Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

44 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
For this game, it can be veri ed that the probability vectors (x =(1 0) T y =(1 0) T ).
(^x = (0 1) T ^y = (0 1) T ) are both equilibrium pairs. The losses from the two equilibrium
pairs (x y), (^x ^y) are distinct, (x y) will be preferred by player I, whereas II
will prefer (^x ^y). Because of this, these equilibrium pairs are unstable. Even if player
Iknows that II will use the strategy ^y, shemay insist on using strategy x rather than
^x, hoping that this will induce II to switch to y. So, in this game, it is di cult to
foresee what will happen. The probability vectors (~x =(4=5 1=5) ~y =(1=5 4=5) T )is
another equilibrium pair. In this problem, knowledge of these equilibrium pairs seems
to have contributed very little towards the development of any \optimum" strategy.
Even though the theory of equilibrium strategies is mathematically elegant, and
algorithms for computing them (through the LCP formulation) are practically e cient,
they have not found many real world applications because of the problems with them
illustrated in the above examples.
1.5 OTHER APPLICATIONS
Besides these applications, LCP has important applications in the nonlinear analysis of
certain elastic-plastic structures such as reinforced concrete beams, in the free boundary
problems for journal bearings, in the study of nance models, and in several other
areas. See references [1.1 to 1.5, 1.8, 1.12, 1.13, 1.19, 1.21, 1.29, 1.32, 1.35].
1.6 THE NONLINEAR
COMPLEMENTARITY PROBLEM
For each j = 1 to n, let fj(z) be a real valued function de ned on R n . Let f(z) =
(f1(z):::fn(z)) T . The problem of nding z 2 R n satisfying
z > = 0 f(z) > = 0
zjfj(z) =0 for each j =1to n
(1:44)
is known as a nonlinear complementarity problem (abbreviated as NLCP). If we de ne
fj(z) = Mj.z + qj for j = 1 to n, it can be veri ed that (1.44) becomes the LCP
(1.1). Thus the LCP is a special case of the NLCP. Often, it is possible to transform
the necessary optimality conditions for a nonlinear program into that of an NLCP and
thereby solve the nonlinear program using algorithms for NLCP. The NLCP can be
transformed into a xed point computing problem, as discussed in Section 2.7.7, and
solved by the piecewise linear simplicial methods presented in Section 2.7. Other than
this, we will not discuss any detailed results on NLCP, but the references [1.14 to 1.16,
1.24, 1.25, 1.39] can be consulted by the interested reader.