Chapter 1 LINEAR COMPLEMENTARITY PROBLEM, ITS ...

38 Chapter 1. Linear Complementarity Problem, Its Geometry, and Applications
for all i 2 J(x) \fk +1:::mg and for any y 6= 0,y 2fy :(rgi(x)) y =0 i 2
J(x)g, yT; @2L(x )
@x2 y>0,
(iii) the initial point x0 is su ciently close to a KKT point for (1.22)
it has been proved (see references [1.44, 1.45]) that the sequence (x r r ) generated by
the algorithm converges superlinearly to (x ) which together satisfy (1.23).
These recursive quadratic programming methods have given outstanding numerical
performance and thereby attracted a lot of attention. However, as pointed out
above, one di culty with this approach is that the quadratic programming problem
(1.26) may be infeasible in some steps, even if the original nonlinear program has an
optimum solution, in addition the modi ed quadratic program (1.33) may have the
optimum solution ~ d =0,in which case the method breaks down. Another di culty is
that constraint gradients need to be computed for each constraint in each step, even for
constraints which are inactive. Yet another di culty is the function f( ) minimized
in the line search routine in each step, which is a non-di erentiable L1-penalty function.
To avoid these and other di culties, the following modi ed sequential quadratic
programming method has been proposed for solving (1.22) by K. Schittkowski [1.50,
1.51].
Choose the initial point x 0 , multiplier vector 0 , B0 = I or some PD symmetric
approximation for @2 L(x 0 0 )
@x2 , 0 2 R 1 , 0 2 R m ( 0 > 0, 0 > 0) and constants ">0,
> 1, 0 < < 1. The choice of " = 10 ;7 , = 0:9, = 100, and suitable positive
values for 0, 0 is reported to work well by K. Schittkowski [1.51]. Evaluate (x 0 ),
gi(x 0 ), rgi(x 0 ), i =1to m and go to stage 1.
General Stage r+1: Let x r , r denote the current solution and Lagrange multiplier
vector. De ne
J1 = f1:::kg[fi : k +1< = i < = m and either gi(x r ) < = " or
J2 = f1:::mgnJ1 :
r
i > 0g
The constraints in (1.22) corresponding to i 2 J1 are treated as the active set of
constraints at this stage, constraints in (1.22) corresponding to i 2 J2 are the current
inactive constraints.
Let Br be the present matrix which is a PD symmetric approximation for @2L(x r r )
@x2 ,
this matrix is updated from step to step using the BFGS quasi-Newton update formula
discussed earlier. The quadratic programming subproblem to be solved at this stage
contains an additional variable, xn+1, to make sureit is feasible. It is the following
; rx 2 n+1
minimize P (d) = 1
2 dT Brd +(r (x r ))d + 1
2
subject to (rgi(x r ))d +(1; xn+1)gi(x r )
=0
>
=
i =1to k
0 i 2 J1
(rgi(x
\fk +1:::mg
si ))d + gi(x r ) >
=
0 i 2 J2
0 < = xn+1 < = 1
(1:36)