We provide a mathematical formulation and develop a computational framework for identifying multiple strains of microorganisms from mixed samples of DNA. Our method is applicable in public health domains where efficient identification of pathogens is paramount, e.g. for the monitoring of disease outbreaks. We formulate strain identification as an inverse problem that aims at simultaneously estimating a binary matrix (encoding presence or absence of mutations in each strain) and a real-valued vector (representing the mixture of strains) such that their product is approximately equal to the measured data vector. The problem at hand has a similar structure to blind deconvolution, except for the presence of binary constraints, which we enforce in our approach. Following a Bayesian approach, we derive a posterior density. We present two computational methods for solving the non-convex maximum a posteriori estimation problem. The first one is a local optimization method that is made efficient and scalable by decoupling the problem into smaller independent subproblems, whereas the second one yields a global minimizer by converting the problem into a convex mixed-integer quadratic programming problem. The decoupling approach also provides an efficient way to integrate over the posterior. This provides useful information about the ambiguity of the underdetermined problem and, thus, the uncertainty associated with numerical solutions. We evaluate the potential and limitations of our framework in silico using synthetic and experimental data with available ground truths.