A Wolfe-type steepest descent algorithm for uncertain quadratic multiobjective optimization problems

  • Shubham Kumar Department of Mathematics, IILM University Greater Noida Uttar Pradesh India.
  • Amar Deep

Abstract

This work develops a Wolfe-type steepest descent algorithm for solving uncertain quadratic multiobjective optimization problems (UQMOPs) by reformulating them into deterministic robust counterparts via objective-wise worst-case criteria. The proposed method incorporates a Wolfe-type inexact line search to obtain more efficient descent directions and improve overall convergence behavior. A Zoutendijk-type condition is established to guarantee linear convergence under standard assumptions.

Downloads

Download data is not yet available.

References

J. Fliege, B. F. Svaiter, Steepest descent methods for multicriteria optimization, Math. Oper. Res., 51(3), 479-494 (2000).

L. G. Drummond, B. F. Svaiter, A steepest descent method for vector optimization, J. Comput. Appl. Math., 175(2), 395-414 (2005).

M. Ehrgott, Multicriteria optimization, Springer Science and Business Media, Vol. (491), (2005).

J. Fliege, L. G. Drummond, B. F. Svaiter, Newton’s method for multiobjective optimization, SIAM J. Optim., 20(2), 602-626 (2009).

D. Ghosh, D. Chakraborty, A new Pareto set generating method for multi-criteria optimization problems, Oper. Res. Lett., 42 (8), 514-521 (2014).

K. Miettinen, Nonlinear multiobjective optimization, Springer Science and Business Media, Vol. (12), (2012).

M. Ehrgott, J. Ide, A. Schobel, Minmax robustness for multiobjective optimization problems, Eur. J. Oper. Res., 239(1), 17-31 (2014).

A. Ben-Tal, L. El Ghaoui, A. Nemirovski, Robust optimization, Princeton University Press, Vol. 28, (2009).

A. Ben-Tal, A. Nemirovski, Selected topics in robust convex optimization, Math. Program., 112(1), 125-158 (2008).

E. H. Fukuda, L. M. G. Drummond, A survey on multiobjective descent methods, Pesqui. Oper., 34, 585-620 (2014).

M. A. T. Ansary, G. Panda, A modified quasi-Newton method for vector optimization problem, Optim., 64(11), 2289-2306 (2015).

S. K. Mishra, G. Panda, M. A. T. Ansary, B. Ram, On q-Newton’s method for unconstrained multiobjective optimization problems, J. Appl. Math. Comput., 63(1), 391-410 (2020).

A. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res., 21, 1154-1157 (1973).

S. Gass, T. Saaty, The computational algorithm for the parametric objective function, Nav. Res. Logist.Q. 2 39-45 (1955).

Y. Haimes, On a bicriterion formulation of the problems of integrated system identification and system optimization, IEEE transactions on systems, man, and cybernetics, (3), 296-297 (1971).

P. C. Fishburn, Lexicographic orders, utilities and decision rules: a survey, Manag. Sci., 20, 1442-1471 (1974).

Z. Povalej, Quasi-Newton’s method for multiobjective optimization, J. Comput. Appl. Math., 255, 765-777(2014).

M. T. Nguyen, T. Cao, A hybrid decision making model for evaluating land combat vehicle system, In 22nd International Congress on Modelling and Simulation, MODSIM2017, Modelling and Simulation Society of Australia and New Zealand, 1399-1405 (2017).

M. T. Nguyen, T. Cao, A multi-method approach to evaluate land combat vehicle system, J. Appl. Decis. Sci., 12(4), 337-360 (2019).

T. D. Chuong, V. H. Mak-Hau, J. Yearwood, R. Dazeley, M.T. Nguyen, and T. Cao, Robust Pareto solutions for convex quadratic multiobjective optimization problems under data uncertainty, Ann. Oper. Res., 1-32 (2022).

D. Kuroiwa, G. M. Lee, On robust multiobjective optimization, Vietnam J. Math, 40(2-3), 305-317 (2012).

J. Ide, A. Schobel, Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts, OR Spectr., 38(1), 235-271 (2016).

A. M. Goberna, J. Vaithilingam, L. Guoyin, J. Vicente-Perez, Guaranteeing highly robust weakly efficient solutions for uncertain multi-objective convex programs, Eur. J. Oper. Res., 270(1), 40-50 (2018).

A. Dhara, and J. Dutta. Optimality conditions in convex optimization: a finite dimensional view, CRC Press, Ist Edition, (2011).

S. Kumar, M. A. T. Ansary, N. K. Mahato, D. Ghosh, and Y. Shehu. Newton’s method for uncertain multiobjective optimization problems under finite uncertainty sets, J. Nonlinear Var. Anal., 7(5), 785-809, (2023).

S. Kumar, M. A. T. Ansary, N. K. Mahato, D. Ghosh: Steepest descent method for uncertain multiobjective optimization problems under finite uncertainty set, Appl. Anal., 104(7), 1163-1184 (2025).

S. Kumar, N. K. Mahato, M. A. T. Ansary, D. Ghosh: A quasi-newton method for uncertain multiobjective optimization problems via robust optimization approach, arXiv preprint arXiv:2310.07226 (2023).

S. Kumar, N. K. Mahato, M. A. T. Ansary, D. Ghosh, S. Treanta: A modified quasinewton method for uncertain multiobjective optimization problems under a finite uncertainty set, Eng. Optim., 57(5), 1392-1421 (2025).

S. Kumar, N. K. Mahato, M. A. T. Ansary, D. Ghosh: Solving an uncertain quadratic multiobjective optimization problem using Newton’s descent method via robust optimization approach, J. Appl. Numer. Optim., 7(2), 1-29 (2025).

S. Kumar, N. K. Mahato, D. Ghosh: A trust region method for uncertain multiobjective optimization: comparative analysis with existing descent methods, Optim. Eng. (2026).

T. D. Chuong, X. Yu, C. Liu: Solving Two-stage Quadratic Multiobjective Problems via Optimality and Relaxations, J. Optim. Theory Appl. 203, 676-713 (2024).

T. D. Chuong, V.H. Mak-Hau, J. Yearwood: Robust Pareto solutions for convex quadratic multiobjective optimization problems under data uncertainty, Ann. Oper. Res. 319, 1533-1564 (2022).

G. Eichfelder, P. Groetzner: A note on completely positive relaxations of quadratic problems in a multiobjective framework, J. Glob. Optim. 82, 615-626 (2022).

L. R. Lucambio Perez, L. F. Prudente: Nonlinear conjugate gradient methods for vector optimization, SIAM J. Optim. 28(3), 2690-2720 (2018).

Published
2026-04-10
Section
Conf. Issue: Non-Linear Analysis and Applied Mathematics