A comprehensive characterization of the robust isolated calmness of Ky Fan $ k $-norm regularized convex matrix optimization problems

Ziran Yin; Chongyang Liu; Xiaoyu Chen; Jihong Zhang; Jinlong Yuan; Ziran Yin; Chongyang Liu; Xiaoyu Chen; Jihong Zhang; Jinlong Yuan

doi:10.3934/math.2025227

AIMS Mathematics

2025, Volume 10, Issue 3: 4955-4969. doi: 10.3934/math.2025227

Previous Article Next Article

Research article Special Issues

A comprehensive characterization of the robust isolated calmness of Ky Fan $ k $-norm regularized convex matrix optimization problems

1.
School of Science, Dalian Maritime University, Dalian, Liaoning, 116026, China
2.
School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong, 264005, China
3.
School of Science, Shenyang Ligong University, Shenyang, Liaoning, 110159, China

Received: 23 January 2025 Revised: 15 February 2025 Accepted: 25 February 2025 Published: 06 March 2025
MSC : 65K10, 90C25, 90C31

This paper extends a result of isolated calmness for nuclear norm regularized convex optimization problems to Ky Fan $ k $-norm regularized convex optimization problems. We find that there exists a certain equivalence relationship among the critical cones of the Ky Fan $ k $-norm function and its conjugate as well as the "sigma term", namely, the conjugate function of the parabolic second-order directional derivative of the Ky Fan $ k $-norm. By establishing the equivalence between the primal (dual) strict Robinson constraint qualification (SRCQ) and the dual (primal) second-order sufficient condition (SOSC), we derive a series of complete characterizations of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) mapping for Ky Fan $ k $-norm regularized convex matrix optimization problems. The obtained results enrich the stability theory of the Ky Fan $ k $-norm regularized convex optimization problems and further enhance the usability of the related algorithms.
- isolated calmness,
- Ky Fan $ k $-norm,
- critical cone,
- second-order sufficient condition,
- strict Robinson constraint qualification
Citation: Ziran Yin, Chongyang Liu, Xiaoyu Chen, Jihong Zhang, Jinlong Yuan. A comprehensive characterization of the robust isolated calmness of Ky Fan $ k $-norm regularized convex matrix optimization problems[J]. AIMS Mathematics, 2025, 10(3): 4955-4969. doi: 10.3934/math.2025227

Related Papers:

Abstract

This paper extends a result of isolated calmness for nuclear norm regularized convex optimization problems to Ky Fan $ k $-norm regularized convex optimization problems. We find that there exists a certain equivalence relationship among the critical cones of the Ky Fan $ k $-norm function and its conjugate as well as the "sigma term", namely, the conjugate function of the parabolic second-order directional derivative of the Ky Fan $ k $-norm. By establishing the equivalence between the primal (dual) strict Robinson constraint qualification (SRCQ) and the dual (primal) second-order sufficient condition (SOSC), we derive a series of complete characterizations of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) mapping for Ky Fan $ k $-norm regularized convex matrix optimization problems. The obtained results enrich the stability theory of the Ky Fan $ k $-norm regularized convex optimization problems and further enhance the usability of the related algorithms.

References

[1]	K. C. Toh, L. N. Trefethen, The Chebyshev polynomials of a matrix, SIAM J. Matrix Anal. A., 20 (1998), 400–419. https://doi.org/10.1137/S0895479896303739 doi: 10.1137/S0895479896303739
[2]	P. Apkarian, D. Noll, Nonsmooth $H_{\infty}$ synthesis, IEEE T. Automat. Contr., 51 (2006), 71–86. https://doi.org/10.1109/TAC.2005.860290 doi: 10.1109/TAC.2005.860290
[3]	V. Bompart, D. Noll, P. Apkarian, Second-order nonsmooth optimization for $H_{\infty}$ synthesis, Numer. Math., 107 (2007), 433–454. https://doi.org/10.1007/s00211-007-0095-9 doi: 10.1007/s00211-007-0095-9
[4]	J. L. Yuan, D. Y. Yang, D. Y. Xun, K. L. Teo, C. Z. Wu, A. Li, et al., Sparse optimal control of cyber-physical systems via PQA approach, Pac. J. Optim., 2024.
[5]	A. Johansson, N. Engsner, C. Strannegård, P. Mostad, Improved spectral norm regularization for neural networks, In: Modeling Decisions for Artificial Intelligence, Cham: Springer, 2023,181–201. https://doi.org/10.1007/978-3-031-33498-6_13
[6]	R. D. Gao, W. K. Yang, X. Sun, Defying forgetting in continual relation extraction via batch spectral norm regularization, In: 2024 International Joint Conference on Neural Networks (IJCNN), 2024, 1–8. https://doi.org/10.1109/IJCNN60899.2024.10651110
[7]	E. J. Candès, B. Recht, Exact matrix completion via convex optimization, Found. Comput. Math., 9 (2009), 717–772. https://doi.org/10.1007/s10208-009-9045-5 doi: 10.1007/s10208-009-9045-5
[8]	B. Recht, M. Fazel, P. A. Parrilo, Guaranteed minimum rank solutions to linear matrix equations via nuclear norm minimization, SIAM Rev., 52 (2010), 471–501. https://doi.org/10.1137/070697835 doi: 10.1137/070697835
[9]	G. A. Watson, On matrix approximation problems with Ky Fan $k$ norms, Numer. Algor., 5 (1993), 263–272. https://doi.org/10.1007/BF02210386 doi: 10.1007/BF02210386
[10]	S. Boyd, P. diaconis, P. A. Parrilo, L. Xiao, Fastest mixing Markov chain on graphs with symmetries, SIAM J. Optim., 20 (2009), 792–819. https://doi.org/10.1137/070689413 doi: 10.1137/070689413
[11]	Z. K. Yao, F. Y. Xu, G. P. Jiang, J. Y. Yao, Data-driven control of hydraulic manipulators by reinforcement learning, IEEE-ASME T. Mech., 29 (2024), 2673–2684. https://doi.org/10.1109/TMECH.2023.3336070 doi: 10.1109/TMECH.2023.3336070
[12]	X. T. Yang, Q. X. Zhu, H. Wang, Exponential stabilization of stochastic systems via novel event-triggered switching controls, IEEE T. Automat. Contr., 69 (2024), 7948–7955. https://doi.org/10.1109/TAC.2024.3406668 doi: 10.1109/TAC.2024.3406668
[13]	Q. X. Zhu, Event-triggered sampling problem for exponential stability of stochastic nonlinear delay systems driven by Lévy processes, IEEE T. Automat. Contr., 70 (2025), 1176–1183. https://doi.org/10.1109/TAC.2024.3448128 doi: 10.1109/TAC.2024.3448128
[14]	C. Ding, Variational analysis of the Ky Fan $k$-norm, Set-Valued Var. Anal., 25 (2017), 265–296. https://doi.org/10.1007/s11228-016-0378-3 doi: 10.1007/s11228-016-0378-3
[15]	C. Ding, An introduction to a class of matrix optimization problems, Singapore: National University of Singapore, 2012.
[16]	Y. L. Liu, S. H. Pan, Locally upper Lipschitz of the perturbed KKT system of Ky Fan k-norm matrix conic optimization problems, 2015. https://doi.org/10.48550/arXiv.1509.00681
[17]	D. R. Han, D. F. Sun, L. W. Zhang, Linear rate convergence of the alternating direction method of multipliers for convex composite programming, Math. Oper. Res., 43 (2018), 622–637. https://doi.org/10.1287/moor.2017.0875 doi: 10.1287/moor.2017.0875
[18]	R. T. Rockafellar, Augmented Lagrangians and applications of the proximal point algorithm in convex programming, Math. Oper. Res., 1 (1976), 97–116. https://doi.org/10.1287/moor.1.2.97 doi: 10.1287/moor.1.2.97
[19]	Y. L. Zhang, L. W. Zhang, On the upper Lipschitz property of the KKT mapping for nonlinear semidefinite optimization, Oper. Res. Lett., 44 (2016), 474–478. https://doi.org/10.1016/j.orl.2016.04.012 doi: 10.1016/j.orl.2016.04.012
[20]	Y. Zhang, L. W. Zhang, J. Wu, K. Wang, Characterizations of local upper Lipschitz property of perturbed solutions to nonlinear second-order cone programs, Optimization, 66 (2017), 1079–1103. https://doi.org/10.1080/02331934.2017.1325887 doi: 10.1080/02331934.2017.1325887
[21]	C. Ding, D. F. Sun, L. W. Zhang, Characterization of the robust isolated calmness for a class of conic programming problems, SIAM J. Optim., 5 (2017), 67–90. https://doi.org/10.1137/16M1058753 doi: 10.1137/16M1058753
[22]	D. R. Han, D. F. Sun, L. W. Zhang, Linear rate convergence of the alternating direction method of multipliers for convex composite quadratic and semi-definite programming, 2015. https://doi.org/10.48550/arXiv.1508.02134
[23]	Y. Cui, D. F. Sun, A complete characterization of the robust isolated calmness of nuclear norm regularized convex optimization problems, J. Comput. Math., 36 (2018), 441–458. https://doi.org/10.4208/jcm.1709-m2017-0034 doi: 10.4208/jcm.1709-m2017-0034
[24]	R. T. Rockafellar, R. J. B. Wets, Variational analysis, Berlin: Heidelberg, 1998. https://doi.org/10.1007/978-3-642-02431-3
[25]	J. F. Bonnans, A. Shapiro, Perturbation analysis of optimization problems, New York: Springer, 2000. https://doi.org/10.1007/978-1-4612-1394-9
[26]	R. T. Rockafellar, Convex analysis, Princeton university press, 1997.
[27]	A. Beck, First-order methods in optimization, Society for Industrial and Applied Mathematics, 2017.
[28]	J. von Neumann, Some matrix-inequalities and metrization of matric-space, 1937.
[29]	K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations I, P. Nat. Acad. Sci. USA, 35 (1949), 652–655. https://doi.org/10.1073/pnas.35.11.652 doi: 10.1073/pnas.35.11.652
[30]	L. W. Zhang, N. Zhang, X. T. Xiao, On the second-order directional derivatives of singular values of matrices and symmetric matrix-valued functions, Set-Valued Var., 21 (2013), 557–586. https://doi.org/557-586.10.1007/s11228-013-0237-4 doi: 10.1007/s11228-013-0237-4
[31]	Z. R. Yin, X. Y. Chen, J. H. Zhang, The robust isolated calmness of spectral norm regularized convex matrix optimization problems, 2024. https://doi.org/10.48550/arXiv.2410.16697

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)