L_1极小化问题的一种Gauss-Seidal算法

Gauss-Seidal Algorithm to L_1 Minimization

采用罚函数法与Gauss-Seidal算法相结合的思想研究求解L1极小化问题的数值算法:把L1正则化问题视为对L1极小化问题的一种罚函数,由于该函数是非光滑函数,采用光滑化函数对其进行光滑逼近;在此基础上,对此无约束光滑极小化问题采用Gauss-Seidal迭代法求其某种形式的非精确解;再通过合理调整罚参数和光滑化参数,使得算法产生点列收敛于L1极小化问题的解;最后,通过数值试验测试文中算法的效果,并从数值计算角度与已有算法进行比较,结果表明,文中算法具有很好的数值效果.

The numerical method for solving the L1 minimization problem is studied. The penalty method and the Gauss-Seidal iteration technique are adopted to develop the method. The L1 regularization problem is regarded as a penalized L1 minimizationproblem. Taking into account that the L1 norm function is nonsmooth,a smoothing function is adopted as an approximation to it. Then a smooth unconstrained optimization problem is solved inexactly via Gauss-Seidal iteration. At last,the penalty parameter is adjusted in an appropriate way so that the generated sequence of iterates converges to a solution of the L1 minimization problem. The proposed method is tested by numerical experiments,and its performance is compared with some existing methods. The results show that the proposed method is practically effective.