AIMS Mathematics, 2020, 5(6): 5521-5540. doi: 10.3934/math.2020354.

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On q-steepest descent method for unconstrained multiobjective optimization problems

1 College of Economics, Shenzhen University, Shenzhen 518060, China
2 Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India
3 Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India
4 Department of Economic Sciences, Indian Institute of Technology Kanpur, Kanpur 208016 India
5 DST-Centre for Interdisciplinary Mathematical Sciences, Institute of Science, Banaras Hindu University, Varanasi 221005, India

The q-gradient is the generalization of the gradient based on the q-derivative. The q-version of the steepest descent method for unconstrained multiobjective optimization problems is constructed and recovered to the classical one as q equals 1. In this method, the search process moves step by step from global at the beginning to particularly neighborhood at last. This method does not depend upon a starting point. The proposed algorithm for finding critical points is verified in the numerical examples.
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Keywords multiobjective; q-calculus; steepest descent; pareto optimality; critical point; algorithms

Citation: Kin Keung Lai, Shashi Kant Mishra, Geetanjali Panda, Md Abu Talhamainuddin Ansary, Bhagwat Ram. On q-steepest descent method for unconstrained multiobjective optimization problems. AIMS Mathematics, 2020, 5(6): 5521-5540. doi: 10.3934/math.2020354

References

  • 1. Marusciac, On Fritz John type optimality criterion in multi-objective optimization, Anal. Numer. Theor. Approx., 11 (1982), 109-114.
  • 2. G. Palermo, C. Silvano, S. Valsecchi, et al. A system-level methodology for fast multi-objective design space exploration, ACM, New York, 2003.
  • 3. M. Tavana, A subjective assessment of alternative mission architectures for the human exploration of mars at NASA using multicriteria decision making, Comput. Oper. Res., 31 (2004), 1147-1164.
  • 4. H. Eschenauer, J. Koski, A. Osyczka, Multicriteria design optimization: Procedures and applications, Springer-Verlag, Berlin, 1990.
  • 5. I. Das, J. E. Dennis, Normal-Boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems, SIAM J. Optim.,8 (1998), 631-657.
  • 6. I. Y. Kim, O. L. de Weck, Adaptive weighted-sum method for bi-objective optimization: Pareto front generation, Struct. Multidisc. Optim., 29 (2005), 149-158.
  • 7. Y. Fu, U. M. Diwekar, An efficient sampling approach to multiobjective optimization, Ann. Oper. Res., 132 (2004), 109-134.
  • 8. A. Jüschke, J. Jahn, A. Kirsch, A bicriterial optimization problem of antenna design, Comput. Optim. Appl., 7 (1997), 261-276.
  • 9. T. M. Leschine, H. Wallenius, W. A. Verdini, Interactive multiobjective analysis and assimilative capacity-based ocean disposal decisions, Eur. J. Oper. Res., 56 (1992), 278-289.
  • 10. J. Fliege, OLAF-A general modeling system to evaluate and optimize the location of an air polluting facility, OR Spektrum, 23 (2001), 117-136.
  • 11. E. Carrizosa, J. B. G. Frenk, Dominating sets for convex functions with some applications, J. Optim. Theory Appl., 96 (1998), 281-295.
  • 12. G. W. Evan, An overview of techniques for solving multiobjective mathematical programs, Manage. Sci., 30 (1984), 1268-1282.
  • 13. D. Prabuddha, J. B. Ghosh, C. E. Wells, On the minimization of completion time variance with a bicriteria extension, Oper. Res., 40 (1992), 1148-1155.
  • 14. D. J. White, Epsilon-dominating solutions in mean-variance portfolio analysis, Eur. J. Oper. Res., 105 (1998), 457-466.
  • 15. M. G. C. Tapia, C. A. C. Coello, Applications of multi-objective evolutionary algorithms in economics and finance: A survey, IEEE Congr. Evol. Comput., (2007), 532-539.
  • 16. E. Zitzler, K. Deb, L. Thiele, Comparison of multiobjective evolutionary algorithms: Empirical results, Evol. Comput., 8 (2000), 173-195.
  • 17. M. Ehrgott, Multicriteria optimization, Springer, Berlin, 2005.
  • 18. K. Miettinen, M. M. Makelä, Interactive bundle-based method for nondifferentiable multiobjeective optimization: NIMBUS, Optimization, 34 (1995), 231-246.
  • 19. L. M. G. Drummond, B. F. Svaiter, A steepest descent method for vector optimization, J. Comput. Appl. Math., 175 (2005), 395-414.
  • 20. J. Fliege, L. M. G. Drummond, B. F. Svaiter, Newton's method for multiobjective optimization, SIAM J. Optim., 20 (2009), 602-626.
  • 21. A. Cauchy, Méthode générale pour la résolution des systèms d'équations simultanées, Comp. Rend. Sci. Paris, 25 (1847), 536-538.
  • 22. H. Zhao, J. Zheng, W. Deng, et al. Semi-supervised broad learning system based on manifold regularization and broad network, IEEE Trans. Circuits Syst., 67 (2020), 983-994.
  • 23. W. Deng, J. Xu, Y. Song, et al. An effective improved co-evolution ant colony optimization algorithm with multi-strategies and its application, Int. J. Bio-Inspir. Com., (2020) 1-10.
  • 24. W. Deng, J. Xu, H. Zhao, An improved ant colony optimization algorithm based on hybrid strategies for scheduling problem, IEEE access, 7 (2019), 20281-20292.
  • 25. W. Deng, H. Liu, J. Xu, et al. An improved quantum-inspired differential evolution algorithm for deep belief network, IEEE Trans. Instrum. Meas., 2020.
  • 26. S. K. Mishra, B. Ram, Introduction to Unconstrained Optimization with R, Springer Nature, Singapore, 2019.
  • 27. J. Fliege, B. F. Svaiter, Steepest descent methods for multicriteria optimization, Math. Method Oper. Res., 51 (2000), 479-494.
  • 28. T. D. Chuong, J. C. Yao, Steepest descent methods for critical points in vector optimization problems, Appl. Anal., 91 (2012), 1811-1829.
  • 29. J. Fliege, A. I. F. Vaz, L. N. Vicente, Complexity of gradient descent for multiobjective optimization, Optim. Method Softw., 34 (2019), 949-959.
  • 30. F. H. Jackson, On q-functions and a certain difference operator, Trans. Roy Soc. Edin., 46 (1908) 253-281.
  • 31. F. H. Jackson, On q-Definite Integrals, Pure Appl. Math. Q., 41 (1910), 193-203.
  • 32. A. Aral, V. Gupta, R. P. Agarwal, Applications of q-calculus in operator theory, Springer, New York, 2013.
  • 33. P. M. Rajković, M. S. Stanković, S. D. Marinković, Mean value theorems in q-calculus, Mat. Vesn., 54 (2002), 171-178.
  • 34. M. E. Ismail, D. Stanton, Applications of q-Taylor theorems, J. Comput. Appl. Math., 153 (2003), 259-272.
  • 35. S. C. Jing, H. Y. Fan, q-Taylor's formula with its q-remainder, Commun. Theor. Phys., 23 (1995), 117.
  • 36. P. M. Rajković, S. D. Marinković, M. S. Stanković, Fractional integrals and derivatives in qcalculus, Appl. Anal. Discrete Math., 1 (2007), 311-323.
  • 37. H. Gauchman, Integral inequalities in q-calculus, Comput. Math. Appl., 47 (2004), 281-300.
  • 38. G. Bangerezako, Variational q-calculus, J. Math. Anal. Appl., 289 (2004), 650-665.
  • 39. L. Abreu, A q-sampling theorem related to the q-Hankel transform, Proc. Am. Math. Soc., 133 (2005), 1197-1203
  • 40. T. H. Koornwinder, R. F. Swarttouw, On q-analogues of the Fourier and Hankel transforms, Trans. Am. Math. Soc., 333 (1992), 445-461.
  • 41. J. Ablinger, A. K. Uncu, qFunctions-A Mathematica package for q-series and partition theory applications, arXiv preprint arXiv:1910.12410, 2019.
  • 42. A. C. Soterroni, R. L. Galski, F. M. Ramos, The q-gradient vector for unconstrained continuous optimization problems, In: Operations research proceedings 2010. Springer-Verlag Berlin Heidelberg, (2011), 365-370.
  • 43. A. C. Soterroni, R. L. Galski, F. M. Ramos, The q-gradient method for global optimization, arXiv:1209.2084, 2012.
  • 44. A. C. Soterroni1, R. L. Galski, M. C. Scarabello1, et al. The q-G method: A q-version of the steepest descent method for global optimization, SpringerPlus, 4 (2015), 647.
  • 45. A. C. Soterroni, R. L. Galski, F. M. Ramos, Satellite constellation design using the q-G global optimization method, Third World Conference on Complex Systems (WCCS), IEEE, (2015), 1-6.
  • 46. U. M. Al-Saggaf, M. Moinuddin, M. Arif, et al. The q-least mean squares algorithm, Signal Process., 111 (2015), 50-60.
  • 47. A. Sadiq, S. Khan, I. Naseem, et al. Enhanced q-least mean square, Circ. Syst. Signal Process., 38 (2019), 4817-4839.
  • 48. P. M. Rajković, S. D. Marinković, M. S. Stanković, On q-Newton-Kantorovich method for solving systems of equations, Appl. Math. Comput., 168 (2005), 1432-1448.
  • 49. S. K. Chakraborty, G. Panda, Newton like line search method using q-calculus, In: D. Giri, R. N. Mohapatra, H. Begehr, M. Obaidat, Communications in Computer and Information Science, Singapore: Springer, 2017.
  • 50. S. K. Mishra, G. Panda, M. A. T. Ansary, et al. On q-Newton's method for unconstrained multiobjective optimization problems, J. Appl. Math. Comput., (2020), 1-20.
  • 51. Ž. Povalej, Quasi-Newton's method for multiobjective optimization, J. Comput. Appl. Math., 255 (2014), 765-777.
  • 52. K. K. Lai, S. K. Mishra, B. Ram, On q-Quasi-Newton's Method for Unconstrained Multiobjective Optimization Problems, Mathematics, 8 (2020), 616.
  • 53. M. A. T. Ansary, G. Panda, A sequential quadratically constrained quadratic programming technique for a multi-objective optimization problem, Eng. Optim., 51 (2019), 22-41.
  • 54. W. E. Karush, Minima of functions of several variables with inequalities as side conditions. PhD thesis, Univ. Chicago, 1939. 55. M. A. Ansary, G. Panda, A modified quasi-Newton method for vector optimization problem, Optimization, 64 (2015), 2289-2306.
  • 55. M. A. T. Ansary, G. Panda, A modified quasi-Newton method for vector optimization problem, Optimization, 64 (2015), 2289-2306.
  • 56. K. Deb, L. Thiele, M. Laumanns, et al. Scalable multi-objective optimization test problems, IEEE, 1 (2002), 825-830.
  • 57. O. P. Ferreira, M. S. Louzeiro, L. F. Prudente, Iteration-complexity and asymptotic analysis of steepest descent method for multiobjective optimization on Riemannian manifolds, J. Optim. Theory Appl., 184 (2020), 507-533.

 

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