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Primal Superlinear Convergence of Sqp Methods in Piecewise Linear-Quadratic Composite Optimization

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Abstract

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.

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Acknowledgements

We thank the two anonymous reviewers whose comments and suggestions helped improve the original presentation of the paper.

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  1. Department of Mathematics, Miami University, Oxford, OH, 45065, USA

    M. Ebrahim Sarabi

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Correspondence to M. Ebrahim Sarabi.

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Sarabi, M.E. Primal Superlinear Convergence of Sqp Methods in Piecewise Linear-Quadratic Composite Optimization. Set-Valued Var. Anal 30, 1–37 (2022). https://doi.org/10.1007/s11228-021-00580-6

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