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Multiobjective BFGS method for optimization on Riemannian manifolds

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Abstract

This paper introduces a Riemannian BFGS method to address multiobjective optimization problems with strongly retraction-convex objective functions. This method is a natural extension of the Euclidean version and produces a sequence of iterates that converges to a Pareto optimal point regardless of the initial point. The main component of our globalization strategy is a generalized Wolfe line search. Numerical experiments demonstrate the superiority and effectiveness of the proposed algorithm.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)

    Book  Google Scholar 

  2. Assunção, P.B., Ferreira, O.P., Prudente, L.F.: Conditional gradient method for multiobjective optimization. Comput. Optim. Appl. 78(3), 741–768 (2021)

    MathSciNet  Google Scholar 

  3. Bello Cruz, J.Y., Lucambio Pérez, L.R., Melo, J.G.: Convergence of the projected gradient method for quasiconvex multiobjective optimization. Nonlinear Anal. Theory Methods Appl. 74(16), 5268–5273 (2011)

    MathSciNet  Google Scholar 

  4. Bento, G.C., Ferreira, O.P., Oliveira, P.R.: Unconstrained steepest descent method for multicriteria optimization on Riemannian manifolds. J. Optim. Theory Appl. 154(1), 88–107 (2012)

    MathSciNet  Google Scholar 

  5. Bento, G.C., Neto, J.C.: A subgradient method for multiobjective optimization on Riemannian manifolds. J. Optim. Theory Appl. 159(1), 125–137 (2013)

    MathSciNet  Google Scholar 

  6. Bento, G.C., Neto, J.C., Meireles, L.V.: Proximal point method for locally Lipschitz functions in multiobjective optimization of Hadamard manifolds. J. Optim. Theory Appl. 179(1), 37–52 (2018)

    MathSciNet  Google Scholar 

  7. Bento, G.C., Neto, J.C., Santos, P.: An inexact steepest descent method for multicriteria optimization on Riemannian manifolds. J. Optim. Theory Appl. 159, 108–124 (2013)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  9. Boumal, N.: An Introduction to Optimization on Smooth Manifolds. Cambridge University Press, Cambridge (2023)

    Google Scholar 

  10. Cai, T., Song, L., Li, G., Liao, M.: Multi-task learning with Riemannian optimization. In: ICIC 2021: Intelligent Computing Theories and Application, volume 12837 of Lecture Notes in Computer Science, pp. 499–509 (2021)

  11. 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)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  13. Chuong, T.D.: Newton-like methods for efficient solutions in vector optimization. Comput. Optim. Appl. 54(3), 495–516 (2013)

    MathSciNet  Google Scholar 

  14. Cocchi, G., Liuzzi, G., Lucidi, S., Sciandrone, M.: On the convergence of steepest descent methods for multiobjective optimization. Comput. Optim. Appl. 77(1), 1–27 (2020)

    MathSciNet  Google Scholar 

  15. Das, I., Dennis, J.E.: A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Optim. 14, 63–69 (1997)

    Google Scholar 

  16. Das, I., Dennis, J.E.: Normal-boundary intersection: a new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631–657 (1998)

    MathSciNet  Google Scholar 

  17. Deb, K.: Multiobjective Optimization using Evolutionary Algorithms. Wiley, New York (2001)

    Google Scholar 

  18. Eschenauer, H., Koski, J., Osyczka, A.: Multicriteria Design Optimization. Springer, Berlin (1990)

    Google Scholar 

  19. Eslami, N., Najafi, B., Vaezpour, S.M.: A trust-region method for solving multicriteria optimization problems on riemannian manifolds. J. Optim. Theory Appl. 196(1), 212–239 (2023)

    MathSciNet  Google Scholar 

  20. Evans, G.W.: An overview of techniques for solving multiobjective mathematical programs. Manage. Sci. 30(11), 1268–1282 (1984)

    MathSciNet  Google Scholar 

  21. Ferreira, O.P., Louzeiro, M.S., Prudente, L.F.: Iteration-complexity and asymptotic analysis of steepest descent method for multiobjective optimization on Riemannian manifolds. J. Optim. Theory Appl. 184(2), 507–533 (2020)

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  25. Fliege, J., Vicente, L.N.: Multicriteria approach to bilevel optimization. J. Optim. Theory Appl. 131(2), 209–225 (2006)

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  28. 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 

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

    MathSciNet  Google Scholar 

  30. Gonçalves, M.L., Lima, F.S., Prudente, L.F.: Globally convergent Newton-type methods for multiobjective optimization. Comput. Optim. Appl. 83(2), 403–434 (2022)

    MathSciNet  Google Scholar 

  31. Gonçalves, M.L., Prudente, L.F.: On the extension of the Hager-Zhang conjugate gradient method for vector optimization. Comput. Optim. Appl. 76(3), 889–916 (2020)

    MathSciNet  Google Scholar 

  32. Graña Drummond, L.M., Iusem, A.: A projected gradient method for vector optimization problems. Comput. Optim. Appl. 28, 5–29 (2004)

    MathSciNet  Google Scholar 

  33. Graña Drummond, L.M., Raupp, F.M.P., Svaiter, B.F.: A quadratically convergent Newton method for vector optimization. Optimization 63(5), 661–677 (2014)

    MathSciNet  Google Scholar 

  34. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals. Springer, Berlin (1993)

    Google Scholar 

  35. Huang, W., Absil, P.A., Gallivan, K.A.: A Riemannian symmetric rank-one trust-region method. Math. Program. 150(2), 179–216 (2015)

    MathSciNet  Google Scholar 

  36. Huang, W., Absil, P.A., Gallivan, K.A.: A Riemannian BFGS method without differentiated retraction for nonconvex optimization problems. SIAM J. Optim. 28(1), 470–495 (2018)

    MathSciNet  Google Scholar 

  37. Huang, W., Gallivan, K.A., Absil, P.A.: A Broyden class of quasi-Newton methods for Riemannian optimization. SIAM J. Optim. 25(3), 1660–1685 (2015)

    MathSciNet  Google Scholar 

  38. Lu, F., Chen, C.R.: Newton-like methods for solving vector optimization problems. Appl. Anal. 93(8), 1567–1586 (2014)

    MathSciNet  Google Scholar 

  39. Lucambio Pérez, L.R., Prudente, L.F.: Nonlinear conjugate gradient methods for vector optimization. SIAM J. Optim. 28(3), 2690–2720 (2018)

    MathSciNet  Google Scholar 

  40. Lucambio Pérez, L.R., Prudente, L.F.: A Wolfe line search algorithm for vector optimization. ACM Trans. Math. Softw. 45(4), 1–23 (2019)

    MathSciNet  Google Scholar 

  41. Najafi, S., Hajarian, M.: Multiobjective conjugate gradient methods on Riemannian manifolds. J. Optim. Theory Appl. 197(3), 1229–1248 (2023)

    MathSciNet  Google Scholar 

  42. Neto, J.C., Da Silva, G.J., Ferreira, O.P., Lopes, J.O.: A subgradient method for multiobjective optimization. Comput. Optim. Appl. 54(3), 461–472 (2013)

    MathSciNet  Google Scholar 

  43. Picheny, V., Wagner, T., Ginsbourger, D.: A benchmark of kriging-based infill criteria for noisy optimization. Struct. Multidiscip. Optim. 48(3), 607–626 (2013)

    Google Scholar 

  44. Prudente, L.F., Souza, D.R.: A quasi-Newton method with Wolfe line searches for multiobjective optimization. J. Optim. Theory Appl. 194(3), 1107–1140 (2022)

    MathSciNet  Google Scholar 

  45. Rosenbrock, H.H.: An automatic method for finding the greatest or least value of a function. Comput. J. 3(3), 175–184 (1960)

    MathSciNet  Google Scholar 

  46. 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  Google Scholar 

  47. Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and its Applications, vol. 297. Springer, Dordrecht (1994)

    Google Scholar 

  48. Wang, X.M., Wang, J.H., Li, C.: Convergence of inexact steepest descent algorithm for multiobjective optimizations on Riemannian manifolds without curvature constraints. J. Optim. Theory Appl. (2023). https://doi.org/10.1007/s10957-023-02235-y

    Article  MathSciNet  Google Scholar 

  49. Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: Empirical results. Evol. Comput. 8(2), 173–195 (2000)

    CAS  PubMed  Google Scholar 

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Acknowledgements

We would like to express our appreciation to the editor and anonymous referees for providing valuable feedback that improved the quality of this paper.

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  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

    Shahabeddin Najafi & Masoud Hajarian

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  1. Shahabeddin Najafi
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Correspondence to Masoud Hajarian.

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Najafi, S., Hajarian, M. Multiobjective BFGS method for optimization on Riemannian manifolds. Comput Optim Appl 87, 337–354 (2024). https://doi.org/10.1007/s10589-023-00522-y

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