Homogeneous Self-Dual Embedding via Perspective Functions

We present a generalization of the well-known homogeneous self-dual embedding model, which is widely used in conic optimization. The new embedding applies to a problem of minimizing the sum of two proper lower-semicontinuous convex functions and can be represented as a single inequality that uses perspectives of these functions and of their conjugates. A solution to the proposed embedding encodes a primal-dual solution to the original problem when available, or an infeasibility certificate otherwise. We then use the Douglas-Rachford algorithm to find a solution to the embedding and discuss its efficient implementation by exploiting the problem structure. The resulting algorithm recovers an existing method for solving quadratic cone programs as a special case. We demonstrate the generality and effectiveness of the algorithm on a class of convex optimization problems with non-smooth objective function and non-conic constraints.