Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variational problems

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve fast convergence despite possible degeneracy of constraints of optimization problems, when the Lagrange multipliers associated to a solution are not unique. Superlinear convergence of sSQP had been previously established under the strong second-order sufficient condition for optimality (without any constraint qualification assumptions). We prove a stronger superlinear convergence result than the above, assuming the usual second-order sufficient condition only. In addition, our analysis is carried out in the more general setting of variational problems, for which we introduce a natural extension of sSQP techniques. In the process, we also obtain a new error bound for Karush–Kuhn–Tucker systems for variational problems that holds under an appropriate second-order condition.