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Issue: Vol. 3 No. 101 (2025)
Section: Articles
Date Published: 27/03/2026
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Pham, M. C., & Ta, Q. S. (2026). On an optimality algorithm based on the subgradient method for locally Lipschitz functions. Scientific Journal of Saigon University, 3(101), 60-75. https://sj.sgu.edu.vn/index.php/sjsgu/article/view/410

On an optimality algorithm based on the subgradient method for locally Lipschitz functions

Manh Chinh Pham1, Quang Son Ta2,
1 Trung tâm Giáo dục Nghề nghiệp - Giáo dục Thường xuyên Quận 12
2 Saigon University

Main Article Content

Abstract

In this paper, we are concerned with a subgradient method based on an iterative sequence involving the Clarke subdifferential. This method is used to construct an algorithm for finding optimal solutions of a nonconvex optimization problem whose objective function is locally Lipschitz. The obtained results regarding the convergence, accuracy, and stopping conditions of the algorithm are presented. The method used in this study extends the framework developed by S. Boyd for convex optimization problems introduced in [1].

Keywords

Subgradient method, Clarke subdifferential, Iterative convergence, Stopping conditions

Article Details

References

[1] S. Boyd, Subgradient Methods, Notes for EE364b, Stanford University, May 2014.
[2] N. Shor, Minimization Methods for Non-differentiable Functions. Berlin, Germany: Springer-Verlag, 1985.
[3] B.T. Polyak, Introduction to Optimization, Optimization Software, Inc., Publications Division, 1987.
[4] R. M. Freund, Subgradient Optimization, Generalized Programming, and Nonconvex
Duality, MIT Open Course Ware, Massachusetts Institute of Technology, 2004.
[5] J. F. Bonnans, J. C. Gilbert, C. Lemaréchal and C. A. Sagastizábal, Numerical
Optimization: Theoretical and Practical Aspects, 2nd ed. Berlin, Germany: Springer- Verlag, 2006.
[6] A. Scagliotti and P.C. Franzone, A Subgradient Method with Constant Step-Size for ℓ₁- Composite Optimization, Bollettino dell'Unione Matematica Italian, vol. 17, no. 3, 2024.
[7] X. Li, L. Zhao, D. Zhu and A. M. So, Revisiting Subgradient Method: Complexity and Convergence Beyond Lipschitz Continuity, Vietnam Journal of Mathematics, vol. 52, no. 3, 2024.
[8] F. Clarke, Optimization and Nonsmooth Analysis. John Wiley & Sons, 1983.
[9] C. K. W. Yu, Y. Hu, X. Yang and S. K. Choy, Abstract convergence theorem for quasi-m convex optimization problems with applications, Optimization, vol. 68, no. 7, 2019.
[10] Y. Nesterov, Lectures on Convex Optimization, 2nd ed. Switzerland: Springer, 2018.
[11] A. Beck, First-Order Methods in Optimization. Philadelphia, SIAM, 2017.
[12] Đặng Đức Trọng, Phạm Hoàng Quân, Đặng Hoàng Tâm và Đinh Ngọc Thanh, Giáo trình Giải tích hàm. NXB Đại học quốc gia TP.HCM, 2011.
[13] Tạ Quang Sơn, Giáo trình Giải tích lồi và Tối ưu. NXB Giáo dục Việt Nam, 2018.
[14] B. Lafferriere, G. Lafferriere and N. M. Nam, Introduction to Mathematical Analysis. Portland State University Library, 2015.