Sharp Primal Superlinear Convergence Results for Some Newtonian Methods for Constrained Optimization - Fernández, D.; Izmailov, A. F.; Solodov, M. V. (SIAM journal on optimization, 2010)
As is well known, Q-superlinear or Q-quadratic convergence of the primal-dual sequence generated by an optimization algorithm does not, in general, imply Q-superlinear convergence of the primal part. Primal convergence, however, is often of particular interest. For the sequential quadratic programming (SQP) algorithm, local primal-dual quadratic convergence can be established under the assumptions of uniqueness of the Lagrange multiplier associated to the solution and the second-order sufficient condition. At the same time, previous primal Q-superlinear convergence results for SQP required strengthening of the first assumption to the linear independence constraint qualification. In this paper, we show that this strengthening of assumptions is actually not necessary. Specifically, we show that once primal-dual convergence is assumed or already established, for primal superlinear rate one needs only a certain error bound estimate. This error bound holds, for example, under the second-order sufficient condition, which is needed for primal-dual local analysis in any case.