Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

46 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
1.8 Write down the LCP corresponding to
1.9 Let
M =
Minimize cx + 1
2 xT Dx
Subject to x > = 0 :
8
>:
9
8
9
;2 1>
q = >:
1 ;2
1>
:
1
Show that the LCP (q M) has four distinct solutions. For n = 3, construct a square
matrix M of order 3 and a q 2 R 3 such that (q M) has eight distinct solutions.
Hint. Try ; M =
1.10 Let
8
>:
2 ;1 ;1
;1 3 ;1
;1 ;1 4
M =
8
>:
9
>
0 0 1
0 0 1
0 0 0
9
>
q =
8
>:
q =
1
1
1
8
>:
9
> or try M = ;Iq > 0 :
9
0
;1>
0
:
Find out a solution of the LCP (q M) by inspection. However, prove that there exists
no complementary feasible basis for this problem.
(L. Watson)
1.11 Test whether the following matrices are PD, PSD, or not PSD by using the
algorithms described in Section 1.3.1
8
>:
9
0 1 ;1
0 0 ;2>
1 2 1
8
>:
9
4 3 ;7
0 0 ;2>
0 0 6
8
>:
9
4 100 2
0 2 10>
0 0 1
8
>:
9
5 ;2 ;2
0 5 ;2>
0 0 5
:
1.12 Let Q(x) =(1=2)x T Dx ; cx. If D is PD, prove that Q(x) is bounded below.
1.13 Let K be a nonempty closed convex polytope in R n . Let f(x) be a real valued
function de ned on R n . If f(x) is a concave function, prove that there exists an
extreme point of K which minimizes f(x) on K.
1.14 Let D be an arbitrary square matrix of order n. Prove that, for every positive
and su ciently large , the function Q (x) =x T (D ; I)x + cx is a concave function
on R n .