唐琦 彭定濤

摘 要：本文考慮無(wú)約束組稀疏回歸問(wèn)題，其損失函數(shù)為凸函數(shù)，正則項(xiàng)為MCP（minimax concave? penalty），主要刻畫(huà)該問(wèn)題的兩類(lèi)穩(wěn)定點(diǎn)。首先，給出d-穩(wěn)定點(diǎn)以及critical點(diǎn)的具體刻畫(huà)，并且證明了這兩類(lèi)穩(wěn)定點(diǎn)的關(guān)系;其次，分析d-穩(wěn)定點(diǎn)與問(wèn)題局部解的關(guān)系;最后，證明了該模型的下界性質(zhì)。

關(guān)鍵詞：組稀疏問(wèn)題;MCP正則;d-穩(wěn)定點(diǎn);critical點(diǎn);下界性質(zhì)

中圖分類(lèi)號(hào)：O224?? 文獻(xiàn)標(biāo)識(shí)碼： A

參考文獻(xiàn)：

[1]YUAN M，? LIN Y.? Model selection and estimation in regression with groupedvariables[J].? Journal of the Royal Statistical Society，? 2006，? 68（1）： 49-67.

[2]HU Y H，? LI C，? MENG K W， et al.? Group sparse optimization via p， q regularization[J].? Journal of Machine Learning Research，? 2017，? 18（1）： 960-1011.

[3]JIAO Y L，? JIN B T，? LU X L.? Groupsparse recovery via the 02 penalty： theory and algorithm[J].? IEEE Transactions on Signal Processing，? 2016，? 65（4）： 998-1012.

[4]FAN J Q，? LI R Z.? Variable selection via nonconcave penalized likelihood and its oracle properties[J].? Journal of the American Statistical Association，? 2001，? 96（456）： 1348-1360.

[5]GRAMFORT A，? KOWALSKI M.? Improving M/EEG source localization with an inter-condition sparse prior[J].? IEEE International Symposium on Biomedical Imaging（ISBI），? 2009： 141-144.

[6]HUANG J Z，? ZHANG T.? The benefit of groupsparsity[J].? Annals of Statistics，? 2010，? 38（4）： 1978-2004.

[7]BIAN W，? CHEN X J.? Optimality and complexity for constrained optimization problems withnonconvex regularization[J].? Mathematics of Operations Research，? 2017，? 42（4）： 1063-1084.

[8]HUANG J，? BREHENY P，? MA S G.? A selective review of group selection inhigh-dimensional models[J].? Statistical Science，? 2012，? 27（4）： 481-499.

[9]LIU H C，? YAO T，? LI R Z，? et al.? Folded concave penalized sparse linear regression： sparsity，? statistical performance，? and algorithmic theory for local solution[J].? Mathematical? Programming，? 2017，? 166（1-2）： 207-240.

[10]AHN M，? PANG J S，? XIN J.? Difference-of-convex learning： directional stationarity，? optimality，? and sparsity[J].? SIAM Journal on optimization，? 2017， 27（3）：1637-1665.

[11]PENG D T，? CHEN X J.? Computation of second-order directional stationary points for group sparse optimization[J].? Optimization Methods and Software，? 2020， 35（2）：1978-2004.

[12]ROCKAFELLAR R T，? WETS R J B. Variational Analysis[M].? Berlin：Springer，? 1998.

[13]ROCKAFELLAR R T.? Convexanalysis[M].? Princeton：Princeton University Press，? 1970.

[14]CLARKE F H.? Optimization andnonsmooth analysis[M].? New York： SIAM，?? 1990.

[15]BIAN W，? CHEN X J.? A smoothing proximal gradient algorithmfor nonsmooth convex regression with cardinality penalty[J].? SIAM Journal on Numerical Analysis，? 2020，? 58（1）： 858-883.

[16]GONG P H，? ZHANG C S，? LU Z S，? et al.? A general iterative shrinkage andthresholding algorithm for nonconvex regularized optimization problems[J].? Proceedings of the 30th International Conference on Machine Learning，? 2013，? 28： 37-45.

[17]AN L T H，? TAO P D，? MINH L H，? et al.? DC approximation approaches for sparse optimization[J].? European Journal of Operational Research，? 2015，? 244： 26-46.

（責(zé)任編輯：曾 晶）

Analysis of Stationary Points for Group Sparse Problems

with the Minimax Concave Penalty

TANG Qi，? PENG Dingtao*

（School of Mathematics and Statistics， Guizhou University，? Guiyang 550025，? China）

Abstract：

In this paper，? we focus on the group sparse problem，? where the loss function is convex，? and the penalty term is defined by the minimax concave penalty.? We discuss two kinds of stationary points of the problem.? First，? we provide concrete description for the d-stationary point and the critical point of the nonconvex regular group sparse problem，? and analyze the relation of d-stationary point with critical point.? Furthermore，? we show that a point is a local minimizer of the relaxation problem，? then it is a d-stationary point.? Whats more，? we obtain the lower bound property of the problem.

Key words：

group sparse problem;MCP;d-stationary point;critical point;lower bound property

收稿日期：2020-01-08

基金項(xiàng)目：國(guó)家自然科學(xué)基金資助項(xiàng)目（11861020）;貴州省高層次留學(xué)人才創(chuàng)新創(chuàng)業(yè)擇優(yōu)資助重點(diǎn)項(xiàng)目（[2018]03）;貴州省科技計(jì)劃資助項(xiàng)目（[2018]5781）;貴州省青年科技人才成長(zhǎng)資助項(xiàng)目（[2018]121）

作者簡(jiǎn)介：唐 琦（1996-），女，在讀碩士，研究方向：稀疏優(yōu)化，Email： qqtang77@163.com.

通訊作者：彭定濤，Email：dingtaopeng@126.com.