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This paper mainly concerns with the primal superlinear convergence of thequasi-Newton sequential quadratic programming (SQP) method for piecewiselinear-quadratic composite optimization problems. We show that the latterprimal superlinear convergence can be justified under the noncriticality ofLagrange multipliers and a version of the Dennis-More condition. Furthermore,we show that if we replace the noncriticality condition with the second-ordersufficient condition, this primal superlinear convergence is equivalent with anappropriate version of the Dennis-More condition. We also recover Bonnans'result in [1] for the primal-dual superlinear of the basic SQP method for thisclass of composite problems under the second-order sufficient condition and theuniqueness of Lagrange multipliers. To achieve these goals, we first obtain anextension of the reduction lemma for convex Piecewise linear-quadraticfunctions and then provide a comprehensive analysis of the noncriticality ofLagrange multipliers for composite problems. We also establish certain primalestimates for KKT systems of composite problems, which play a significant rolein our local convergence analysis of the quasi-Newton SQP method.

Primal superlinear convergence of SQP methods in piecewise linear-quadratic composite optimization