This paper aims to provide an optimization framework for coded caching that accounts for various heterogeneous aspects of practical systems. An optimization theoretic perspective on the seminal work on the fundamental limits of caching by Maddah Ali and Niesen is first developed, whereas it is proved that the coded caching scheme presented in that work is the optimal scheme among a large, non-trivial family of possible caching schemes. The optimization framework is then used to develop a coded caching scheme capable of handling simultaneous non-uniform file length, non-uniform file popularity, and non-uniform user cache size. Although the resulting full optimization problem scales exponentially with the problem size, this paper shows that tractable simplifications of the problem that scale as a polynomial function of the problem size can still perform well compared to the original problem. By considering these heterogeneities both individually and in conjunction with one another, insights into their interactions and influence on optimal cache content are obtained.