GAMS/PATH User Guide Version 4.3

the heuristic that if a variable enters the basis more that a given number of
times, we are cycling. The number of times the variable has entered is reset
whenever t increases beyond its previous maximum or an artificial variable
leaves the basis. If cycling is detected, we terminate the linear solver at the
largest value of t and return this point.
Another heuristic is added when the linear code terminates on a ray.
The returned point in this case is not the base of the ray. We move a
slight distance up the ray and return this new point. If we fail to solve the
linear subproblem five times in a row, a Lemke ray start will be performed
in an attempt to solve the linear subproblem. Computational experience
has shown this to be an effective heuristic and generally results in solving
the linear model. Using a Lemke ray start is not the default mode, since
typically many more pivots are required.
For time when a Lemke start is actually used in the code, an advanced
ray can be used. We basically choose the “closest” extreme point of the
polytope and choose a ray in the interior of the normal cone at this point.
This helps to reduce the number of pivots required. However, this can fail
when the basis corresponding to the cell is not invertible. We then revert to
the Lemke start.
Since the EXPAND pivot rules are used, some of the variable may be
nonbasic, but slightly infeasible, as the solution. Whenever the linear code
finisher, the nonbasic variables are put at their bounds and the basic variable
are recomputed using the current factorization. This procedure helps to find
the best possible solution to the linear system.
The resulting linear solver as modified above is robust and has the desired
property that we start from (ˆz, ˆw, ˆv) and construct a path to a solution.
3.2.5 Other Features
Some other heuristics are incorporated into the code. During the first iteration,
if the linear solver fails to find a Newton point, a Lemke start is used.
Furthermore, under repeated failures during the linear solve, a Lemke starts
will be attempted. A gradient step can also be used when we fail repeatedly.
The proximal perturbation is shrunk each major iteration. However, when
numerical difficulties are encountered, it will be increase to a fraction of the
current merit function value. These are determined as when the linear solver
returns the Reset or Singular status.
Spacer steps are taken every major iteration, in which the iterate is chosen
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