Scientific Program

Semi-plenary Lecture

Program -> Plenary and Semi-Plenary -> Fri.9:00.H 0104 title only | abstract | bio sketch

Friday, 9:00 - 9:50 h, H 0104

Xiaojun Chen
Nonsmooth, Nonconvex Regularized Optimization for Sparse Approximations

Chair: Ya-xiang Yuan

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 Lp 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 ε-stationary point from any starting point with complexity O(ε-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.



Program -> Plenary and Semi-Plenary -> Fri.9:00.H 0104 title only | abstract | bio sketch

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