Convergence of a stabilized SQP method for equality constrained optimization

Abstract

We herein present a stabilized sequential programming method for equality constrained programming. The proposed method uses the concepts of proximal point methods and primal-dual regularization. A sequence of regularized problems are approximately solved with the regularization parameter approaching zero. At each iteration, a regularized QP subproblem is solved to obtain a primal-dual search direction. Further, a trust-funnel-like line search scheme is used to globalize the algorithm, and a global convergence under the weak assumption of cone-continuity property is shown. To achieve a fast local convergence, a specially designed second-order correction (SOC) technique is adopted near a solution. Under the second-order sufficient condition and some weak conditions (among which no constraint qualification is involved), the regularized QP subproblem transits to a stabilized QP subproblem in the limit. By possibly combining with the SOC step, the full step will be accepted in the limit and hence the superlinearly local convergence is achieved. Preliminary numerical results are reported, which are encouraging.