Math. Program., Ser. A (2008) 112:275–301
DOI 10.1007/s1010700600170
FULL
LENGTH
PAPER
A sparse proximal implementation of the LP dual
active set algorithm
Timothy A. Davis
·
William W. Hager
Received: 13 August 2003 / Accepted: 21 June 2006 / Published online: 18 August 2006
© SpringerVerlag 2006
Abstract
We present an implementation of the LP Dual Active Set Algorithm
(LP DASA) based on a quadratic proximal approximation, a strategy for drop
ping inactive equations from the constraints, and recently developed algorithms
for updating a sparse Cholesky factorization after a lowrank change. Although
our main focus is linear programming, the first and secondorder proximal tech
niques that we develop are applicable to general concave–convex Lagrangians
and to linear equality and inequality constraints. We use Netlib LP test prob
lems to compare our proximal implementation of LP DASA to Simplex and
Barrier algorithms as implemented in CPLEX.
Keywords
Dual active set algorithm
·
Linear programming
·
Simplex
method
·
Barrier method
·
Interior point method
·
Equation dropping
Mathematics Subject Classification (2000)
90C05
·
90C06
·
65Y20
This material is based upon work supported by the National Science Foundation under Grant
No. 0203270.
T. A. Davis
Department of Computer and Information Science and Engineering,
University of Florida, Gainesville, PO Box 116120, FL 326116120, USA
email: [email protected]
URL: http://www.cise.ufl.edu/
∼
davis
W. W. Hager(
B
)
Department of Mathematics, University of Florida,
Gainesville, PO Box 118105, FL 326118105, USA
email: [email protected]
URL: http://www.math.ufl.edu/
∼
hager
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T. A. Davis, W. W. Hager
1 Introduction
We present an implementation of the LP Dual Active Set Algorithm (LP
DASA) [23] for solving linear programming problems. Global convergence
is established and comparisons with Simplex and Barrier algorithms, as imple
mented in CPLEX, are reported. Since our quadratic proximal approach can
be applied to general concave–convex Lagrangians, we develop the theory in
an abstract setting, and then apply it to the linear programming problem.
Given a function
L
:
×
X
→
R
, where
⊂
R
m
and
X
⊂
R
n
are closed and
convex, we consider the problem
sup
λ
∈
inf
x
∈
X
L
(
λ
,
x
)
.
(1)
We refer to
L
in (1) as the “Lagrangian,” while the “dual function,” defined on
, is given by
L
(
λ
)
=
inf
x
∈
X
L
(
λ
,
x
)
.
(2)
Hence, the maximin problem (1) is equivalent to the maximization of the dual
function.
Let
c
T
denote the transpose of a vector
c
. The primal linear program (LP)
min
c
T
x
subject to
Ax
=
b
,
x
≥
0
,
(3)
where
A
is
m
by
n
and all vectors are of compatible size, corresponds to the
following choices in the dual formulation (1):
X
= {
x
∈
R
n
:
x
≥
0
}
,
=
R
m
,
and
(4)
L
(
λ
,
x
)
=
c
T
x
+
λ
T
(
b
−
Ax
)
.
(5)
Any LP can be written in the canonical form (3) [37]. If the primal problem (3)
has a solution
x
∗
, then dual problem (1) has a solution; if
λ
∗
is a solution of (1),
then
x
∗
minimizes
L
(
λ
∗
,
·
)
over
X
.
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 Linear Programming, Optimization, LP, dual active set, active set algorithm

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