The smoothed duality gap as a stopping criterion

We optimize the running time of the primal-dual algorithms by optimizing their stopping criteria for solving convex optimization problems under affine equality constraints, which means terminating the algorithm earlier with fewer iterations. We study the relations between four stopping criteria and show under which conditions they are accurate to detect optimal solutions. The uncomputable one: "Optimality gap and Feasibility error", and the computable ones: the "Karush–Kuhn–Tucker error", the "Projected Duality Gap", and the "Smoothed Duality Gap". Assuming metric sub-regularity or quadratic error bound, we establish that all of the computable criteria provide practical upper bounds for the optimality gap, and approximate it effectively. Furthermore, we establish comparability between some of the computable criteria under certain conditions. Numerical experiments on basis pursuit, and quadratic programs with(out) non-negative weights corroborate these findings and show that the smoothed duality gap is more widely applicable than the rest.