This paper mainly concerns with the primal superlinear convergence of the quasi-Newton sequential quadratic programming (SQP) method for piecewise linear-quadratic composite optimization problems. We show that the latter primal superlinear convergence can be justified under the noncriticality of Lagrange multipliers and a version of the Dennis-Moré condition. Furthermore, we show that if we replace the noncriticality condition with the second-order sufficient condition, this primal superlinear convergence is equivalent with an appropriate version of the Dennis-Moré condition. We also recover Bonnans’ result in (Appl. Math. Optim. 29, 161–186, 1994) for the primal-dual superlinear of the basic SQP method for this class of composite problems under the second-order sufficient condition and the uniqueness of Lagrange multipliers. To achieve these goals, we first obtain an extension of the reduction lemma for convex Piecewise linear-quadratic functions and then provide a comprehensive analysis of the noncriticality of Lagrange multipliers for composite problems. We also establish certain primal estimates for KKT systems of composite problems, which play a significant role in our local convergence analysis of the quasi-Newton SQP method.

本文主要针对分段线性-二次复合优化问题的拟牛顿序贯二次规划（SQP）方法的原始超线性收敛。我们表明，在拉格朗日乘子和Dennis-Moré条件的非临界状态下，可以证明后者的原始超线性收敛是合理的。此外，我们表明，如果将非临界条件替换为二阶充分条件，则该原始超线性收敛与Dennis-Moré条件的适当形式等效。我们还恢复Bonnans'结果（应用数学。的Optim。29（161-186，1994年），这是在二阶充分条件和Lagrange乘子唯一性的基础上，针对此类复合问题的基本SQP方法的原始对偶超线性。为了实现这些目标，我们首先获得凸分段分段线性二次函数的归约引理的扩展，然后对拉格朗日乘子对复合问题的非临界性进行全面分析。我们还为复合问题的KKT系统建立了某些原始估计，这在我们对拟牛顿SQP方法的局部收敛分析中起着重要作用。