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Bilevel derivative-free optimization and its application to robust optimization
Conn, Andrew R.; Vicente, L. N.We address bilevel programming problems when the derivatives of both the upper and the lower level objective functions are unavailable. The core algorithms used for both levels are trust-region interpolation-based methods, using minimum Frobenius norm quadratic models when the number of points is smaller than the number of basis components. We take advantage of the problem structure to derive conditions (relate...
Global convergence of general derivative-free trust-region algorithms to first ...
Conn, Andrew R.; Scheinberg, Katya; Vicente, Luís NunesIn this paper we prove global convergence for first and second-order stationarity points of a class of derivative-free trust-region methods for unconstrained optimization. These methods are based on the sequential minimization of linear or quadratic models built from evaluating the objective function at sample sets. The derivative-free models are required to satisfy Taylor-type bounds but, apart from that, the ...
Geometry of sample sets in derivative free optimization. Part II: polynomial re...
Conn, Andrew R.; Scheinberg, Katya; Vicente, Luís NunesIn the recent years, there has been a considerable amount of work in the development of numerical methods for derivative free optimization problems. Some of this work relies on the management of the geometry of sets of sampling points for function evaluation and model building. In this paper, we continue the work developed in [7] for complete or determined interpolation models (when the number of interpolation ...
Error estimates and poisedness in multivariate polynomial interpolation
Conn, Andrew R.; Scheinberg, Katya; Vicente, Luís NunesWe show how to derive error estimates between a function and its interpolating polynomial and between their corresponding derivatives. The derivation is based on a new de nition of well-poisedness for the interpolation set, directly connecting the accuracy of the error estimates with the geometry of the points in the set. This de nition is equivalent to the boundedness of Lagrange polynomials, but it provides n...
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