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Exact Computation of Basic Solutions for Linear Programming

Introduction

Exact linear programming has many applications such as finding provably optimal
solutions in standard application areas, computer assisted proofs and subroutines
for exact integer programming. The recent proof of the Kepler Conjecture by
Thomas Hales relies on solving thousands of linear programs exactly.

The fastest known technique for solving linear programs exactly is performing the
simplex algorithm with floating point arithmetic, then computing, verifying and
correcting the final solution with symbolic computation. This leaves solving sparse
rational systems of equations as the major bottleneck. We investigate methods to
solve sparse rational systems of linear equations exactly, and study their behavior
computationally on a large test set arising from linear programming problems.

Exact Linear Programming

Linear programming is a technique for
optimizing a linear objective function
over a set of linear inequalities. Geometrically,
the inequalities represent a convex
polyhedron.

Standard form LP

Simplex Method

The most commonly used technique for linear programming is the simplex method.
A simplified geometric description is:
•Start at a vertex of the polyhedron
•Follow edges of the polyhedron to better vertices
Terminate at locally optimal vertex

At each vertex of the polyhedron the corresponding solution vector x can be determined
by solving a system of linear equations. This is called a basic solution, and
the subsystem of A defining this solution is called a basis.

If the simplex method is performed using floating point computation, the basis of
the final solution provides a structural description of the solution as the solution to
a system of equations. The coefficients of these equations come from the original
problem description, so the final solution can be computed exactly.

Optimality Verification

Linear programming duality theory provides a tool to certify optimality of an exact
solution. Every linear programming problem has a related dual problem and if
either is feasible they share the same optimal objective value .

Linear Program (Primal)

Dual Problem

If x and y are feasible, then they are optimal if and only if cTx = bTy. An optimal
basis of the primal problem (vertex of the polyhedron) structurally describes both
the primal optimal solution, and the dual optimal solution y. Given the optimal
basis, calculating and certifying the exact optimal solution involves solving two
systems of linear equations.

QSopt ex: A Rational LP Solver

The QSopt ex software [1] (by Applegate, et al.) does the following.
•Apply simplex in floating point arithmetic
•Compute basic solution exactly from final basis
•If solution is not optimal, increase precision
•Repeat procedure in higher precision, starting at current basis

This strategy proves very effective. Much of the simplex algorithm is done in floating
point. Even when a suboptimal basis is returned due to floating point errors,
it is often close to the optimal basis, reducing the extended precision computation .
The final output includes the optimal solution and an exact certificate of optimality.

Solving Sparse Rational Systems

Several methods are known for solving sparse systems of rational equations Ax = b
exactly, beyond applying standard techniques using rational arithmetic [3, 4]. If an
accurate enough approximate solution is calculated, the exact rational solution can
be reconstructed using diophantine approximation. If the problem is scaled to have
integer coefficients and solved modulo a large enough integer, the exact rational solution
can be calculated using rational number reconstruction . Diophantine approximation
and rational number reconstruction are performed by calculating continued
fraction convergents using the Extended Euclidean Algorithm.

Exact LU Decomposition

•Solve system directly using an LU decomposition over Q

Sparse decomposition algorithms focus on permuting rows and columns of the system
to avoid fill-in and arithmetic operations.

Iterative Refinement

•Compute floating point LU decomposition of A
•Iteratively refine approximate solution
•Use rational number reconstruction to find exact solution

Using mixed precision iterative refinement an approximate solution is computed,
suitable for rational reconstruction/diophantine approximation to be applied.

Wiedemann

•Use Wiedemann’s method to solve Ax = b over finite field
•Use p-adic lifting to find solution modulo large number
•Use rational number reconstruction to find exact solution

Wiedemann’s method [5] for solving systems of equations over a finite field is a
black box algorithm, it only accesses the matrix for matrix-vector multiplication.

Dixon

Compute LU decomposition of A over Zp
Use p-adic lifting to find solution modulo large number
Use rational number reconstruction to find exact solution

The finite field solves are done using a sparse LU factorization, designed to minimize
fill-in.

Computations

Implementation

We implemented the four methods in C. We rely on the GNU Multiple Precision
Library to store rational numbers and large integers. The codes for LU factorization
over the rational numbers, finite fields and double precision floating point numbers
all share a common structure and were adapted from the QSopt software, which is
designed for very sparse matrices. The rational reconstruction codes employ output
sensitive lifting, Lehmer’s algorithm and other techniques to increase speed.

Test Problems

The linear-programming research community is fortunate to have several publicly-available
libraries of test instances. We collected instances from several test libraries,
including NETLIB, MIPLIB 3.0 and many others. These collections are
comprised of instances gathered from business and industrial applications, and from
academic studies. The instances were were given to QSopt ex and the optimal basis
matrix, or the last basis encountered after 24 hours was recorded. After some
preprocessing and reduction the final set contains 276 instances, with dimensions
ranging from 100 to over 50,000. For each instance we have both the square basis
matrix and the corresponding right-hand-side vector.

Results

We have given a performance profile comparing the relationships . A performance
profile plots the number of instances solved within a factor x of the fastest solve
time among the methods. The vertical axis represents the number of instances.
The horizontal axis gives the solve-time ratios. Ratios are compared instead of
averaging solution times , which ranged from hours to seconds.

This table gives the solve time ratios on subsets of the problems partitioned by dimension,
solution size and density (nonzero entries per row). The solution bitsize
is the largest number of bits required to represent a component of the solution vector.
We consider the ratio with respect to the Dixon method, and give the geometric
average of the ratios. This table allows us to observe the effect of various problem
characteristics on the relative performance of the methods.

Conclusions

We evaluated several methods for symbolically solving large sparse rational systems
of linear equations arising from a wide range of linear programming problems.
Dixon’s method and iterative refinement had similar performance, beating
the other methods. Iterative refinement was slightly faster in our experience, but
Dixon’s method avoids numerical difficulties by using finite field arithmetic. We
found Wiedemann’s algorithm to be much slower than the other methods tested on
this problem set. This set of real world problems also provides a useful test set for
future studies. Further computational results and analysis are contained in [2].

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