On local convergence of sequential quadratically-constrained quadratic-programming type methods,with an extension to variational problems |
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Authors: | Damián Fernández Mikhail Solodov |
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Affiliation: | (1) Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botanico, Rio de Janeiro, RJ, 22460-320, Brazil |
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Abstract: | We consider the class of quadratically-constrained quadratic-programming methods in the framework extended from optimization to more general variational problems. Previously, in the optimization case, Anitescu (SIAM J. Optim. 12, 949–978, 2002) showed superlinear convergence of the primal sequence under the Mangasarian-Fromovitz constraint qualification and the quadratic growth condition. Quadratic convergence of the primal-dual sequence was established by Fukushima, Luo and Tseng (SIAM J. Optim. 13, 1098–1119, 2003) under the assumption of convexity, the Slater constraint qualification, and a strong second-order sufficient condition. We obtain a new local convergence result, which complements the above (it is neither stronger nor weaker): we prove primal-dual quadratic convergence under the linear independence constraint qualification, strict complementarity, and a second-order sufficiency condition. Additionally, our results apply to variational problems beyond the optimization case. Finally, we provide a necessary and sufficient condition for superlinear convergence of the primal sequence under a Dennis-Moré type condition. Research of the second author is partially supported by CNPq Grants 300734/95-6 and 471780/2003-0, by PRONEX–Optimization, and by FAPERJ. |
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Keywords: | Quadratically constrained quadratic programming Karush-Kuhn-Tucker system Variational inequality Quadratic convergence Superlinear convergence Dennis-Moré condition |
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