Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

26 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
for (1.11), is said to be a Karush-Kuhn-Tucker point (or abbreviated as a KKT
point) if there exist Lagrange multiplier vectors y, u, such that x, y, u together satisfy
(1.20) or the equivalent (1.19). So the LCP (1.19) is the problem of nding a KKT
point for (1.11). We now have the following results.
Theorem 1.13 If x is an optimum solution for (1.11), x mustbeaKKT point for
it, whether Q(x) is convex or not.
Proof. Follows from Theorem 1.12 and Corollary 1.1.
Thus (1.20) or equivalently (1.19) provide the necessary optimality conditions for a
feasible solution x of (1.11) to be optimal. Or, in other words, every optimum solution
for (1.11) must be a KKT point for it. However, given a KKT point for (1.11) we
cannot guarantee that it is optimal to (1.11) in general. In the special case when D
is PSD, every KKT point for (1.11) is optimal to (1.11), this is proved in Theorem
1.14 below. Thus for convex quadratic programs, (1.20) or equivalently (1.19) provide
necessary and su cient optimality conditions.
Theorem 1.14 If D is PSD and x is a KKT point of (1.11), x is an optimum
feasible solution of (1.11).
Proof. >From the de nition of a KKT point and the results in Section 1.2, if x is a
KKT point for (1.11), it must be an optimum feasible solution of the LP (1.18). Let x
be any feasible solution of (1.11).
Q(x) ; Q(x) =(c + x T D)(x ; x)+ 1
2 (x ; x)T D(x ; x) :
The rst term on the right-hand side expression is nonnegative since x is an optimal
feasible solution of (1.18). The second term in that expression is also nonnegative since
D is PSD. Hence, Q(x) ; Q(x) > = 0forall feasible solutions, x, of (1.11). This implies
that x is an optimum feasible solution of (1.11).
Clearly (1.19) is an LCP. An optimum solution of (1.11) must be a KKT point for
it. Solving (1.19) provides a KKT point for (1.11) and if D is PSD, this KKT point is
an optimum solution of (1.11). [If D is not PSD and if a KKT point is obtained when
(1.19) is solved, it may not be an optimum solution of (1.11).]
Example 1.7 Minimum Distance Problem.
Let K denote the shaded convex polyhedral region in Figure 1.7. Let P0 be the point
(;2 ;1). Find the point in K that is closest to P0 (in terms of the usual Euclidean
distance). Such problems appear very often in operations research applications.