This work investigates the potential of exploiting movable antennas (MAs) to enhance the performance of a multi-user downlink integrated sensing and communication (ISAC) system. Specifically, we formulate an optimization problem to maximize the transmit beampattern gain for sensing while simultaneously meeting each user's communication requirement by jointly optimizing antenna positions and beamforming design. The problem formulated is highly non-convex and involves multivariate-coupled constraints. To address these challenges, we introduce a series of auxiliary random variables and transform the original problem into an augmented Lagrangian problem. A double-loop algorithm based on a penalty dual decomposition framework is then developed to solve the problem. Numerical results validate the effectiveness of the proposed design, demonstrating its superiority over MA designs based on successive convex approximation optimization and other baseline approaches in ISAC systems. The results also highlight the advantages of MAs in achieving better sensing performance and improved beam control, especially for sparse arrays with large apertures.

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