Minimization over the $$\ell _1$$-ball using an active-set non-monotone projected gradient
提出一种活动集策略嵌入非单调一阶方法，用于高效求解ℓ1球上的最小化问题，证明全局收敛，数值实验显示算法有效。
Abstract The $$\ell _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> -ball is a nicely structured feasible set that is widely used in many fields (e.g., machine learning, statistics and signal analysis) to enforce some sparsity in the model solutions. In this paper, we devise an active-set strategy for efficiently dealing with minimization problems over the $$\ell _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> -ball and embed it into a tailored algorithmic scheme that makes use of a non-monotone first-order approach to explore the given subspace at each iteration. We prove global convergence to stationary points. Finally, we report numerical experiments, on two different classes of instances, showing the effectiveness of the algorithm.