP, Ferreira OP, et al. A subgradient method for multiobjective optimization. Comput Optim Appl. 2013;54(3):461–472.
  • Désidéri J-A. Multiple-gradient descent algorithm (MGDA); 2009 (Technical Report 6953, INRIA Research Report).
  • Mäkelä MM, Eronen V-P, Karmitsa N. On nonsmooth multiobjective optimality conditions with generalized convexities. In: Rassias TM, Floudas CA, Butenko S, editors. Optimization in science and engineering. New York (NY): Springer; 2014. p. 333–357.
  • Rockafellar RT. Convex analysis. Princeton (NJ): Princeton University Press; 1970.
  • Bazaraa MS, Sherali HD, Shetty CM. Nonlinear programming: theory and algorithms. 3rd ed. Hoboken (NJ): Wiley; 2006.
  • Shor NZ. Minimization methods for non-differentiable functions. Berlin: Springer-Verlag; 1985.
  • Vlček J, Lukšan L. Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization. J Optim Theory Appl. 2001;111(2):407–430.
  • Kiwiel KC. An aggregate subgradient method for nonsmooth convex minimization. Math Program. 1983;27(3):320–341.
  • Lukšan L, Vlček J. Test problems for nonsmooth unconstrained and linearly constrained optimization. Technical Report 798, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague; 2000.
  • Mäkelä MM. Multiobjective proximal bundle method for nonconvex nonsmooth optimization: Fortran subroutine MPBNGC 2.0; 2003 (Technical Report B 13/2003). Reports of the Department of Mathematical Information Technology, Series B, Scientific computing, University of Jyväskylä, Jyväskylä.
  • Armijo L. Minimization of functions having Lipschitz continuous first partial derivatives. Pac J Math. 1966;16(1):1–3.
  • Lukšan L. Dual method for solving a special problem of quadratic programming as a subproblem at linearly constrained nonlinear minimax approximation. Kybernetika. 1984;20:445–457.
  • Dolan ED, Moré JJ. Benchmarking optimization software with performance profiles. Math Program. 2002;91(2):201–213.

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