In this paper, we consider a class of minimization problems with the objective functions having a form of summation of a penalized differentiable convex function, and a weighted $\ell _1$-norm. However, different from the common assumption of positive weights in existing studies, we shall address a general case where the weights can be either positive or negative, motivated by the fact that negative weights are also capable of inducing sparsity, and even achieving outstanding performance. To deal with the resulting problem, a generalized fixed-point continuation (GFPC) method is introduced, and an accelerated variant is developed. More importantly, the convergence of this algorithm is analyzed in detail, and its application to compressing sensing problems that employ the Shannon entropy function (SEF) for sparsity promotion is also studied. Numerical examples are carried out to demonstrate the effectiveness of the GFPC algorithm.

中文翻译：

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广义定点延拓法：收敛与应用

在本文中，我们考虑一类具有惩罚可微凸函数求和形式的目标函数和加权$\ell_1$范数的最小化问题。然而，与现有研究中常见的正权重假设不同，我们将解决权重可以为正或负的一般情况，其动机是负权重也能够引起稀疏性，甚至实现出色的性能。为了解决由此产生的问题，引入了广义定点延续 (GFPC) 方法，并开发了加速变体。更重要的是，详细分析了该算法的收敛性，并研究了其在利用香农熵函数（SEF）进行稀疏提升的压缩感知问题中的应用。