Browsing by Author "Fontecilla, Rodrigo"

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    A Convergence Theory for a Class of Quasi-Newton Methods for Constrained Optimization
    (1983-05) Fontecilla, Rodrigo; Steihaug, Trond; Tapia, Richard A.
    In this paper we develop a general convergence theory for a class of quasi-Newton methods for equality constrained optimization. The theory is set in the framework of the diagonalized multiplier method defined by Tapia and is an extension of the theory developed by Glad. We believe that this framework is flexible and amenable to convergence analysis and generalizations. A key ingredient of a method in this class is a multiplier update. Our theory is tested by showing that a straightforward application gives the best known convergence results for several known multiplier updates. Also a characterization of q-superlinear convergence is presented. It is shown that in the special case when the diagonalized multiplier method is equivalent to the successive quadratic programming approach, our general characterization result gives the Boggs, Tolle and Wang characterization.
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    The Lack of Positive Definiteness in the Hessian in Constrained Optimization
    (1983-06) Fontecilla, Rodrigo
    The use of the DFP or the BFGS secant updates requires the Hessian at the solution to be positive definite. The second order sufficiency conditions insure the positive definiteness only in a subspace of R^n. Conditions are given so we can safely update with either update. A new class of algorithms is proposed which generate a sequence {xk} converging 2-step q-superlinearly. We also propose two specific algorithms: One converges q-superlinearly if the Hessian is positive definite in R^n and converges 2-step q-superlinearly if the Hessian is positive definite only in a subspace; the second one generates a sequence converging 1-step q-superlinearly. While the former costs one extra gradient evaluation, the latter costs one extra gradient evaluation and one extra function evaluation on the constraints.