Movable antenna (MA) systems have attracted growing interest in wireless communications due to their ability to reshape wireless channels via local antenna movement within a confined region. However, optimizing antenna positions to enhance communication performance turns out to be challenging due to the highly nonlinear relationship between wireless channels and antenna positions. Existing approaches, such as gradient-based and heuristic algorithms, often suffer from high computational complexity or undesired local optima. To address the above challenge, this letter proposes a general and low-complexity optimization framework for MA position optimization. Specifically, we discretize the antenna movement region into a set of sampling points, thereby transforming the continuous optimization problem into a discrete point selection problem. Next, we sequentially update the optimal sampling point for each MA over multiple rounds. To avoid convergence to poor local optima, a Gibbs sampling (GS) phase is introduced between rounds to explore adjacent and randomly generated candidate solutions. As a case study, we investigate joint precoding and antenna position optimization for an MA-enhanced broadcast system by applying the proposed framework. Numerical results demonstrate that the proposed algorithm achieves near-optimal performance and significantly outperforms existing benchmarks.