A Comparative Numerical Study between Minorant Functions and Line Search Methods in Penalty Methods for Linear Optimization
Received: 16 November 2022 | Revised: 27 November 2022 | Accepted: 2 December 2022 | Online: 5 February 2023
Corresponding author: Assma Leulmi
Abstract
The aim of this paper is to present a comparative numerical study between the minorant functions and line search methods in computing the step size in the penalty method for linear optimization. The minorant functions were confirmed by many interesting numerical experimentations to be more beneficial than the classical line search methods.
Keywords:
penalty methods, line search, minorant function, linear optimizationDownloads
References
M. Achache, "A polynomial-time weighted path-following interior-point algorithm for linear optimization," Asian-European Journal of Mathematics, vol. 13, no. 2, Mar. 2020, Art. no. 2050038. DOI: https://doi.org/10.1142/S1793557120500382
P. Armaos, "A Study of Joint Cost Inclusion in Linear Programming Optimization," Engineering, Technology & Applied Science Research, vol. 3, no. 4, pp. 473–478, Aug. 2013. DOI: https://doi.org/10.48084/etasr.327
B. Badri-Koohi, R. Tavakkoli-Moghaddam, and M. Asghari, "Optimizing Number and Locations of Alternative-Fuel Stations Using a Multi-Criteria Approach," Engineering, Technology & Applied Science Research, vol. 9, no. 1, pp. 3715–3720, Feb. 2019. DOI: https://doi.org/10.48084/etasr.2474
M. Bouafia, D. Benterki, and A. Yassine, "A new efficient short-step projective interior point method for linear programming," Operations Research Letters, vol. 46, no. 3, pp. 291–294, May 2018. DOI: https://doi.org/10.1016/j.orl.2018.02.004
J.-P. Crouzeix and B. Merikhi, "A logarithm barrier method for semi-definite programming," RAIRO - Operations Research, vol. 42, no. 2, pp. 123–139, Apr. 2008. DOI: https://doi.org/10.1051/ro:2008005
J.-P. Chehab and M. Raydan, "Geometrical properties of the Frobenius condition number for positive definite matrices," Linear Algebra and its Applications, vol. 429, no. 8, pp. 2089–2097, Oct. 2008. DOI: https://doi.org/10.1016/j.laa.2008.06.006
R. M. Freund and S. Mizuno, "Interior Point Methods: Current Status and Future Directions," in High Performance Optimization, H. Frenk, K. Roos, T. Terlaky, and S. Zhang, Eds. Boston, MA, USA: Springer US, 2000, pp. 441–466. DOI: https://doi.org/10.1007/978-1-4757-3216-0_18
N. Karmarkar, "A new polynomial-time algorithm for linear programming," in Proceedings of the sixteenth annual ACM symposium on Theory of computing, New York, NY, USA, Sep. 1984, pp. 302–311.
A. Leulmi and S. Leulmi, "Logarithmic Barrier Method Via Minorant Function for Linear Programming | Journal of Siberian Federal University," Journal of Siberian Federal University. Mathematics & Physics, vol. 12, no. 2, pp. 191–201, 2019. DOI: https://doi.org/10.17516/1997-1397-2019-12-2-191-201
A. Leulmi, B. Merikhi, and D. Benterki, "Study of a Logarithmic Barrier Approach for Linear Semidefinite Programming," Journal of Siberian Federal University. Mathematics & Physics, vol. 11, no. 3, pp. 1–13, 2018. DOI: https://doi.org/10.17516/1997-1397-2018-11-3-300-312
N. Karmarkar, "A new polynomial-time algorithm for linear programming," in Proceedings of the sixteenth annual ACM symposium on Theory of computing, New York, NY, USA, Sep. 1984, pp. 302–311. DOI: https://doi.org/10.1145/800057.808695
H. Mansouri and M. Zangiabadi, "An adaptive infeasible interior-point algorithm with full-Newton step for linear optimization," Optimization, vol. 62, no. 2, pp. 285–297, Feb. 2013. DOI: https://doi.org/10.1080/02331934.2011.611881
M. R. Rezoug, M. Benaouadj, D. Taibi, and R. Chenni, "A New Optimization Approach for a Solar Tracker Based on an Inertial Measurement Unit," Engineering, Technology & Applied Science Research, vol. 11, no. 5, pp. 7542–7550, Oct. 2021. DOI: https://doi.org/10.48084/etasr.4330
H. Wolkowicz and G. P. H. Styan, "Bounds for eigenvalues using traces," Linear Algebra and its Applications, vol. 29, pp. 471–506, Feb. 1980. DOI: https://doi.org/10.1016/0024-3795(80)90258-X
J. F. Bonnans, J. C. Gilbert, C. Lemaréchal, and C. A. Sagastizábal, Numerical Optimization: Theoretical and Practical Aspects (Universitext). Berlin, Heidelberg, Germany: Springer-Verlag, 2006.
R. T. Rockafellar, Convex Analysis: (PMS-28), vol. 30. Princeton, NJ, USA: Princeton University Press, 1970.
Downloads
How to Cite
License
Copyright (c) 2022 Assma Leulmi
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain the copyright and grant the journal the right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) after its publication in ETASR with an acknowledgement of its initial publication in this journal.
