On the Relationship between Bilevel Decomposition Algorithms and Direct Interior-Point Methods

Angel-Victor DeMiguel (avmiguellondon.edu)
Francisco Javier Nogales (FcoJavier.Nogalesuc3m.es)

Abstract: Engineers have been using \emph{bilevel decomposition algorithms} to solve certain nonconvex large-scale optimization problems arising in engineering design projects. These algorithms transform the large-scale problem into a bilevel program with one upper-level problem (the master problem) and several lower-level problems (the subproblems). Unfortunately, there is analytical and numerical evidence that some of these commonly used bilevel decomposition algorithms may fail to converge even when the starting point is very close to the minimizer. In this paper, we establish a relationship between a particular bilevel decomposition algorithm, which only performs one iteration of an interior-point method when solving the subproblems, and a direct interior-point method, which solves the problem in its original (integrated) form. Using this relationship, we formally prove that the bilevel decomposition algorithm converges locally at a superlinear rate. The relevance of our analysis is that it bridges the gap between the incipient local convergence theory of bilevel decomposition algorithms and the mature theory of direct interior-point methods.

Keywords: Bilevel programming, decomposition algorithms, interior-point methods, local convergence

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Applications -- Science and Engineering (Multidisciplinary Design Optimization )

Citation: London Business School working paper 2002

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Entry Submitted: 04/29/2004
Entry Accepted: 04/29/2004
Entry Last Modified: 04/29/2004

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