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A Modified q-BFGS Algorithm for Unconstrained Optimization

Author

Listed:
  • Kin Keung Lai

    (International Business School, Shaanxi Normal University, Xi’an 710119, China)

  • Shashi Kant Mishra

    (Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India)

  • Ravina Sharma

    (Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India)

  • Manjari Sharma

    (Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India)

  • Bhagwat Ram

    (Centre for Digital Transformation, Indian Institute of Management, Ahmedabad 380015, India)

Abstract

This paper presents a modification of the q -BFGS method for nonlinear unconstrained optimization problems. For this modification, we use a simple symmetric positive definite matrix and propose a new q -quasi-Newton equation, which is close to the ordinary q -quasi-Newton equation in the limiting case. This method uses only first order q -derivatives to build an approximate q -Hessian over a number of iterations. The q -Armijo-Wolfe line search condition is used to calculate step length, which guarantees that the objective function value is decreasing. This modified q -BFGS method preserves the global convergence properties of the q -BFGS method, without the convexity assumption on the objective function. Numerical results on some test problems are presented, which show that an improvement has been achieved. Moreover, we depict the numerical results through the performance profiles.

Suggested Citation

  • Kin Keung Lai & Shashi Kant Mishra & Ravina Sharma & Manjari Sharma & Bhagwat Ram, 2023. "A Modified q-BFGS Algorithm for Unconstrained Optimization," Mathematics, MDPI, vol. 11(6), pages 1-24, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1420-:d:1098026
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    References listed on IDEAS

    as
    1. Aline Cristina Soterroni & Roberto Luiz Galski & Fernando Manuel Ramos, 2011. "The q-Gradient Vector for Unconstrained Continuous Optimization Problems," Operations Research Proceedings, in: Bo Hu & Karl Morasch & Stefan Pickl & Markus Siegle (ed.), Operations Research Proceedings 2010, pages 365-370, Springer.
    2. Kin Keung Lai & Shashi Kant Mishra & Bhagwat Ram, 2020. "On q -Quasi-Newton’s Method for Unconstrained Multiobjective Optimization Problems," Mathematics, MDPI, vol. 8(4), pages 1-14, April.
    3. Borges, Ernesto P., 2004. "A possible deformed algebra and calculus inspired in nonextensive thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 95-101.
    4. Gouvêa, Érica J.C. & Regis, Rommel G. & Soterroni, Aline C. & Scarabello, Marluce C. & Ramos, Fernando M., 2016. "Global optimization using q-gradients," European Journal of Operational Research, Elsevier, vol. 251(3), pages 727-738.
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