Superlinear Convergence of the Sequential Quadratic Method in Constrained Optimization
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Superlinear Convergence of the Sequential Quadratic Method in Constrained Optimization Ashkan Mohammadi1 · Boris S. Mordukhovich1 · M. Ebrahim Sarabi2 Received: 16 October 2019 / Accepted: 13 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper pursues a twofold goal. Firstly, we aim at deriving novel second-order characterizations of important robust stability properties of perturbed Karush–Kuhn– Tucker systems for a broad class of constrained optimization problems generated by parabolically regular sets. Secondly, the obtained characterizations are applied to establish well-posedness and superlinear convergence of the basic sequential quadratic programming method to solve parabolically regular constrained optimization problems. Keywords Variational analysis · Constrained optimization · KKT systems · Metric subregularity and calmness · Critical and noncritical multipliers · SQP methods · Superlinear convergence Mathematics Subject Classification 90C31 · 65K99 · 49J52 · 49J53
1 Introduction This paper is devoted to the second-order variational analysis and numerical applications for a general class of constrained optimization problems formulated in Sect. 2. It has been well recognized in optimization theory and its applications that second-order
Communicated by Marcin Studniarski.
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Boris S. Mordukhovich [email protected] Ashkan Mohammadi [email protected] M. Ebrahim Sarabi [email protected]
1
Wayne State University, Detroit, MI, USA
2
Miami University, Oxford, OH, USA
123
Journal of Optimization Theory and Applications
analysis concerning both qualitative/theoretical and quantitative/numerical aspects of constrained optimization requires certain second-order regularity conditions. In this paper, we explore a novel one, which is the parabolic regularity of the constraint set at the point in question. Actually the notion of parabolic regularity for extended-real-valued functions was first formulated by Rockafellar and Wets [1, Definition 13.65], but was not studied there except for fully amenable compositions with the immediate application to the unconstrained format of second-order optimality; see also [2, Subsection 3.3.5]. We are not familiar with any other publications where parabolic regularity was either further investigated, or applied to structural problems of constrained optimization. Very recently [3], a systematic study of parabolically regular sets (both convex and nonconvex) has been conducted in our paper, where we reveal a fundamental role of this concept in second-order variational analysis and its important applications to optimization theory. In particular, it is shown in [3] that parabolic regularity is more general than known second-order regularity notions, is preserved under various operations on sets, ensures—among other significant results—twice epi-differentiability of set indicator functions, precise/equality type calculi of second subderivatives and second-order tangents, etc. Furthermore, parabolic regularity married
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