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/k²). Further, unlike any existing first-order methods, PIPG also improves the convergence rate of the constraint violation to O(1/k³). We demonstrate the application of PIPG in constrained optimal control problems.