Presolver
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The geometer defines a polyhedral (semiinfinite) cone <math>\,\mathcal{K}</math> in <math>\,\reals^m</math> as a set | The geometer defines a polyhedral (semiinfinite) cone <math>\,\mathcal{K}</math> in <math>\,\reals^m</math> as a set | ||
<center><math>\mathcal{K}\triangleq\{A_{}x~|~x\succeq0\}</math></center> | <center><math>\mathcal{K}\triangleq\{A_{}x~|~x\succeq0\}</math></center> | ||
| - | that is closed and convex but not necessarily pointed ( | + | that is closed and convex but not necessarily pointed (might not have a vertex). |
[[Image:Polycone2.jpg|thumb|right|200px|A pointed polyhedral cone (truncated)]] | [[Image:Polycone2.jpg|thumb|right|200px|A pointed polyhedral cone (truncated)]] | ||
Revision as of 21:40, 11 August 2011
Contents |
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 central idea in our presolving 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, think of 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 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).
positivity
An optimization problem in can always be written with nonnegativity constraints
...
assumptions
polyhedral cone is pointed...
point belongs to cone boundary...