On local convergence of sequential quadratically-constrained quadratic-programming type methods, with an extension to variational problems
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.
KeywordsQuadratically constrained quadratic programming Karush-Kuhn-Tucker system Variational inequality Quadratic convergence Superlinear convergence Dennis-Moré condition
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- 7.Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003) Google Scholar
- 10.Josephy, N.H.: Newton’s method for generalized equations. Technical Summary Report 1965, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin (1979) Google Scholar
- 11.Kruk, S., Wolkowicz, H.: Sequential, quadratically constrained, quadratic programming for general nonlinear programming. In: Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.) Handbook of Semidefinite Programming, pp. 563–575. Kluwer Academic, Dordrecht (2000) Google Scholar
- 13.Maratos, N.: Exact penalty function algorithms for finite dimensional and control optimization problems. Ph.D. thesis, Imperial College, University of London (1978) Google Scholar
- 15.Nesterov, Y.E., Nemirovskii, A.S.: Interior Point Polynomial Methods in Convex Programming: Theory and Applications. SIAM, Philadelphia (1993) Google Scholar
- 18.Powell, M.J.D.: Variable metric methods for constrained optimization. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming: The State of Art, pp. 288–311. Springer, Berlin (1983) Google Scholar