Presolver
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Revision as of 23:51, 4 December 2011
Introduction
Presolving conventionally means quick elimination of some variables and constraints prior to numerical solution of an optimization problem.
Presented with constraints for example,
a presolver is likely to check whether constant vector
is positive; for if so,
variable
can have only the trivial solution.
The effect of such tests is to reduce the problem dimensions.
Most commercial optimization problem solvers incorporate presolving. Particular reductions can be proprietary or invisible, while some control or selection may be given to a user. But all presolvers have the same motivation: to make an optimization problem smaller and (ideally) easier to solve. There is profit potential because a solver can then compete more effectively in the marketplace for large-scale problems.
We present a method for reducing variable dimension based upon geometry of constraints in the problem statement:
where is a real
matrix,
represents the integers,
the real numbers,
is some predetermined set,
and
is some possibly empty index set.
A caveat to use of our proposed method for presolving is that it is not fast.
(One would incorporate this method only when a problem is too big to be solved;
that is, when solver software chronically exits with error or hangs.)
That it can be decomposed into independent parallel subproblems is its saving grace;
speedup becomes proportional to number of parallel processors.
Geometry of Constraints
The idea central to our presolve method is best understood geometrically. Constraints
suggest that a polyhedral cone comes into play.
Geometers in convex analysis regard cones as convex Euclidean bodies semiinfinite in extent.
Finite circular cones hold ice cream and block road traffic in daily life.
Each of the great Pyramids of Egypt is a finite polyhedral cone.
The geometer defines a polyhedral (semiinfinite) cone in
as a set
that is closed and convex but not necessarily pointed (might not have a vertex).
To visualize a pointed polyhedral cone in three dimensions,
imagine one Egyptian Pyramid continuing into the ground
and then indefinitely out into Space from the opposite side of Earth.
Its four edges correspond to four columns from matrix .
Those four columns completely describe the semiinfinite Pyramid
per definition of
.
But
can have more than four columns and still describe the same Pyramid.
For such a fat
, each additional column resides anywhere in
:
either interior to the cone
or on one of its faces (the vertex, an edge, or a facet).
Polyhedral cones have infinite variety. Most are not so regularly shaped as a Pyramid, and they can have any number of edges and facets. We assume a pointed polyhedral cone throughout, so there can be only one vertex (which resides at the origin in Euclidean space by definition).
Assumptions
1) Pointedness of cones.
Our presolve method applies only to pointed polyhedral cones,
namely cones that contain no line.
(Recall that a line is a 1-dimensional Euclidean body that extends infinitely in two opposite directions.)
A cone contains no line if and only if the cone
and its polar cone
intersect only at the origin:
This condition is necessary and sufficient to establish pointedness. Halfspace figure as example of cone that contains line...
It is now prudent to ask whether a given set of generators for
(the columns of
)
describe a pointed cone.
A practical test for pointedness therefore attempts to establish existence of a line.
The test we choose is proof by counterexample:
given matrix
having no zero columns,
This test says that if contains a line, then some conic combination of its generators can be made to point opposite in direction to another
. Constraint
precludes the trivial solution.
If there is an
solving this convex feasibility problem, then the cone described by
cannot be pointed.
2) on Boundary.
Success of our method relies on point
belonging
to the boundary of cone
.
One way to think of a pointed closed convex cone is as a kind
of coordinate system whose basis is generally nonorthogonal and overcomplete;
a conic system, very much like the familiar Cartesian system
whose analogous cone is the first quadrant (the nonnegative orthant)
and whose generators are the Cartesian coordinate axes.
In the context of pointed polyhedral cones,
overcomplete means an abundance of linearly dependent generators in .
Consequently, nonnegative coordinates
of
are not unique;
there are, generally, an infinite number of coordinates
.
We can discern a smaller set of coordinate axes among the columns of
by finding
the smallest face of cone
to which
belongs.
Then the generators of that smallest face constitute a subset of conic coordinate axes
that can describe
. Once found, remaining coordinates may be assumed zero.
This assumption, that resides on the boundary
of cone
, is realistic. This phenomenon occurs whenever
has sparse representation in its coordinates.
It is therefore prudent to check, before presolving, whether
resides on the boundary:
For full-dimensional pointed closed convex cone
described by given matrix
this linear feasibility problem has a solution iff .
The requirement that
be full-dimensional is equivalent to the condition that fat matrix
be full rank.
This feasibility problem is a statement of boundary membership relation from convex analysis []...