The need to solve discrete ill-posed problems arises in many areas of science and engineering. Solutions of these problems, if they exist, are very sensitive to perturbations in the available data. Regularization replaces the original problem by a nearby regularized problem, whose solution is less sensitive to the error in the data. The regularized problem contains a fidelity term and a regularization term. Recently, the use of a p-norm to measure the fidelity term and a q-norm to measure the regularization term has received considerable attention. The balance between these terms is determined by a regularization parameter. In many applications, such as in image restoration, the desired solution is known to live in a convex set, such as the nonnegative orthant. It is natural to require the computed solution of the regularized problem to satisfy the same constraint(s). This paper shows that this procedure induces a regularization method and describes a modulus-based iterative method for computing a constrained approximate solution of a smoothed version of the regularized problem. Convergence of the iterative method is shown, and numerical examples that illustrate the performance of the proposed method are presented. Submitted to: Inverse Problems Modulus-based iterative methods for constrained lp-lq minimization 2

中文翻译：

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约束$\ell_p$-$\ell_q$最小化的基于模数的迭代方法

许多科学和工程领域都需要解决离散的不适定问题。这些问题的解决方案（如果存在）对可用数据中的扰动非常敏感。正则化用附近的正则化问题代替原始问题，其解决方案对数据中的错误不太敏感。正则化问题包含保真度项和正则化项。最近，使用 p 范数来衡量保真度项和使用 q 范数来衡量正则化项受到了相当多的关注。这些项之间的平衡由正则化参数决定。在许多应用中，例如在图像恢复中，已知所需的解存在于凸集中，例如非负 orthant。很自然地要求正则化问题的计算解决方案满足相同的约束。本文表明，该过程引入了一种正则化方法，并描述了一种基于模数的迭代方法，用于计算正则化问题的平滑版本的约束近似解。显示了迭代方法的收敛性，并给出了说明所提出方法性能的数值例子。提交给：逆问题基于模数的迭代方法，用于约束 lp-lq 最小化 2 并给出了说明所提出方法的性能的数值例子。提交给：逆问题基于模数的迭代方法，用于约束 lp-lq 最小化 2 并给出了说明所提出方法的性能的数值例子。提交给：逆问题基于模数的迭代方法，用于约束 lp-lq 最小化 2