A restarted three-term vector conjugate gradient method with a diagonal variable metric for nonconvex unconstrained vector optimization

Yuchang Zhang; Jing Gao; Chongyang He; Yuchang Zhang; Jing Gao; Chongyang He

doi:10.3934/math.20261224

AIMS Mathematics

2026, Volume 11, Issue 9: 30904-30943. doi: 10.3934/math.20261224

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Research article

A restarted three-term vector conjugate gradient method with a diagonal variable metric for nonconvex unconstrained vector optimization

1.
School of Mathematics and Statistics, Beihua University, Jilin 132013, China
2.
School of Engineering, RMIT University, Melbourne, VIC 3000, Australia

Received: 26 July 2026 Revised: 08 September 2026 Accepted: 17 September 2026 Published: 21 September 2026
MSC : 65K05, 90C26, 90C29

This paper proposes a restarted three-term vector conjugate gradient method with a diagonal variable metric for nonconvex unconstrained vector optimization. The method incorporates a regularized diagonal Barzilai–Borwein-type variable metric into the vector steepest descent subproblem. A restart mechanism and a safeguarding correction strategy are introduced to ensure that the generated search directions satisfy a uniform sufficient descent condition. Under standard assumptions and a vector Wolfe line search, the global subsequential convergence of the method is established, and the sequence generated by the algorithm is shown to have at least one $ K $-critical accumulation point. Numerical experiments demonstrate that the proposed method achieves good computational efficiency and numerical stability. The results also indicate that the diagonal variable metric can effectively improve the overall performance of the algorithm.
- nonconvex vector optimization,
- diagonal variable metric,
- Barzilai–Borwein-type update,
- restarted three-term structure,
- sufficient descent,
- global convergence
Citation: Yuchang Zhang, Jing Gao, Chongyang He. A restarted three-term vector conjugate gradient method with a diagonal variable metric for nonconvex unconstrained vector optimization[J]. AIMS Mathematics, 2026, 11(9): 30904-30943. doi: 10.3934/math.20261224

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Abstract

This paper proposes a restarted three-term vector conjugate gradient method with a diagonal variable metric for nonconvex unconstrained vector optimization. The method incorporates a regularized diagonal Barzilai–Borwein-type variable metric into the vector steepest descent subproblem. A restart mechanism and a safeguarding correction strategy are introduced to ensure that the generated search directions satisfy a uniform sufficient descent condition. Under standard assumptions and a vector Wolfe line search, the global subsequential convergence of the method is established, and the sequence generated by the algorithm is shown to have at least one $ K $-critical accumulation point. Numerical experiments demonstrate that the proposed method achieves good computational efficiency and numerical stability. The results also indicate that the diagonal variable metric can effectively improve the overall performance of the algorithm.

References

[1]	J. Jahn, A. Kirsch, C. Wagner, Optimization of rod antennas of mobile phones, Math. Meth. Oper. Res., 59 (2004), 37–51. https://doi.org/10.1007/s001860300318 doi: 10.1007/s001860300318
[2]	T. S. Hong, D. L. Craft, F. Carlsson, T. R. Bortfeld, Multicriteria optimization in intensity-modulated radiation therapy treatment planning for locally advanced cancer of the pancreatic head, Int. J. Radiat. Oncol., 72 (2008), 1208–1214. https://doi.org/10.1016/j.ijrobp.2008.07.015 doi: 10.1016/j.ijrobp.2008.07.015
[3]	Y. Q. Wang, J. Zhu, H. P. Huang, F. Xiao, Bi-objective ant colony optimization for trajectory planning and task offloading in UAV-assisted MEC systems, IEEE T. Mobile Comput., 23 (2024), 12360–12377. https://doi.org/10.1109/TMC.2024.3408603 doi: 10.1109/TMC.2024.3408603
[4]	M. Tavana, A subjective assessment of alternative mission architectures for the human exploration of Mars at NASA using multicriteria decision making, Comput. Oper. Res., 31 (2004), 1147–1164. https://doi.org/10.1016/S0305-0548(03)00074-1 doi: 10.1016/S0305-0548(03)00074-1
[5]	T. M. Leschine, H. Wallenius, W. A. Verdini, Interactive multiobjective analysis and assimilative capacity-based ocean disposal decisions, Eur. J. Oper. Res., 56 (1992), 278–289. https://doi.org/10.1016/0377-2217(92)90228-2 doi: 10.1016/0377-2217(92)90228-2
[6]	D. J. White, Epsilon-dominating solutions in mean-variance portfolio analysis, Eur. J. Oper. Res., 105 (1998), 457–466. https://doi.org/10.1016/S0377-2217(97)00056-8 doi: 10.1016/S0377-2217(97)00056-8
[7]	G. Eichfelder, Multiobjective bilevel optimization, Math. Program., 123 (2010), 419–449. https://doi.org/10.1007/s10107-008-0259-0 doi: 10.1007/s10107-008-0259-0
[8]	J. Fliege, L. N. Vicente, Multicriteria approach to bilevel optimization, J. Optim. Theory Appl., 131 (2006), 209–225. https://doi.org/10.1007/s10957-006-9136-2 doi: 10.1007/s10957-006-9136-2
[9]	P. De, J. B. Ghosh, C. E. Wells, On the minimization of completion time variance with a bicriteria extension, Oper. Res., 40 (1992), 1148–1155. https://doi.org/10.1287/opre.40.6.1148 doi: 10.1287/opre.40.6.1148
[10]	C. B. Qin, K. L. Sun, A. Sun, D. H. Zhang, J. S. Zhang, G. H. Wang, et al., Dynamic event-triggered optimal safety control of interconnected nonlinear systems with discontinuous state constraints by adaptive dynamic programming, Neurocomputing, 669 (2026), 132467. https://doi.org/10.1016/j.neucom.2025.132467 doi: 10.1016/j.neucom.2025.132467
[11]	C. B. Qin, M. Y. Pang, Z. W. Wang, S. Y. Hou, Q. Qiu, ADP-based dynamic event-triggered fault-tolerant control for nonlinear systems with actuator failures and full-state time-varying constraints, Commun. Nonlinear Sci., 157 (2026), 109676. https://doi.org/10.1016/j.cnsns.2026.109676 doi: 10.1016/j.cnsns.2026.109676
[12]	Y. Y. Hu, J. Q. Chen, C. B. Qin, Event-triggered prescribed performance control of the multiplayer game nonlinear system via integral reinforcement learning, Appl. Math. Comput., 511 (2026), 129716. https://doi.org/10.1016/j.amc.2025.129716 doi: 10.1016/j.amc.2025.129716
[13]	C. B. Qin, Z. W. Wang, S. Y. Hou, M. Y. Pang, G. H. Wang, Y. Wang, et al., Mixed zero-sum game based dynamic event-triggered optimal control of multi-input nonlinear system with safety constraints, Commun. Nonlinear Sci., 152 (2026), 109376. https://doi.org/10.1016/j.cnsns.2025.109376 doi: 10.1016/j.cnsns.2025.109376
[14]	R. S. Burachik, C. Y. Kaya, M. M. Rizvi, A new scalarization technique and new algorithms to generate Pareto fronts, SIAM J. Optim., 27 (2017), 1010–1034. https://doi.org/10.1137/16M1083967 doi: 10.1137/16M1083967
[15]	G. Eichfelder, An adaptive scalarization method in multiobjective optimization, SIAM J. Optim., 19 (2009), 1694–1718. https://doi.org/10.1137/060672029 doi: 10.1137/060672029
[16]	A. M. Geoffrion, Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22 (1968), 618–630. https://doi.org/10.1016/0022-247X(68)90201-1 doi: 10.1016/0022-247X(68)90201-1
[17]	J. Fliege, B. F. Svaiter, Steepest descent methods for multicriteria optimization, Math. Meth. Oper. Res., 51 (2000), 479–494. https://doi.org/10.1007/s001860000043 doi: 10.1007/s001860000043
[18]	J. Fliege, L. M. Graña Drummond, B. F. Svaiter, Newton's method for multiobjective optimization, SIAM J. Optim., 20 (2009), 602–626. https://doi.org/10.1137/08071692X doi: 10.1137/08071692X
[19]	E. H. Fukuda, L. M. Graña Drummond, Inexact projected gradient method for vector optimization, Comput. Optim. Appl., 54 (2013), 473–493. https://doi.org/10.1007/s10589-012-9501-z doi: 10.1007/s10589-012-9501-z
[20]	H. Bonnel, A. N. Iusem, B. F. Svaiter, Proximal methods in vector optimization, SIAM J. Optim., 15 (2005), 953–970. https://doi.org/10.1137/S1052623403429093 doi: 10.1137/S1052623403429093
[21]	L. C. Ceng, B. S. Mordukhovich, J. C. Yao, Hybrid approximate proximal method with auxiliary variational inequality for vector optimization, J. Optim. Theory Appl., 146 (2010), 267–303. https://doi.org/10.1007/s10957-010-9667-4 doi: 10.1007/s10957-010-9667-4
[22]	L. R. Lucambio Pérez, L. F. Prudente, Nonlinear conjugate gradient methods for vector optimization, SIAM J. Optim., 28 (2018), 2690–2720. https://doi.org/10.1137/17M1126588 doi: 10.1137/17M1126588
[23]	L. R. Lucambio Pérez, L. F. Prudente, A Wolfe line search algorithm for vector optimization, ACM T. Math. Software, 45 (2019), 37. https://doi.org/10.1145/3342104 doi: 10.1145/3342104
[24]	M. L. N. Gonçalves, L. F. Prudente, On the extension of the Hager–Zhang conjugate gradient method for vector optimization, Comput. Optim. Appl., 76 (2020), 889–916. https://doi.org/10.1007/s10589-019-00146-1 doi: 10.1007/s10589-019-00146-1
[25]	M. L. N. Gonçalves, F. S. Lima, L. F. Prudente, A study of Liu–Storey conjugate gradient methods for vector optimization, Appl. Math. Comput., 425 (2022), 127099. https://doi.org/10.1016/j.amc.2022.127099 doi: 10.1016/j.amc.2022.127099
[26]	Q. R. He, C. R. Chen, S. J. Li, Spectral conjugate gradient methods for vector optimization problems, Comput. Optim. Appl., 86 (2023), 457–489. https://doi.org/10.1007/s10589-023-00508-w doi: 10.1007/s10589-023-00508-w
[27]	J. W. Chen, Y. S. Bai, X. L. Qin, X. Q. Ou, X. L. Qin, A PRP type conjugate gradient method without truncation for nonconvex vector optimization, J. Optim. Theory Appl., 204 (2025), 13. https://doi.org/10.1007/s10957-024-02571-7 doi: 10.1007/s10957-024-02571-7
[28]	Z. Yang, L. Ma, Adaptive step size rules for stochastic optimization in large-scale learning, Stat. Comput., 33 (2023), 45. https://doi.org/10.1007/s11222-023-10218-2 doi: 10.1007/s11222-023-10218-2
[29]	W. W. Wang, J. Gao, An efficient three-term conjugate gradient algorithm with restart strategy and image restoration, Sci. Rep., 15 (2025), 35451. https://doi.org/10.1038/s41598-025-19329-4 doi: 10.1038/s41598-025-19329-4
[30]	H. Pan, J. W. Chen, K. P. Liu, Y. Lv, A modified Dai–Yuan conjugate gradient method for vector optimization problems, Optimization, 2025 (2025), 1–24. https://doi.org/10.1080/02331934.2025.2547719 doi: 10.1080/02331934.2025.2547719
[31]	J. W. Peng, J. W. Zhang, J. C. Yao, A novel hybrid conjugate gradient method for multiobjective optimization problems, Optimization, 75 (2026), 773–792. https://doi.org/10.1080/02331934.2024.2373903 doi: 10.1080/02331934.2024.2373903
[32]	M. A. T. Ansary, G. Panda, A modified quasi-Newton method for vector optimization problem, Optimization, 64 (2015), 2289–2306. https://doi.org/10.1080/02331934.2014.947500 doi: 10.1080/02331934.2014.947500
[33]	S. Huband, P. Hingston, L. Barone, L. While, A review of multiobjective test problems and a scalable test problem toolkit, IEEE T. Evolut. Comput., 10 (2006), 477–506. https://doi.org/10.1109/TEVC.2005.861417 doi: 10.1109/TEVC.2005.861417
[34]	Y. C. Jin, M. Olhofer, B. Sendhoff, Dynamic weighted aggregation for evolutionary multi-objective optimization: Why does it work and how, In: Proceedings of the 3rd Annual Conference on Genetic and Evolutionary Computation (GECCO'01), San Francisco: Morgan Kaufmann Publishers Inc., 2001, 1042–1049. Available from: https://dl.acm.org/doi/10.5555/2955239.2955427.
[35]	A. Lovison, Singular continuation: Generating piecewise linear approximations to Pareto sets via global analysis, SIAM J. Optim., 21 (2011), 463–490. https://doi.org/10.1137/100784746 doi: 10.1137/100784746
[36]	J. J. Moré, B. S. Garbow, K. E. Hillstrom, Testing unconstrained optimization software, ACM T. Math. Software, 7 (1981), 17–41. https://doi.org/10.1145/355934.355936 doi: 10.1145/355934.355936
[37]	O. Schütze, M. Laumanns, C. A. Coello Coello, M. Dellnitz, E.-G. Talbi, Convergence of stochastic search algorithms to finite-size Pareto set approximations, J. Glob. Optim., 41 (2008), 559–577. https://doi.org/10.1007/s10898-007-9265-7 doi: 10.1007/s10898-007-9265-7
[38]	J. Thomann, G. Eichfelder, Numerical results for the multiobjective trust region algorithm MHT, Data Brief, 25 (2019), 104103. https://doi.org/10.1016/j.dib.2019.104103 doi: 10.1016/j.dib.2019.104103
[39]	Ph. L. Toint, Test problems for partially separable optimization and results for the routine PSPMIN, Technical Report 83/4, Department of Mathematics, University of Namur, Belgium, 1983. Available from: https://perso.unamur.be/phtoint/pubs/TR83-04.pdf.

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