The development of optical metamaterials in recent years has enabled the design of novel optical devices with exciting properties and applications ranging across many fields, including in scientific instrumentation for space missions. This in turn has led to demand for computational methods which can produce efficient device designs. Traditional optical devices admit a closed-form solution for this inverse design problem. However, in the presence of strong multiple scattering, which is often the case when considering optical metamaterials, the inverse problem becomes ill-posed. As a result, many optimization and machine learning techniques have been applied towards discovering good solutions. In this MSc thesis project, several of the most promising of these techniques are applied to a specific problem, the discovery of silicon metamaterial lens designs for the CoPILOT high-altitude balloon project. Ultimately, a software tool capable of producing effective and admissible designs is produced and demonstrated. First, an overview of the CoPILOT design problem is presented. Next, relevant background material topics, including properties of metamaterials and computational methods for simulating them, are covered in some detail. After this, methods used to solve optical design problems in past literature are described and contrasted. Then, a comprehensive explanation of the method developed and used for this project, including important design considerations, is given. The best solutions found using this lens optimization method are shown and compared. Finally, fruitful areas of future work on this topic are listed.