Fast SDP solver in C/C++. Currently, there is no freely available SDP solver written in C which would be very useful as it could be used in all the popular numerical programming environments (Matlab/R/Python/Julia).

A new programming language for optimization. Existing optimization frameworks (e.g. CVX) are bolted onto general purpose languages which is not ideal in terms of both syntax and semantics. A new language would enable new functionality as well as avoid single platform lock in.

Interior point methods with CHOLMOD and GPUs.

Implementation of an automatic differentiation library, e.g. to automatically generate a first-order solver from a symbolic representation of the objective function.