Conference Program

PLENARY AND SEMI-PLENARY TALKS

 (Add to Calendar)  Friday, August 24, 09:00 – 09:50 h, H0104:


Xiaojun Chen: Nonsmooth, Nonconvex Regularized Optimization for Sparse Approximations

Chair: Ya-xiang Yuan


Abstract:

Minimization problems with nonsmooth, nonconvex, perhaps even non-Lipschitz regularization terms have wide applications in image restoration, signal reconstruction and variable selection, but they seem to lack optimization theory. On L_p non-Lipschitz regularized minimization, we show that finding a global optimal solution is strongly NP-hard. On the other hand, we present lower bounds of nonzero entries in every local optimal solution without assumptions on the data matrix. Such lower bounds can be used to classify zero and nonzero entries in local optimal solutions and select regularization parameters for desirable sparsity of solutions. Moreover, we show smoothing methods are efficient for solving such regularized minimization problems. In particular, we introduce a smoothing SQP method which can find an affine scaled epsilon-stationary point from any starting point with complexity O(epsilon-2), and a smoothing trust region Newton method which can find a point satisfying the affine scaled second order necessary condition from any starting point. Examples with six widely used nonsmooth nonconvex regularization terms are presented to illustrate the theory and algorithms. Joint work with W. Bian, D. Ge, L. Niu, Z. Wang, Y. Ye, Y. Yuan.

 

Biographical sketch:

Professor Xiaojun Chen received her PhD degree in Computational Mathematics from Xi'an Jiaotong University, China in 1987 and PhD degree in Applied Mathematics from Okayama University of Science, Japan in 1991. She was a postdoctoral fellow at the University of Delaware, an Australia Research Fellow in the University of New South Wales and a Professor in Hirosaki University, Japan. She joined the Hong Kong Polytechnic University as a Professor in 2007. Her current research interests include nonsmooth nonconvex optimization, stochastic equilibrium problems and numerical approximation methods on the sphere with important applications in engineering and economics. She has published over 80 papers in major international journals in operations research and computational mathematics. Prof. Chen has won many grants as a principal investigator from several government funding agencies and organized several important international conferences. She serves in the editorial boards of five mathematical journals including SIAM Journal on Numerical Analysis.

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