We describe the Microbial Community Reconstruction (MCR) Problem, which is fundamental for microbiome analysis. In this problem, the goal is to reconstruct the identity and frequency of species comprising a microbial community, using short sequence reads from Massively Parallel Sequencing (MPS) data obtained for specified genomic regions. We formulate the problem mathematically as a convex optimization problem and provide sufficient conditions for identifiability, namely the ability to reconstruct species identity and frequency correctly when the data size (number of reads) grows to infinity. We discuss different metrics for assessing the quality of the reconstructed solution, including a novel phylogenetically-aware metric based on the Mahalanobis distance, and give upper-bounds on the reconstruction error for a finite number of reads under different metrics. We propose a scalable divide-and-conquer algorithm for the problem using convex optimization, which enables us to handle large problems (with ∼ 106 species). We show using numerical simulations that for realistic scenarios, where the microbial communities are sparse, our algorithm gives solutions with high accuracy, both in terms of obtaining accurate frequency, and in terms of species phylogenetic resolution.