This paper presents a differentiable optimization framework for the design of constrained Linear Differential Microphone Arrays (LDMAs). The proposed method leverages a non-uniform delay-and-sum beamformer as a light-weight base system model, proving its ability to achieve the optimal beampattern of LDMAs by jointly optimizing microphone positions and filter weights. The formulation enables the optimized design of a filter with a distortion-free constraint in the desired sound direction, while also imposing constraints on microphone positioning to ensure consistent performance. Through evaluation on multiple metrics, including Mean Squared Error (MSE), Directivity Index (DI), White Noise Gain (WNG), and computation time, and comparison with state-of-the-art methods, this approach demonstrates a flexible, directive, robust, and hardware-efficient design.