296 research outputs found
SDPNAL: A Majorized Semismooth Newton-CG Augmented Lagrangian Method for Semidefinite Programming with Nonnegative Constraints
In this paper, we present a majorized semismooth Newton-CG augmented
Lagrangian method, called SDPNAL, for semidefinite programming (SDP) with
partial or full nonnegative constraints on the matrix variable. SDPNAL is a
much enhanced version of SDPNAL introduced by Zhao, Sun and Toh [SIAM Journal
on Optimization, 20 (2010), pp.~1737--1765] for solving generic SDPs. SDPNAL
works very efficiently for nondegenerate SDPs but may encounter numerical
difficulty for degenerate ones. Here we tackle this numerical difficulty by
employing a majorized semismooth Newton-CG augmented Lagrangian method coupled
with a convergent 3-block alternating direction method of multipliers
introduced recently by Sun, Toh and Yang [arXiv preprint arXiv:1404.5378,
(2014)]. Numerical results for various large scale SDPs with or without
nonnegative constraints show that the proposed method is not only fast but also
robust in obtaining accurate solutions. It outperforms, by a significant
margin, two other competitive publicly available first order methods based
codes: (1) an alternating direction method of multipliers based solver called
SDPAD by Wen, Goldfarb and Yin [Mathematical Programming Computation, 2 (2010),
pp.~203--230] and (2) a two-easy-block-decomposition hybrid proximal
extragradient method called 2EBD-HPE by Monteiro, Ortiz and Svaiter
[Mathematical Programming Computation, (2013), pp.~1--48]. In contrast to these
two codes, we are able to solve all the 95 difficult SDP problems arising from
the relaxations of quadratic assignment problems tested in SDPNAL to an
accuracy of efficiently, while SDPAD and 2EBD-HPE successfully solve
30 and 16 problems, respectively.Comment: 43 pages, 1 figure, 5 table
An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming
Powerful interior-point methods (IPM) based commercial solvers, such as
Gurobi and Mosek, have been hugely successful in solving large-scale linear
programming (LP) problems. The high efficiency of these solvers depends
critically on the sparsity of the problem data and advanced matrix
factorization techniques. For a large scale LP problem with data matrix
that is dense (possibly structured) or whose corresponding normal matrix
has a dense Cholesky factor (even with re-ordering), these solvers may require
excessive computational cost and/or extremely heavy memory usage in each
interior-point iteration. Unfortunately, the natural remedy, i.e., the use of
iterative methods based IPM solvers, although can avoid the explicit
computation of the coefficient matrix and its factorization, is not practically
viable due to the inherent extreme ill-conditioning of the large scale normal
equation arising in each interior-point iteration. To provide a better
alternative choice for solving large scale LPs with dense data or requiring
expensive factorization of its normal equation, we propose a semismooth Newton
based inexact proximal augmented Lagrangian ({\sc Snipal}) method. Different
from classical IPMs, in each iteration of {\sc Snipal}, iterative methods can
efficiently be used to solve simpler yet better conditioned semismooth Newton
linear systems. Moreover, {\sc Snipal} not only enjoys a fast asymptotic
superlinear convergence but is also proven to enjoy a finite termination
property. Numerical comparisons with Gurobi have demonstrated encouraging
potential of {\sc Snipal} for handling large-scale LP problems where the
constraint matrix has a dense representation or has a dense
factorization even with an appropriate re-ordering.Comment: Due to the limitation "The abstract field cannot be longer than 1,920
characters", the abstract appearing here is slightly shorter than that in the
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