Abstract
Linear Complementarity Problems (LCPs) belong to the class of \({\mathbb{NP}}\) -complete problems. Therefore we cannot expect a polynomial time solution method for LCPs without requiring some special property of the coefficient matrix. Our aim is to construct interior point algorithms which, according to the duality theorem in EP (Existentially Polynomial-time) form, in polynomial time either give a solution of the original problem or detects the lack of property \({\mathcal{P}_*(\tilde\kappa)}\) , with arbitrary large, but apriori fixed \({\tilde\kappa}\)). In the latter case, the algorithms give a polynomial size certificate depending on parameter \({\tilde{\kappa}}\) , the initial interior point and the input size of the LCP). We give the general idea of an EP-modification of interior point algorithms and adapt this modification to long-step path-following interior point algorithms.
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The research of Tibor Illés and Marianna Nagy has been supported by the Hungarian National Research Fund OTKA No. T 049789 and by the Hungarian Science and Technology Foundation TÉT SLO-4/2005. Supported by an NSERC Discovery grant, MITACS and the Canada Research Chair program.
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Illés, T., Nagy, M. & Terlaky, T. A polynomial path-following interior point algorithm for general linear complementarity problems. J Glob Optim 47, 329–342 (2010). https://doi.org/10.1007/s10898-008-9348-0
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DOI: https://doi.org/10.1007/s10898-008-9348-0