Computation of second-order directional stationary points for group sparse optimization

ABSTRACT

We consider a nonconvex and nonsmooth group sparse optimization problem where the penalty function is the sum of compositions of a folded concave function and the ${\ell }_2$ vector norm for each group variable. We show that under some mild conditions a first-order directional stationary point is a strict local minimizer that fulfils the first-order growth condition, and a second-order directional stationary point is a strong local minimizer that fulfils the second-order growth condition. In order to compute second-order directional stationary points, we construct a twice continuously differentiable smoothing problem and show that any accumulation point of the sequence of second-order stationary points of the smoothing problem is a second-order directional stationary point of the original problem. We give numerical examples to illustrate how to compute a second-order directional stationary point by the smoothing method.

KEYWORDS:

• Group sparse optimization
• nonconvex and nonsmooth optimization
• composite folded concave penalty
• directional stationary point
• smoothing method

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 90C26
• 90C46

Acknowledgments

The authors would like to thank two referees for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper is partially supported by the Hong Kong Research Grant Council PolyU, PolyU153000/17P, NSFC (11861020), the Growth Project of Education Department of Guizhou Province for Young Talents in Science and Technology ([2018]121), and the Foundation for Selected Excellent Project of Guizhou Province for High-level Talents Back from Overseas ([2018]03)

Notes on contributors

Dingtao Peng

Dingtao Peng received a PhD degree from Beijing Jiaotong University, Beijing, China, in 2013. He is currently a Professor with the School of Mathematics and Statistics, Guizhou University, China. His current research interests include algorithms for nonsmooth and nonconvex optimization, nonlinear analysis, and game theory.

Xiaojun Chen

Xiaojun Chen received a PhD degree from Xi'an Jiaotong University in 1987. She is a Chair Professor with the Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China. Her current research interests include nonsmooth and nonconvex optimization, stochastic variational inequalities, and approximations on spheres.