等式与不等式约束下组稀疏优化问题的最优性条件研究

数学物理学报 ›› 2026, Vol. 46 ›› Issue (5): 2040-2054.

等式与不等式约束下组稀疏优化问题的最优性条件研究

董烨霏(), 高彩霞^*()

内蒙古大学数学科学学院呼和浩特 010030

收稿日期:2025-05-13 修回日期:2025-09-30 出版日期:2026-10-26 发布日期:2026-09-07
通讯作者: 高彩霞 E-mail:dyffz@foxmail.com;gaocx0471@163.com
作者简介:董烨霏, E-mail: dyffz@foxmail.com
基金资助:
国家自然科学基金(12461057)

Research on Optimality Conditions for Group Sparse Optimization Problems with Equality and Inequality Constraints

Yefei Dong(), Caixia Gao^*()

School of Mathematical Sciences, Inner Mongolia University, Huhhot 010030

Received:2025-05-13 Revised:2025-09-30 Online:2026-10-26 Published:2026-09-07
Contact: Caixia Gao E-mail:dyffz@foxmail.com;gaocx0471@163.com
Supported by:
NSFC(12461057)

摘要/Abstract

摘要：

该文基于组稀疏支撑投影, 研究等式和不等式约束下组稀疏优化问题 (NL-GSCO) 的最优性理论. 该文建立了强 $\gamma$-{Lagrange} 稳定点, 刻画其一阶最优性条件, 探讨该稳定点与 $\alpha$-稳定点、F-KKT 点之间的等价性及递进关系. 但组支撑投影的不可微性导致问题求解困难, 又考虑到不等式约束互补松弛条件的计算负担, 因此研究等式约束下组稀疏优化问题 (EC-GSCO) 的可微形式的拉格朗日方程, 在稳定点局部邻域内证明了其雅可比矩阵的非奇异性. 最后, 改进单稀疏的梯度投影牛顿追踪算法(GPNP) 为组稀疏 GPNP 算法, 在一定条件下收敛于强 $\gamma$-Lagrange 稳定点. 针对 EC-GSCO 问题进行数值实验, 具有优良的计算性能.

关键词: 组稀疏约束优化, 最优性条件, 稳定点, 拉格朗日方程, 组稀疏 GPNP 算法

Abstract:

The study investigates the optimality theory for group-sparse optimization problems with equality and inequality constraints (NL-GSCO) based on group-sparse support projections. It establishes the concept of strong $\gamma$-Lagrange stationary points, characterizes their first-order optimality conditions, and explores the equivalence and progressive relationships between such points and $\alpha$-stationary points as well as F-KKT points. However, the non-differentiability of the group-sparse projection poses challenges for solving the problem. Additionally, considering the computational burden induced by the complementary slackness conditions of inequality constraints, the study focuses on the differentiable Lagrangian equation form of the equality-constrained group-sparse optimization problem (EC-GSCO). Within a local neighborhood of a stationary point, the nonsingularity of the corresponding Jacobian matrix is proven. Finally, the existing gradient projection Newton pursuit (GPNP) algorithm designed for single sparsity is extended to a group-sparse GPNP algorithm, which is shown to converge to a strong $\gamma$-Lagrange stationary point under certain conditions. Numerical experiments conducted on group-sparse nonlinear optimization problems demonstrate its excellent computational performance.

Key words: group-sparse constrained optimization, optimality conditions, stationary points, lagrangian equations, group-sparse GPNP Algorithm

中图分类号:

O224

董烨霏, 高彩霞. 等式与不等式约束下组稀疏优化问题的最优性条件研究[J]. 数学物理学报, 2026, 46(5): 2040-2054.

Yefei Dong, Caixia Gao. Research on Optimality Conditions for Group Sparse Optimization Problems with Equality and Inequality Constraints[J]. Acta mathematica scientia,Series A, 2026, 46(5): 2040-2054.

图/表 2

参考文献 25

[1]	Candès E J, Tao T. Decoding by linear programming. IEEE Transactions on Information Theory, 2005, 51(12): 4203-4215 doi: 10.1109/TIT.2005.858979
[2]	Donoho D L. Compressed sensing. IEEE Transactions on Information Theory, 2006, 52(4): 1289-1306 doi: 10.1109/TIT.2006.871582
[3]	Candès E J, Romberg J, Tao T. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Transactions on Information Theory, 2006, 52(2): 489-509 doi: 10.1109/TIT.2005.862083
[4]	Scardapane S, Comminiello D, Hussain A, et al. Group sparse regularization for deep neural networks. Neurocomputing, 2017, 241: 81-89 doi: 10.1016/j.neucom.2017.02.029
[5]	Shi Y, Tang A, Niu L, et al. Sparse optimization guided pruning for neural networks. Neurocomputing, 2024, 574: Art 127280 doi: 10.1016/j.neucom.2024.127280
[6]	Qin J, Hu Y, Yao J C, et al. Cell fate conversion prediction by group sparse optimization method utilizing single-cell and bulk OMICs data. Briefings in Bioinformatics, 2021, 22(6): bbab311
[7]	Yuan M, Lin Y. Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society Series B: Statistical Methodology, 2006, 68(1): 49-67 doi: 10.1111/j.1467-9868.2005.00532.x
[8]	Jenatton R, Audibert J Y, Bach F. Structured variable selection with sparsity-inducing norms. The Journal of Machine Learning Research, 2011, 12: 2777-2824
[9]	Beck A, Eldar Y C. Sparsity constrained nonlinear optimization: Optimality conditions and algorithms. SIAM Journal on Optimization, 2013, 23(3): 1480-1509 doi: 10.1137/120869778
[10]	Pan L L, Xiu N H, Zhou S L. On solutions of sparsity constrained optimization. Journal of the Operations Research Society of China, 2015, 3: 421-439 doi: 10.1007/s40305-015-0101-3
[11]	Lämmel S, Shikhman V. On nondegenerate $m$-stationary points for sparsity constrained nonlinear optimization. Journal of Global Optimization, 2022, 82(2): 219-242 doi: 10.1007/s10898-021-01070-7
[12]	Beck A, Hallak N. On the minimization over sparse symmetric sets: projections, optimality conditions, and algorithms. Mathematics of Operations Research, 2016, 41(1): 196-223 doi: 10.1287/moor.2015.0722
[13]	Lu Z S. Optimization over sparse symmetric sets via a nonmonotone projected gradient method. arXiv:1509.08581, 2015
[14]	Chen X J, Pan L L, Xiu N H. Solution sets of three sparse optimization problems for multivariate regression. Journal of Global Optimization, 2023, 87(2): 347-371 doi: 10.1007/s10898-021-01124-w
[15]	Lu Z S, Zhang Y. Sparse approximation via penalty decomposition methods. SIAM Journal on Optimization, 2013, 23(4): 2448-2478 doi: 10.1137/100808071
[16]	Pan L L, Xiu N H, Fan J. Optimality conditions for sparse nonlinear programming. Science China Mathematics, 2017, 60: 759-776 doi: 10.1007/s11425-016-9010-x
[17]	Zhao C, Xiu N H, Qi H D, Luo Z Y. A lagrange-newton algorithm for sparse nonlinear programming. Mathematical Programming, 2022, 195(1): 903-928 doi: 10.1007/s10107-021-01719-x
[18]	Beck A, Hallak N. Optimization problems involving group sparsity terms. Mathematical Programming, 2019, 178(1): 39-67 doi: 10.1007/s10107-018-1277-1
[19]	Wu W Y, Peng D T. Optimality conditions for group sparse constrained optimization problems. Mathematics, 2021, 9(1): Art 84 doi: 10.3390/math9010084
[20]	胡珊珊, 贺素香. 非负组稀疏约束优化问题的最优性条件, 数学物理学报, 2024, 44A(2): 500-512
	Hu S S, He S X. Optimality conditions for non-negative group sparse constrained optimization problems. Acta Math Sci, 2024, 44A(2): 500-512
[21]	Yi S Y, Peng D T, and Zhang X. Optimality conditions for group sparsity constrained optimization problems with equality and inequality constraints. Optimization, 2025, 1-32
[22]	Červinka M, Kanzow C, Schwartz A. Constraint qualifications and optimality conditions for optimization problems with cardinality constraints. Mathematical Programming, 2016, 160: 353-377 doi: 10.1007/s10107-016-0986-6
[23]	Pan L L, Luo Z Y, Xiu N H. Restricted robinson constraint qualification and optimality for cardinality-constrained cone programming. Journal of Optimization Theory and Applications, 2017, 175: 104-118 doi: 10.1007/s10957-017-1166-4
[24]	Rockafellar R T, Wets R J. Variational Analysis. Berlin: Springer, 2009
[25]	Zhou S L. Gradient projection Newton pursuit for sparsity constrained optimization. Applied and Computational Harmonic Analysis, 2022, 61: 75-100 doi: 10.1016/j.acha.2022.06.002