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Mathematics Department

Operations Research and Statistics Seminar

Fall 2021

  • Oct
  • Integer programming for kidney exchange
    Sommer Gentry
    Location: CH 320 and
    Time: 12:00 PM

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    Kidney paired donation matches one patient and his or her incompatible donor with another pair in the same situation for an exchange. We represent patient-donor pairs as vertices of a directed graph G, with edges connecting pairs if the donor of the source is compatible with the recipient of the sink, and maximize the sum of edge weights on disjoint cycles. Because a maximum edge-weight matching might not have the maximum cardinality; there is a risk of an unpredictable trade-off between quality and quantity of paired donations. The number of paired donations is within a multiplicative factor of the maximum possible donations, where the factor depends on the edge weighting. We design an edge weighting of G which guarantees that every matching with maximum weight also has maximum cardinality, and also maximizes the number of transplants for an exceptional subset of recipients, while favoring immunologic concordance. We will also survey various exponential-sized and polynomial-sized integer programming formulations proposed for this problem.
  • Sep
  • Predictor-Corrector Interior-Point Algorithms for Sufficient Linear Complementarity Problems
    Tibor Illes
    Corvinus Centre for Operations Research at Corvinus University, Budapest, Hungary
    Time: 12:00 PM

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    The algebraic equivalent transformation (AET) of the system which defines the central path has been introduced by Darvay (2003) for linear programming problems resulting in new search directions for interior point algorithms (IPA). Generalization of AET for some types of IPAs for sufficient linear complementarity problems (SU-LCP) can cause difficulties, especially for predictor-corrector (PC) ones. After overcoming these difficulties, we introduce new PC-IPAs for SU-LCPs. Moreover, we present a unified discussion of the effect of different AETs on proposing and analysing new IPAs for SU-LCPs. These new PC IPAs from complexity analysis point of view possess similar properties like IPAs with the best known complexity, namely have polynomial iteration complexity in κ, the dimension n and the bit length L of the problem. There are several known results (Fukuda and Terlaky, 1992; Hertog et al. 1993; Valiaho, 1996; Fukuda et al. 1998; P. Tseng 2000; de Klerk and Eisenberg-Nagy 2011) that discuss properties of the sufficient matrices and explain why it is difficult to generate, recognize (decide) and use these matrices. Furthermore, some algorithms (Csizmadia and Illés, 2006; Illés et al. 2000, 2007, 2009, 2010; Csizmadia et al. 2013, etc.) have been developed that do not need a priori information about the matrix properties. These algorithms could be applied either to solve the LCPs or its dual efficiently or give a polynomial size certificate that the matrix is not sufficient. Recently, Illés and Morapitiye (2018), and Eisaenber-Nagy (2020) have introduced, some new ways of generating sufficient matrices making possible, for the first time, to test the computational performance of IPAs on sufficient LCPs, with positive κ parameter. Our new PC IPAs have been tested on a well-known, triangular, P-matrix designed by Zs. Csizmadia (2011) with κ = 22n-8 – 0.25, for n ≥4. (The exact κ has been computed by M. Eisenberg-Nagy, 2019.) Computational performance of a variant of our new PC IPA has not been affected by this fact, namely the number of iterations for this interior point algorithm is not exponential in the dimension n. Some preliminary computational studies related to sufficient LCPs will be presented, as well.
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