Downsampling is a commonly-used technique in 3D point cloudprocessing for saving storage space, transmission bandwidth andcomputational complexity. The classic downsampling methods,such as farthest point sampling and Poisson disk sampling, thoughwidely used for decades, are independent of the subsequent applications, since the downsampled point cloud may compromisetheir performance severely. This paper explores the problem oftask-oriented downsampling, which aims to downsample a pointcloud and maintain the performance of subsequent applications asmuch as possible. We propose MOPS-Net, a novel end-to-end deepneural network which is designed from the perspective of matrixoptimization, making it fundamentally different from the existing deep learning-based methods. Due to its discrete and combinatorial nature, it is difficult to solve the downsampling problem directly. To tackle this challenge, we relax the binary constraint of each variableand lift 3D points to a higher dimensional feature space, in which a constrained and differentiable matrix optimization problem withan implicit objective function is formulated. We then propose a deep neural network architecture to mimic the matrix optimization problem by exploring both the local and the global structures of the input data. With a task network, MOPS-Net can be end-to-endtrained. Moreover, it is permutation-invariant, making it robust to input data. We also extend MOPS-Net such that a single network after one-time training is capable of handling arbitrary downsam-pling ratios. Extensive experimental results show that MOPS-Net can achieve favorable performance against state-of-the-art deeplearning-based method over various tasks, including point cloud classification, retrieval and reconstruction.