Interior Point Algorithms for Solving Linear Complementarity Problems

Abstract

In chapter ten we employed a variant of the standard simplex routine, called the complementary pivot method, to generate a solution to the linear complementarity problem LCP(q,M), which we shall now express as: find an (X,Y) € R²ⁿ satisfying

$$\begin{array}{*{20}{c}}{(a){\rm{ Y = MX + q}}}\\{(b){\rm{ (X,Y)}} \ge {\rm{(0,0)}}}\\{(c){\rm{ }}{x_i}{y_i} = 0,{\rm{ }}i = 1, \ldots ,n,}\end{array}$$

(14.1)

where M is of order (n x n) and q € Rⁿ. Here the feasible region K associated with (14.1), its relative interior K°, and the set of all solutions to (14.1) will be denoted as:

$$K = \{ \left( {X,Y} \right)|\left( {X,Y} \right) \ge \left( {0,0} \right),Y = MX + q\} ,$$

$$K = \{ \left( {X,Y} \right)|\left( {X,Y} \right) \ge \left( {0,0} \right),Y = MX + q\} ,$$

and

$${K^0} = \cap R_{ + + }^{2n} = \left\{ {\left( {X,Y} \right) \in R_{ + + }^{2n}|Y = MX + q} \right\},$$

respectively.