GAMS/PATH User Guide Version 4.3

Furthermore, underlying model convexity is important. For example, if
we have the linear program
minx cT x
subject to Ax = b, x ≥ 0
we can write the first order optimality conditions as either
0 ≤ x ⊥ −A T µ + c
µ free ⊥ Ax − b
or, equivalently,
0 ≤ x ⊥ −AT µ + c
µ free ⊥ b − Ax
because we have an equation. The former is a linear complementarity problem
with a positive semidefinite matrix, while the latter is almost certainly
indefinite. Also, if we need to perturb the problem because of numerical
problems, the former system will become positive definite, while the later
becomes highly nonconvex and unlikely to solve.
Finally, users are strongly encouraged to match equations and free variables
when the matching makes sense for their application. Structure and
convexity can be destroyed if it is left to the solver to perform the matching.
For example, in the above example, we could loose the positive semidefinite
matrix with an arbitrary matching of the free variables.
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