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 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.
Acknowledgments
The authors would like to thank two referees for their helpful comments.
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No potential conflict of interest was reported by the authors.
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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.