Proportional-Integral Projected Gradient Method for Conic Optimization

Conic optimization is the minimization of a differentiable convex objective function subject to conic constraints. We propose a novel primal-dual first-order method for conic optimization, named proportional-integral projected gradient method (PIPG). PIPG ensures that both the primal-dual gap and the constraint violation converge to zero at the rate of $O(1/k)$, where $k$ is the number of iterations. If the objective function is strongly convex, PIPG improves the convergence rate of the primal-dual gap to $O(1/k^2)$. Further, unlike any existing first-order methods, PIPG also improves the convergence rate of the constraint violation to $O(1/k^3)$. We demonstrate the application of PIPG in constrained optimal control problems.