Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

10 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
8
9
>: u>
;
v
8
>: u
9
> >
v =
0
8
>:
8
9
9
0 ;AT >
A 0
8 >: x>
=
y
>: x
9 8
> >
y =
0 and >: u
v
8
>: cT
9
;b
9
>
9
> T 8 >: x
y
> =0:
(1:10)
Conversely, if u, v, x, y together satisfy all the conditions in (1.10), x is an optimum
solution of (1.9). In (1.10) all the vectors and matrices are written in partitioned
form. For example, ; u
v is the vector (u1:::uNv1:::vm) T . If n = m + N,
w =
8
>: u
v
9
> z =
8
>: x
y
9
> M =
8
>:
0 ;AT
A 0
9
> q =
8
>: cT
9
>
;b
(1.10) is seen to be an LCP of order n of the type (1.6) to (1.8). Solving the LP (1.9)
can be achieved by solving the LCP (1.10).
Also, the various complementary pairs of variables in the LCP (1.10) are exactly
those in the pair of primal, dual LPs (1.9) and its dual. As an example consider the
following LP.
Minimize ;13x1 +42x2
Subject to 8x1 ; x2 + 3x2 > = ;16
; 3x1 + 2x2 ; 13x3 > = 12
xj > =
0 j = 1 2 3:
Let (v1y1), (v2y2) denote the nonnegative slack variable, dual variable respectively,
associated with the two primal constraints in that order. Let u1, u2, u3 denote the
nonnegative dual slack variable associated with the dual constraint corresponding to
the primal variable x1, x2, x3, in that order. Then the primal and dual systems
together with the complementary slackness conditions for optimality are
8x1 ; x2 + 3x3 ; v1 = ;16
;3x1 + 2x2 ; 13x3 ; v2 = 12
8y1 ; 3y2 + u1 = ;13
; y1 + 2y2 + u2 = 42
3y1 ; 13y2 + u3 = 0:
xjujyivi > =
0 for all i j.
xjuj = yivi =0 for all i j.