IJBTT

A Symmetric Cone Proximal Multiplier Algorithm

A Symmetric Cone Proximal Multiplier Algorithm
	© 2023 by IJETT Journal
	Volume-71 Issue-1
	Year of Publication : 2023
	Author : Erik Alex Papa Quiroz, Miguel Angel Cano Lengua, Julio Cesar Lopez Luis, Rolando Ichpas Tapia
	DOI : 10.14445/22315381/IJETT-V71I1P223

How to Cite?

Erik Alex Papa Quiroz, Miguel Angel Cano Lengua, Julio Cesar Lopez Luis, Rolando Ichpas Tapia, "A Symmetric Cone Proximal Multiplier Algorithm," International Journal of Engineering Trends and Technology, vol. 71, no. 1, pp. 257-270, 2023. Crossref, https://doi.org/10.14445/22315381/IJETT-V71I1P223

Abstract
This paper introduces a proximal multipliers algorithm to solve separable convex symmetric cone minimization problems subject to linear constraints. The algorithm is motivated by the method proposed by Sarmiento et al. (2016, optimization v.65, 2, 501-537), but we consider in the finite-dimensional vectorial spaces, further to an inner product, a Euclidean Jordan Algebra. Under some natural assumptions on convex analysis, it is demonstrated that all accumulation points of the primal-dual sequences generated by the algorithm are solutions to the problem and assuming strong assumptions on the generalized distances; we obtain the global convergence to a minimize point. To show the algorithm's functionality, we provide an application to find the optimal hyperplane in Support Vector Machine (SVM) for binary classification.

Keywords
Symmetric convex cone optimization, Separable techniques, Proximal distances, Proximal method of multipliers, Support vector machine.

References
[1] Alizadeh, F., and Goldfarb, D., “Second-Order Cone Programming,” Mathematical Programming, vol. 95, pp. 3-51, 2012. Crossref, Crossref, https://doi.org/10.1007/s10107-002-0339-5
[2] Alvarez, F, L´Opez, J., and Ram´Irez C., H., “Interior Proximal Algorithm with Variable Metric for Second-Order Cone Programming: Applications to Structural Optimization and Support Vector Machines,” Optimization Methods Software, vol. 25, no. 6, pp. 859-881, 2012. Crossref, https://doi.org/10.1080/10556780903483356
[3] Alfred Auslender, and Marc Teboulle, “Entropic Proximal Decomposition Methods for Convex Programs and Variational Inequalities,” Mathematical Programming, vol. 91, no. 1, pp. 33-47, 2001. https://doi.org/10.1007/s101070100241
[4] Alfred Auslender, and Marc Teboulle, “Interior Gradient and Proximal Methods for Convex and Conic Optimization,” SIAM Journal on Optimization, vol. 16, no. 3, pp. 697-725, 2006. Crossref, https://epubs.siam.org/doi/10.1137/S1052623403427823
[5] Alfred Auslender, Marc Teboulle, and Sami Ben-Tiba, “A Logarithmic-Quadratic Proximal Method for Variational Inequalities,” Computational Optimization and Applications, vol. 12, pp. 31-40, 1999. Crossref, https://doi.org/10.1023/A:1008607511915
[6] J. Frédéric Bonnans et al., “Numerical Optimization: Theoretical and Practical Aspects,” Springer, 2006.
[7] Gong Chen, and Marc Teboulle, “A Proximal-Based Decomposition Method for Convex Minimization Problems,” vol. 64, pp. 81-101, 1994. Crossref, https://doi.org/10.1007/BF01582566
[8] Corinna Cortes, and Vladimir Vapnik "Support-Vector Networks,” Machine Learning, vol. 20, pp. 273-297, 1995. Crossref, https://doi.org/10.1007/BF00994018
[9] DanielGabay, and BertrandMercier “A Dual Algorithm for the Solution of Nonlinear Variational Problems Via Finite Element Approximation,” Computers & Mathematics with Applications, vol. 1, no. 2, pp.17 - 40, 1976. Crossref, https://doi.org/10.1016/0898- 1221(76)90003-1
[10] Donald Goldfarb, Shiqian Ma, and Katya Scheinberg, “Fast Alternating Linearization Methods for Minimizing the Sum of Two Convex Functions,” Mathematical Programming, vol. 141, pp. 349-382, 2013. Crossref, https://doi.org/10.1007/s10107-012-0530-2
[11] M. Kyono, and M. Fukushima, “Nonlinear Proximal Decomposition Method for Convex Programming,” Journal of Optimization Theory and Applications, vol. 106, pp. 357-372, 2000. Crossref, https://doi.org/10.1023/A:1004655531273
[12] Jacques Faraut, and Adam Korányi, “Analysis on Symmetric Cones, Oxford Mathematical Monographs,” The Clarendon Press,  Oxford University Press, New York, 1994.
[13] Osman Güler, “On the Convergence of the Proximal Point Algorithm for Convex Minimization,” SIAM Journal on Control and Optimization, vol. 29, no. 2, pp. 403-419, 1991. Crossref, https://doi.org/10.1137/0329022
[14] Klerk, E. De, “Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications,” Kluwer Academic Publishers, Series Applied Optimization, vol. 65, 2002.
[15] Kiwiel, K.C., “Proximal Minimization Methods with Generalized Bregman Functions,” SIAM Journal on Control and Optimization, vol. 35, no. 4, pp.1142–1168, 1997. Crossref, https://doi.org/10.1137/S0363012995281742
[16] Li, A., and Tuncel, L., “Some Applications of Symmetric Cone Programming in Financial Mathematics,” Transactions in Operational Research, vol. 17, pp. 1-19, 2006.
[17] Miguel Sousa Lobo et al., “Applications of Second-Order Cone Programming,” Linear Algebra and Its Applications, vol. 284, pp. 193- 228, 1998. Crossref, https://doi.org/10.1016/S0024-3795(98)10032-0
[18] López J. and Papa Quiroz E.A.,”Construction of Proximal Distances Over Symmetric Cones,” Optimization, vol. 66, no. 8, pp. 1301-21, 2017. Crossref, https://doi.org/10.1080/02331934.2016.1277998
[19] ShaohuaPan, and Jein-Shan Chen, “Interior Proximal Methods and Central Paths for Convex Second-Order Cone Programming,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 9, pp. 3083 - 3100, 2010. Crossref, https://doi.org/10.1016/j.na.2010.06.079
[20] Sarrmiento O, Papa Quiroz E.A, and Oliveira P.R., “A Proximal Multiplier Method for Separable Convex Minimization,” Optimization, vol. 65, no. 2, pp. 501-537, 2015. Crossref, https://doi.org/10.1080/02331934.2015.1062009
[21] Abdel Radi Abdel Rahman Abdel Gadir, and Samia Abdallah Yagoub Ibrahim, "Comparison Between Some Analysis Solutions for Solving Fredholm Integral Equation of Second Kind," International Journal of Mathematics Trends and Technology, vol. 66, no. 6 pp. 245-260, 2020. Crossref, https://doi.org/10.14445/22315373/IJMTT-V66I6P525
[22] Spingarn, J.E., “Applications of the Method of Partial Inverses to Convex Programming: Decomposition,” Mathematical Programming, vol. 32, pp.199-223, 1985.
[23] Rockafellar, R.T., “Convex Analysis,” Princeton University Press, Princeton Mathematical Series, p. 472, 1970.
[24] Katya Scheinberg, Shiqian Ma, and Donald Goldfarb, “Sparse Inverse Covariance Selection Via Alternating Linearization Methods,” 2010.
[25] Lieven Vandenberghe, and Stephen Boyd, “Semidefinite Programming,” SIAM Review, vol. 38, pp. 49-95, 1994. Crossref. https://doi.org/10.1137/1038003
[26] Miguel Angel Cano Lengua, and Erik Alex Papa Quiroz, “A Systematic Literature Review on Support Vector Machines Applied to Classification,” Proceedings of the 2020 IEEE Engineering International Research Conference, EIRCON, 2020.
[27] Daniel Mavilo Calderon Nieto, Erik Papa, and Miguel Cano, “A Systematic Literature Review on Support Vector Machines Applied to Regression” Proceedings of the 2021 IEEE Sciences and Humanities International Research Conference, 2021. Crossref, https://doi.org/10.1109/SHIRCON53068.2021.9652268
[28] Erik Alex Papa Quiroz et al., “A Proximal Multiplier Method for Convex Separable Symmetric Cone Optimization,” Proceeding of 5th International Conference on Multimedia Systems and Signal Processing, pp. 92-97, 2020. Crossref, https://doi.org/10.1145/3404716.3404734
[29] Rina Mahakud, Binod Kumar Pattanayak, and Bibudhendu Pati, "A Hybrid Multi-Class Classification Model for the Detection of Leaf Disease Using Xgboost and SVM," International Journal of Engineering Trends and Technology, vol. 70, no. 10, pp. 298-306, 2022. Crossref, https://doi.org/10.14445/22315381/IJETT-V70I10P229
[30] Ibrahima Sory Keita et al., "Classification of Benign and Malignant Mris Using SVM Classifier for Brain Tumor Detection," International Journal of Engineering Trends and Technology, vol. 70, no. 3, pp. 234-240, 2022. Crossref, https://doi.org/10.14445/22315381/IJETT-V70I2P226
[31] Maamar Ali Saud AL tobi et al.," Machinery Faults Diagnosis Using Support Vector Machine (SVM) and Naïve Bayes Classifiers," International Journal of Engineering Trends and Technology, vol. 70, no. 12, pp. 26-34, 2022. Crossref, https://doi.org/10.14445/22315381/IJETT-V70I12P204
[32] Schmieta, S.H., and Alizadeh, F., “Extension of Primal-Dual Interior Point Algorithms to Symmetric Cones,” Mathematical Programming, vol. 96, pp. 409-438, 2003. Crossref, https://doi.org/10.1007/s10107-003-0380-z