Dual Interior-Point Optimization Learning

In many practical applications of constrained optimization, scale and solving time limits make traditional optimization solvers prohibitively slow.  Thus, the research question of how to design optimization proxies -- machine learning models that produce high-quality solutions -- has recently received significant attention. Complementary to this research thread, which has primarily focused on learning primal solutions, this paper studies how to provide learning-based optimality guarantees by learning dual feasible solutions. The paper makes two distinct contributions. First, to train dual optimization proxies, the paper proposes a smoothed self-supervised loss function that augments the traditional self-supervised loss with a dual regularization term. Second, the paper derives closed-form dual completion strategies for several classes of dual regularizations, eliminating the need for computationally-heavy implicit layers. Numerical results are presented on large optimal power flow problems and demonstrate the effectiveness of the proposed approach. The proposed dual completion outperforms methods for learning optimization proxies which do not exploit the structure of the dual problem.  Compared to commercial optimization solvers, the learned dual proxies achieve optimality gaps below 1% and several orders of magnitude speedups.