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On the relationship between bilevel decomposition algorithms and direct interior-point methods

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  • Miguel, Angel Víctor de
  • Nogales Martín, Francisco Javier

Abstract

Engineers have been using 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 upperlevel 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.

Suggested Citation

  • Miguel, Angel Víctor de & Nogales Martín, Francisco Javier, 2004. "On the relationship between bilevel decomposition algorithms and direct interior-point methods," DES - Working Papers. Statistics and Econometrics. WS ws042509, Universidad Carlos III de Madrid. Departamento de Estadística.
    • Handle: RePEc:cte:wsrepe:ws042509
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    References listed on IDEAS

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    1. Arjan Berkelaar & Cees Dert & Bart Oldenkamp & Shuzhong Zhang, 2002. "A Primal-Dual Decomposition-Based Interior Point Approach to Two-Stage Stochastic Linear Programming," Operations Research, INFORMS, vol. 50(5), pages 904-915, October.
    2. Blomvall, Jorgen & Lindberg, Per Olov, 2002. "A Riccati-based primal interior point solver for multistage stochastic programming," European Journal of Operational Research, Elsevier, vol. 143(2), pages 452-461, December.
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    Cited by:

    1. Adejuyigbe O. Fajemisin & Laura Climent & Steven D. Prestwich, 2021. "An analytics-based heuristic decomposition of a bilevel multiple-follower cutting stock problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 43(3), pages 665-692, September.

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