Summary: Maximizing non-monotone submodular functions is one of the most important problems in submodular optimization. Let \(\mathbf{B}=(B_1, B_2,\ldots, B_n)\in{\mathbb{Z}}_+^n\) be an integer vector and \([\mathbf{ B}]=\{(x_1,\dots ,x_n) \in{\mathbb{Z}}_+^n: 0\le x_k \le B_k, \forall 1\le k\le n\}\) be the set of all non-negative integer vectors not greater than \(\mathbf{B}\). A function \(f:[\mathbf{ B}] \rightarrow{\mathbb{R}}\) is said to be weak-submodular if \(f(\mathbf{x}+\delta \mathbf{1}_k)-f(\mathbf{x})\ge f(\mathbf{y}+\delta \mathbf{1}_k)-f(\mathbf{y})\) for any \(k\in \{1,\dots ,n\}\), any pair of \(\mathbf{x}, \mathbf{y}\in [\mathbf{ B}]\) such that \(\mathbf{x}\le \mathbf{y}\) and \(x_k =y_k\), and any \(\delta \in{\mathbb{Z}}_+\) satisfying \(\mathbf{y}+\delta \mathbf{1}_k\in [\mathbf{ B}]\). Here \(\mathbf{1}_k\) is the vector with the \(k\) th component equal to 1 and each of the others equals to 0. In this paper we consider the problem of maximizing a non-monotone and non-negative weak-submodular function on the bounded integer lattice without any constraint. We present an randomized algorithm with an approximation guarantee \(\frac{1}{2}\) for the problem.