Deep neural networks are often computationally expensive, during both the training stage and inference stage. Training is always expensive, because back-propagation requires high-precision floating-point multiplication and addition. However, various mathematical optimizations may be employed to reduce the computational cost of inference. Optimized inference is important for reducing power consumption and latency and for increasing throughput. This chapter introduces the central approaches for optimizing deep neural network inference: pruning “unnecessary” weights, quantizing weights and inputs, sharing weights between layer units, compressing weights before transferring from main memory, distilling large high-performance models into smaller models, and decomposing convolutional filters to reduce multiply and accumulate operations. In this chapter, using a unified notation, we provide a mathematical and algorithmic description of the aforementioned deep neural network inference optimization methods.