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Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods

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Abstract

A family of complementarity problems is defined as extensions of the well-known linear complementarity problem (LCP). These are:

  1. (i)

    second linear complementarity problem (SLCP), which is an LCP extended by introducing further equality restrictions and unrestricted variables;

  2. (ii)

    minimum linear complementarity problem (MLCP), which is an LCP with additional variables not required to be complementary and with a linear objective function which is to be minimized;

  3. (iii)

    second minimum linear complementarity problem (SMLCP), which is an MLCP, but the nonnegative restriction on one of each pair of complementary variables is relaxed so that it is allowed to be unrestricted in value.

A number of well-known mathematical programming problems [namely, quadratic programming (convex, nonconvex, pseudoconvex, nonconvex), linear variational inequalities, bilinear programming, game theory, zero-one integer programming, fixed-charge problem, absolute value programming, variable separable programming] are reformulated as members of this family of four complementarity problems. A brief discussion of the main algorithms for these four problems is presented, together with some computational experience.

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Author information

Authors and Affiliations

  1. Departamento de Matematica, Universidade de Coimbra, Coimbra, Portugal

    J. J. Judice (Reader)

  2. Department of Mathematics and Statistics, Brunel University, West London, Uxbridge, Middlesex, England

    G. Mitra (Professor)

Authors
  1. J. J. Judice
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  2. G. Mitra
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Additional information

Communicated by R. W. H. Sargent

The first author was supported by a NATO scholarship during his period of stay at Brunel University. The authors would also like to thank Professor R. Sargent of Imperial College, who found an error in an earlier version of the reformulation of the variable separable programming as an SMLCP.

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Judice, J.J., Mitra, G. Reformulation of mathematical programming problems as linear complementarity problems and investigation of their solution methods. J Optim Theory Appl 57, 123–149 (1988). https://doi.org/10.1007/BF00939332

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