GAMS/PATH User Guide Version 4.3

Mi =Ψ(x k + tkd k ) for all i ∈ [1,...,m]. This prevents future iterates from
returning to the same problem area.
A watchdog strategy [5] is also available for use in the code. The method
employed allows steps to be accepted when they are “close” to the current
iterate. Nonmonotonic decrease is enforced every m iterations, where m is
set by the nms mstep frequency option.
3.2.4 Linear Complementarity Problems
PATH solves a linear complementarity problem each major iteration. Let
M ∈ℜ n×n , q ∈ℜ n ,andB =[l, u] begiven.(¯z, ¯w, ¯v) solves the linear mixed
complementarity problem defined by M, q, andB if and only if it satisfies
the following constrained system of equations:
Mz − w + v + q =0 (3.1)
w T (z − l) = 0 (3.2)
v T (u − z) = 0 (3.3)
z ∈ B,w ∈ℜ n + ,v ∈ℜn + , (3.4)
where x + ∞ = ∞ for all x ∈ℜand 0 ·∞ = 0 by convention. A triple,
(ˆz, ˆw, ˆv), satisfying equations (3.1) - (3.3) is called a complementary triple.
The objective of the linear model solver is to construct a path from a
given complementary triple (ˆz, ˆw, ˆv) toasolution(¯z, ¯w, ¯v). The algorithm
used to solve the linear problem is identical to that given in [9]; however,
artificial variables are incorporated into the model. The augmented system
is then:
(1 − t)
Mz − w + v + Da + (sr)+q =0 (3.5)
s
w T (z − l) = 0 (3.6)
v T (u − z) = 0 (3.7)
z ∈ B,w ∈ℜ n + ,v ∈ℜn + ,a≡ 0,t∈ [0, 1] (3.8)
where r is the residual, t is the path parameter, and a is a vector of artificial
variables. The residual is scaled by s to improve numerical stability.
The addition of artificial variables enables us to construct an initial invertible
basis consistent with the given starting point even under rank deficiency.
The procedure consists of two parts: constructing an initial guess as to the
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