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486 PATH 4.6
2.6 Preprocessing
The preprocessor is work in progress. The exact output in the final version may differ from that
given below.
The purpose of a preprocessor is to reduce the size and complexity of a model to achieve improved performance
by the main algorithm. Another benefit of the analysis performed is the detection of some provably unsolvable
problems. A comprehensive preprocessor has been incorporated into PATHC as developed in [18].
The preprocessor reports its finding with some additional output to the log file. This output occurs before the
initial point statistics. An example of the preprocessing on the forcebsm model is presented below.
Zero: 0 Single: 112 Double: 0 Forced: 0
Preprocessed size: 72
The preprocessor looks for special polyhedral structure and eliminates variables using this structure. These are
indicated with the above line of text. Other special structure is also detected and reported.
On exit from the algorithm, we must generate a solution for the original problem.
postsolve. Following the postsolve, the residual using the original model is reported.
Postsolved residual: 1.0518e-10
This is done during the
This number should be approximately the same as the final residual reported on the presolved model.
2.6.1 Constrained Nonlinear Systems
Modelers typically add bounds to their variables when attempting to solve nonlinear problems in order to restrict
the domain of interest. For example, many square nonlinear systems are formulated as
F (z) = 0, l ≤ z ≤ u,
where typically, the bounds on z are inactive at the solution. This is not an MCP, but is an example of a
“constrained nonlinear system” (CNS). It is important to note the distinction between MCP and CNS. The MCP
uses the bounds to infer relationships on the function F . If a finite bound is active at a solution, the corresponding
component of F is only constrained to be nonnegative or nonpositive in the MCP, whereas in CNS it must be
zero. Thus there may be many solutions of MCP that do not satisfy F (z) = 0. Only if z ∗ is a solution of MCP
with l < z ∗ < u is it guaranteed that F (z ∗ ) = 0.
Internally, PATHC reformulates a constrained nonlinear system of equations to an equivalent complementarity
problem. The reformulation adds variables, y, with the resulting problem written as:
This is the MCP model passed on to the PATH solver.
l ≤ x ≤ u ⊥ −y
y free ⊥ F (x).
3 Advanced Topics
This chapter discusses some of the difficulties encountered when dealing with complementarity problems. We
start off with a very formal definition of a complementarity problem which is used in later sections on merit
functions and ill-defined, poorly-scaled, and singular models.
3.1 Formal Definition of MCP
The mixed complementarity problem is defined by a function, F : D → R n where D ⊆ R n is the domain of F ,
and possibly infinite lower and upper bounds, l and u. Let C := {x ∈ R n | l ≤ x ≤ u}, a Cartesian product of