DR (Diminishing Return)-submodular functions, which capture diminishing marginal benefits, are widely applicable, especially when incorporating a linear regularization term to effectively address individual costs and sparsity in the objective maximization. This paper is to maximize a regularized DR-submodular function \(F = H + L: [0,1]^n \mapsto \mathbb R\) over a compact, convex and solvable set \(\mathcal Y \ni \textbf{0}\) , where the function H is non-negative, differentiable and DR-submodular, while function L is linear. When H is monotone, we propose an approximation algorithm achieving a bi-factor type approximation ratio of \(\left( 1-\frac{1}{\textrm{e}},1\right) \) . When H is non-monotone, we present two corresponding approximation algorithms for both case that L is non-positive and unrestricted, with approximation ratios as \(\left( \frac{1}{4}, \log 2\right) \) and \(\left( \frac{1}{4}, \frac{1}{2}\right) \) , respectively. Finally, we present numerical experiments that support the approximate guarantee conclusions of this paper and validate the effectiveness of the proposed algorithms compared to existing methods.
