Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

50 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
Extend this method into one for nding the nearest point in; = fx : x > = 0 x< =
g to x 0 , where is a given number, assuming that ; 6= .
(W. Oettli [1.30])
1.28 Let M be a square matrix of order n and q 2 R n . Let z 2 R n be a vector of
variables. De ne: fi(z) = minimum fziMi.z + qig, that is
for each i =1to n.
fi(z) = Ii.z if (Mi. ; Ii.)z + qi > = 0
= Mi.z + qi if (Mi. ; Ii.)z + qi < = 0
(a) Show that fi(z) is a piecewise linear concave function de ned on R n
(b) Consider the system of equations
fi(z) =0 i =1to n:
Let z be a solution of this system. Let w = Mz + q. Prove that (w z) is a
complementary feasible solution of the LCP (q M).
(c) Using (b) show that every LCP is equivalent to solving a system of piecewise linear
equations.
(R. Saigal)
1.29 For j = 1 to n de ne x +
j = Maximum f0xjg, x ;
j = ; Minimum f0xjg. Let
x =(xj) 2 R n , x + =(x +
j ), x; =(x ;
j ). Given the square matrix M of order n, de ne
the piecewise linear function
TM(x) =x + ; Mx ; :
Show that TM(x) is linear in each orthant of R n . Prove that (w = x + z = x ; ) solves
the LCP (q M) i q = TM (x).
(R. E. Stone [3.71])
1.30 Let D be a given square matrix of order n, andf(x) =x T Dx. Prove thatthere
exists a nonsingular linear transformation: y = Ax (where A is a square nonsingular
matrix of order n) such that
f(x) =y 2 1 + :::+ y 2 p ; y 2 p+1 ; :::; y 2 r
where 0 < = p < = r < = n. Discuss an e cient method for nding such a matrix A, given
D.
Find such a transformation for the quadratic form f(x1x2x3) =x 2 1 +x 2 2 +x 2 3 ;2x1x2;
2x1x3 ; 2x2x3 (this dates back to Lagrange in 1759, see D. E. Knuth [10.20]).