Application of Newton-Gauss-Seidel method for solving multi-objective constrained optimization problems

Peng Cheng; Jumat Sulaiman; Khadizah Ghazali; Majid Khan Majahar Ali; Ming Ming Xu.

Many problems in life are composed of conflicting and influencing multiple objectives, and people often encounter the optimization problem of simultaneously optimizing multiple objectives in each area, which is called multi-objective optimization problem. Therefore, solving such problems has important scientific research value and practical significance. This paper proposes a Newton Gauss-Seidel iteration method for solving multi-objective constrained optimization problems by constructing Newton directions and introducing Gauss-Seidel (GS) iterative method for solving linear equations. The solution of this combination between Newton, Gauss-seidel and Lagrange multiplier method involves two stages: objective function and constraint condition processing stage. In the first stage, the original multi-objective function is scalarized, and only the decision-maker needs to give each objective function a weight, by transforming it into a single objective constrained optimization problem. Then the Lagrange multiplier method was used to transform the constrained optimization problem into an unconstrained optimization problem. The second stage is to use the Newton-Gauss-Seidel (NGS) iterative method to solve the transformed constrained optimization problem. Finally, numerical experiments showed that our proposed algorithm can achieve good results.

KEYWORDS: Multi-objective constrained optimization; Weighting method; Lagrange multiplier; Newton's method; Gauss-Seidel iteration



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REFERENCES

1. Atkinson, K. 1985. Elementary Numerical Analysis. Chichester: John Wiley& Sons, Inc.
2. El-Abbasy, M. S., Elazouni, A. & Zayed, T. 2020. Finance-based scheduling multi-objective optimization: benchmarking of ecolutionary algorithms. Automation in Construction, 120, Article ID 103392.
3. Ghazali, K., Sulaiman, J., Dasril, Y. & Gabda, D. 2019. Application of Newton-4egsor Iteration for Solving Large Scale Unconstrained Optimization Problems with a Tridiagonal Hessian Matrix. In: Alfred, R., Lim, Y., Ibrahim, A. & Anthony, P. (eds). Computational Science and Technology. Lecture Notes in Electrical Engineering 481. Singapore: Springer.
4. Ghazali, K., Sulaiman, J., Dasril, Y. & Gabda, D. 2019. Newton-SOR Iteration for Solving Large-Scale Unconstrained Optimization Problems with an Arrowhead Hessian Matrices. Journal of Physics: Conference Series, 1358(1), 012054.
5. Hu, Y. D. 1990. Shiyong duomubiao zuiyouhua [Practical multi-objective optimization], Shanghai: Shanghai Feasible Computing Press.
6. Hassan, M. & Baharum, A. 2019. A new Logarithmic Penalty Function Approach for Nonlinear Constrained Optimization Problem. Decision Science Letters, 8(3), 353-362.
7. Jozefowiez, N., Senmet, F. & Talbi, E. G.2008. Multi-objective vehicle routing problems. European Journal of Operational Research, 189(2), 293-309.
8. Liu, Y. P. 2017. Gaowei duomubiao jinhua lilun yu suanfa [Many-Objective Evolutionary Optimization Theory and Method]. PhD thesis, China University of Mining and Technology, China.
9. Lin, C. Y. & Dong J. L. 1992. Duomubiao youhua de fangfa he lilun [Methods and Theory of Multi-objective Optimization]. Jilin Education Press.
10. Laptin, Y. P. & Bardadym, T. 2019. Problems related to estimating the coefficients of exact penalty functions. Cybernetics and Systems Analysis, 55(1), 400–412.
11. Miettinen, K., 2004. Nonlinear Multiobjective Optimization. New York: Springer Science+Business Media.
12. Nguyen, V. H. & Strodiot J. J. 1979. On the convergence rate for a penalty function method of exponential type. Journal of Optimization Theory and Applications, 27(4), 495-508.
13. Napituplu, H., Sukono, Mohd B. I.,Hidayat, Y. & Suplan. S. 2018. Steepest descent method implementation on unconstrained optimization problem using C++ program. Materials Science and Engineering, 332, 012024.
14. Rockafellar, R. T. 1973. The multiplier method of Hestenes and Powell applied to convex programming. Journal of Optimization Theory and Applications, 12(6), 555-562.
15. Rosen, J. B. 1960. The gradient projection method for nonlinear programming.Part1:Linear constraints. Society for Industrial and Applied Mathematics, 8(1), 181-217.
16. Stanimirovic, I. P., Zlatanovic, M. L. & Petkovic, M. D. 2011. On the linear weighted sum method for multi-objective optimization. Facta Universitatis (NIS) Ser. Math. Inform. 26(2011), 49–63.
17. Sulaiman, J., Hasan, M.K., Othman, M. & Karim, S.A.A. 2014. Fourth-order solutions of nonlinear two-point boundary value problems by Newton-HSSOR iteration. AIP Conference Proceedings, 1602, 69-75.
18. Sulaiman, J., Hasan, M. K., Othman, M. & Karim, S. A. A. 2015. Application of Block Iterative Methods with Newton Scheme for Fisher’s Equation by Using Implicit Finite Difference. Journal Kalam, 8(1), 039-46.
19. Xiao, S. 2010. Qiujie duomubiao yueshuyouhua wenti de jiaohushi niudunfa [Interactive Newton Method for Solving Multiobjective Constrained Optimization Problems]. MSc Thesis, Shanghai Jiao Tong University, China.
20. Yuan, S. Q.& Li Z.M. 2005. Descending Dimension Algorithm for Multi-objective Programming with Linear Equality Constraints. Operations Research Transactions, 9(1), 70-74.
21. Zhu, S. Y. & Zhu, F. 2020. Multi-objective bike-way network design problem with space-time accessibility constraint.Transportation, 47(5), 2479-2503.
22. Zoutendijk, G. 1970. Some Algorithms Based on the Principle of Feasible Directions. Nonlinear Programming, 5(4), 93-121.