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An augmented Lagrangian algorithm for multi-objective optimization

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Abstract

In this paper, we propose an adaptation of the classical augmented Lagrangian method for dealing with multi-objective optimization problems. Specifically, after a brief review of the literature, we give a suitable definition of Augmented Lagrangian for equality and inequality constrained multi-objective problems. We exploit this object in a general computational scheme that is proved to converge, under mild assumptions, to weak Pareto points of such problems. We then provide a modified version of the algorithm which is more suited for practical implementations, proving again convergence properties under reasonable hypotheses. Finally, computational experiments show that the proposed methods not only do work in practice, but are also competitive with respect to state-of-the-art methods.

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References

  1. Bertsekas, D.P.: Nonlinear programming. J. Oper. Res. Soc. 48(3), 334–334 (1997)

    Google Scholar 

  2. Binh, T.T., Mobes, U. Korn.: A multiobjective evolution strategy for constrained optimization problems. In: The Third International Conference on Genetic Algorithms (Mendel 97), vol. 25, p. 27 (1997)

  3. Birgin, E.G., Martinez, J.M.: Practical Augmented Lagrangian Methods for Constrained Optimization, vol. 10. SIAM, Philadelphia (2014)

    MATH  Google Scholar 

  4. Bonnel, H., Iusem, A.N., Svaiter, B.F.: Proximal methods in vector optimization. SIAM J. Optim. 15(4), 953–970 (2005)

    MathSciNet  MATH  Google Scholar 

  5. Campana, E.F., Diez, M., Liuzzi, G., Lucidi, S., Pellegrini, R., Piccialli, V., Rinaldi, F., Serani, A.: A multi-objective direct algorithm for ship hull optimization. Comput. Optim. Appl. 71(1), 53–72 (2018)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    MathSciNet  MATH  Google Scholar 

  7. Carrizosa, E., Frenk, J.B.G.: Dominating sets for convex functions with some applications. J. Optim. Theory Appl. 96(2), 281–295 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Costa, L.A., Espírito-Santo, I.A., Oliveira, P.: A scalarized augmented Lagrangian algorithm (SCAL) for multi-objective optimization constrained problems. In: ICORES, pp. 335–340 (2018)

  9. Custódio, A.L., Madeira, J.A., Vaz, A.I.F., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM J. Optim. 21(3), 1109–1140 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Deb, K., Agrawal, S., Pratap, A., Meyarivan, T.: A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In: International Conference on Parallel Problem Solving from Nature, pp. 849–858. Springer (2000)

  11. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    MathSciNet  MATH  Google Scholar 

  12. Drummond, L.G., Iusem, A.N.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28(1), 5–29 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Eichfelder, G.: Adaptive Scalarization Methods in Multiobjective Optimization, vol. 436. Springer, Berlin (2008)

    MATH  Google Scholar 

  14. El Maghri, M., Elboulqe, Y.: Reduced Jacobian method. J. Optim. Theory Appl. 179(3), 917–943 (2018)

    MathSciNet  MATH  Google Scholar 

  15. El Moudden, M., El Ghali, A.: A new reduced gradient method for solving linearly constrained multiobjective optimization problems. Comput. Optim. Appl. 71(3), 719–741 (2018)

    MathSciNet  MATH  Google Scholar 

  16. Fazzio, N.S.: Teoría y métodos para problemas de optimización multiobjetivo. Ph.D. Thesis, Universidad Nacional de La Plata (2018)

  17. Fliege, J.: OLAF—a general modeling system to evaluate and optimize the location of an air polluting facility. OR-Spektrum 23(1), 117–136 (2001)

    MATH  Google Scholar 

  18. Fliege, J.: Gap-free computation of Pareto-points by quadratic scalarizations. Math. Methods Oper. Res. 59(1), 69–89 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Fliege, J., Drummond, L.G., Svaiter, B.F.: Newton’s method for multiobjective optimization. SIAM J. Optim. 20(2), 602–626 (2009)

    MathSciNet  MATH  Google Scholar 

  20. Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51(3), 479–494 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Fliege, J., Vaz, A.I.F.: A method for constrained multiobjective optimization based on SQP techniques. SIAM J. Optim. 26(4), 2091–2119 (2016)

    MathSciNet  MATH  Google Scholar 

  22. Fu, Y., Diwekar, U.M.: An efficient sampling approach to multiobjective optimization. Ann. Oper. Res. 132(1–4), 109–134 (2004)

    MATH  Google Scholar 

  23. Fukuda, E.H., Drummond, L.G.: On the convergence of the projected gradient method for vector optimization. Optimization 60(8–9), 1009–1021 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Fukuda, E.H., Drummond, L.G.: Inexact projected gradient method for vector optimization. Comput. Optim. Appl. 54(3), 473–493 (2013)

    MathSciNet  MATH  Google Scholar 

  25. Fukuda, E.H., Drummond, L.G., Raupp, F.M.: An external penalty-type method for multicriteria. Top 24(2), 493–513 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Fukuda, E.H., Drummond, L.G., Raupp, F.M.: A barrier-type method for multiobjective optimization. Optimization (2019). https://doi.org/10.1080/02331934.2019.1576667

    Article  Google Scholar 

  27. Gass, S., Saaty, T.: The computational algorithm for the parametric objective function. Naval Res. Logist. Q. 2(1–2), 39–45 (1955)

    MathSciNet  Google Scholar 

  28. Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22(3), 618–630 (1968)

    MathSciNet  MATH  Google Scholar 

  29. Gravel, M., Martel, J.M., Nadeau, R., Price, W., Tremblay, R.: A multicriterion view of optimal resource allocation in job-shop production. Eur. J. Oper. Res. 61(1–2), 230–244 (1992)

    Google Scholar 

  30. Jüschke, A., Jahn, J., Kirsch, A.: A bicriterial optimization problem of antenna design. Comput. Optim. Appl. 7(3), 261–276 (1997)

    MathSciNet  MATH  Google Scholar 

  31. Kanzow, C., Steck, D.: An example comparing the standard and safeguarded augmented Lagrangian methods. Oper. Res. Lett. 45(6), 598–603 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Kasperska, R., Ostwald, M., Rodak, M.: Bi-criteria optimization of open cross section of the thin-walled beams with flat flanges. In: PAMM: Proceedings in Applied Mathematics and Mechanics, vol. 4, pp. 614–615. Wiley Online Library (2004)

  33. Konak, A., Coit, D.W., Smith, A.E.: Multi-objective optimization using genetic algorithms: a tutorial. Reliab. Eng. Syst. Saf. 91(9), 992–1007 (2006)

    Google Scholar 

  34. Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining convergence and diversity in evolutionary multiobjective optimization. Evol. Comput. 10(3), 263–282 (2002)

    Google Scholar 

  35. Leschine, T.M., Wallenius, H., Verdini, W.A.: Interactive multiobjective analysis and assimilative capacity-based ocean disposal decisions. Eur. J. Oper. Res. 56(2), 278–289 (1992)

    Google Scholar 

  36. Liuzzi, G., Lucidi, S., Parasiliti, F., Villani, M.: Multiobjective optimization techniques for the design of induction motors. IEEE Trans. Magn. 39(3), 1261–1264 (2003)

    Google Scholar 

  37. Liuzzi, G., Lucidi, S., Rinaldi, F.: A derivative-free approach to constrained multiobjective nonsmooth optimization. SIAM J. Optim. 26(4), 2744–2774 (2016)

    MathSciNet  MATH  Google Scholar 

  38. Morovati, V., Basirzadeh, H., Pourkarimi, L.: Quasi-newton methods for multiobjective optimization problems. 4OR 16(3), 261–294 (2018)

    MathSciNet  MATH  Google Scholar 

  39. Mostaghim, S., Branke, J., Schmeck, H.: Multi-objective particle swarm optimization on computer grids. In: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation, pp. 869–875. ACM (2007)

  40. Palermo, G., Silvano, C., Valsecchi, S., Zaccaria, V.: A system-level methodology for fast multi-objective design space exploration. In: Proceedings of the 13th ACM Great Lakes symposium on VLSI, pp. 92–95. ACM (2003)

  41. Pellegrini, R., Campana, E., Diez, M., Serani, A., Rinaldi, F., Fasano, G., Iemma, U., Liuzzi, G., Lucidi, S., Stern, F.: Application of derivative-free multi-objective algorithms to reliability-based robust design optimization of a high-speed catamaran in real ocean environment. In: Rodrigues et al.(Eds.), Engineering Optimization IV, p. 15 (2014)

  42. Povalej, Ž.: Quasi-Newton’s method for multiobjective optimization. J. Comput. Appl. Math. 255, 765–777 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Qu, S., Goh, M., Chan, F.T.: Quasi-Newton methods for solving multiobjective optimization. Oper. Res. Lett. 39(5), 397–399 (2011)

    MathSciNet  MATH  Google Scholar 

  44. Shan, S., Wang, G.G.: An efficient Pareto set identification approach for multiobjective optimization on black-box functions. J. Mech. Des. 127(5), 866–874 (2005)

    Google Scholar 

  45. Steuer, R.E., Choo, E.-U.: An interactive weighted Tchebycheff procedure for multiple objective programming. Math. Program. 26(3), 326–344 (1983)

    MathSciNet  MATH  Google Scholar 

  46. Sun, Y., Ng, D.W.K., Zhu, J., Schober, R.: Multi-objective optimization for robust power efficient and secure full-duplex wireless communication systems. IEEE Trans. Wirel. Commun. 15(8), 5511–5526 (2016)

    Google Scholar 

  47. Tanabe, H., Fukuda, E.H., Yamashita, N.: Proximal gradient methods for multiobjective optimization and their applications. Comput. Optim. Appl. 72(2), 339–361 (2019)

    MathSciNet  MATH  Google Scholar 

  48. Tavana, M.: A subjective assessment of alternative mission architectures for the human exploration of Mars at NASA using multicriteria decision making. Comput. Oper. Res. 31(7), 1147–1164 (2004)

    MATH  Google Scholar 

  49. White, D.: Epsilon-dominating solutions in mean-variance portfolio analysis. Eur. J. Oper. Res. 105(3), 457–466 (1998)

    MATH  Google Scholar 

  50. White, D.J.: Multiobjective programming and penalty functions. J. Optim. Theory Appl. 43(4), 583–599 (1984)

    MathSciNet  MATH  Google Scholar 

  51. Zadeh, L.: Optimality and non-scalar-valued performance criteria. IEEE Trans. Autom. Control 8(1), 59–60 (1963)

    Google Scholar 

  52. Zhang, Q., Zhou, A., Zhao, S., Suganthan, P., Liu, W., Tiwari, S.: Multiobjective optimization test instances for the CEC 2009 special session and competition. Mechanical Engineering, 01 (2008)

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Acknowledgements

The authors are very grateful to professor Marco Sciandrone for the precious suggestions and the encouragement to carry on this work. We would also like to thank Tommaso Levato for the useful discussions. We finally express our gratitude to the anonymous referees and the editor, whose comments helped us to improve the paper.

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  1. Department of Information Engineering, Università degli Studi di Firenze, Via di Santa Marta 3, 50139, Florence, Italy

    G. Cocchi & M. Lapucci

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  1. G. Cocchi
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Cocchi, G., Lapucci, M. An augmented Lagrangian algorithm for multi-objective optimization. Comput Optim Appl 77, 29–56 (2020). https://doi.org/10.1007/s10589-020-00204-z

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