6,130 research outputs found
Enumeration of PLCP-orientations of the 4-cube
The linear complementarity problem (LCP) provides a unified approach to many
problems such as linear programs, convex quadratic programs, and bimatrix
games. The general LCP is known to be NP-hard, but there are some promising
results that suggest the possibility that the LCP with a P-matrix (PLCP) may be
polynomial-time solvable. However, no polynomial-time algorithm for the PLCP
has been found yet and the computational complexity of the PLCP remains open.
Simple principal pivoting (SPP) algorithms, also known as Bard-type algorithms,
are candidates for polynomial-time algorithms for the PLCP. In 1978, Stickney
and Watson interpreted SPP algorithms as a family of algorithms that seek the
sink of unique-sink orientations of -cubes. They performed the enumeration
of the arising orientations of the -cube, hereafter called
PLCP-orientations. In this paper, we present the enumeration of
PLCP-orientations of the -cube.The enumeration is done via construction of
oriented matroids generalizing P-matrices and realizability classification of
oriented matroids.Some insights obtained in the computational experiments are
presented as well
Combinatorial Characterizations of K-matrices
We present a number of combinatorial characterizations of K-matrices. This
extends a theorem of Fiedler and Ptak on linear-algebraic characterizations of
K-matrices to the setting of oriented matroids. Our proof is elementary and
simplifies the original proof substantially by exploiting the duality of
oriented matroids. As an application, we show that a simple principal pivot
method applied to the linear complementarity problems with K-matrices converges
very quickly, by a purely combinatorial argument.Comment: 17 pages; v2, v3: clarified proof of Thm 5.5, minor correction
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Reformulations of mathematical programming problems as linear complementarity problems
A family of complementarity problems are defined as extensions of the well known Linear Complementarity Problem (LCP). These are
(i.) Second Linear Complementarity Problem (SLCP) which is an LCP extended by introducing further equality restrictions and unrestricted variables,
(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,
(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), bilinear programming, game theory, zero-one integer programming, the fixed charge problem, absolute value programming, variable separable programming are reformulated as members of this family of four complementarity problems
Computing Integral Solutions of Complementarity Problems
AMS classifications: 90C33, 90C26, 91B50.Discrete set;complementarity problem;algorithm;triangulation
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