In a digital communication system, information is sent from one place to another over a noisy communication channel. It may be possible to detect and correct errors that occur during the transmission if one encodes the original information by adding redundant bits. Low-density parity-check (LDPC) convolutional codes, a member of the LDPC code family, encode the original information to improve error correction capability. In practice these codes are used to decode very long information sequences, where the information arrives in subsequent packets over time, such as video streams. We consider the problem of decoding the received information with minimum error from an optimization point of view and investigate integer programming-based exact and heuristic decoding algorithms for its solution. In particular, we consider relax-and-fix heuristics that decode information in small windows. Computational results indicate that our approaches identify near-optimal solutions significantly faster than a commercial solver in high channel error rates. Our proposed algorithms can find higher quality solutions compared with commonly used iterative decoding heuristics.