  Primal-Dual Interior-Point methods are capable of solving constrained convex
optimization problems to tight tolerances in a fast and robust manner. The
derivatives of the primal-dual solution with respect to the problem matrices
can be computed using the implicit function theorem, enabling efficient
differentiation of these optimizers for a fraction of the cost of the total
solution time. In the presence of active inequality constraints, this technique
is only capable of providing discontinuous subgradients that present a
challenge to algorithms that rely on the smoothness of these derivatives. This
paper presents a technique for relaxing primal-dual solutions with a
logarithmic barrier to provide smooth derivatives near active inequality
constraints, with the ability to specify a uniform and consistent amount of
smoothing. We pair this with an efficient primal-dual interior-point algorithm
for solving an always-feasible $\ell_1$-penalized variant of a convex quadratic
program, eliminating the issues surrounding learning potentially infeasible
problems. This parallelizable and smoothly differentiable solver is
demonstrated on a range of robotics tasks where smoothing is important. An open
source implementation in JAX is available at github.com/kevin-tracy/qpax.
