505 research outputs found
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
PDF fil
MARS: A second-order reduction algorithm for high-dimensional sparse precision matrices estimation
Estimation of the precision matrix (or inverse covariance matrix) is of great
importance in statistical data analysis. However, as the number of parameters
scales quadratically with the dimension p, computation becomes very challenging
when p is large. In this paper, we propose an adaptive sieving reduction
algorithm to generate a solution path for the estimation of precision matrices
under the penalized D-trace loss, with each subproblem being solved by
a second-order algorithm. In each iteration of our algorithm, we are able to
greatly reduce the number of variables in the problem based on the
Karush-Kuhn-Tucker (KKT) conditions and the sparse structure of the estimated
precision matrix in the previous iteration. As a result, our algorithm is
capable of handling datasets with very high dimensions that may go beyond the
capacity of the existing methods. Moreover, for the sub-problem in each
iteration, other than solving the primal problem directly, we develop a
semismooth Newton augmented Lagrangian algorithm with global linear convergence
on the dual problem to improve the efficiency. Theoretical properties of our
proposed algorithm have been established. In particular, we show that the
convergence rate of our algorithm is asymptotically superlinear. The high
efficiency and promising performance of our algorithm are illustrated via
extensive simulation studies and real data applications, with comparison to
several state-of-the-art solvers
An efficient sieving based secant method for sparse optimization problems with least-squares constraints
In this paper, we propose an efficient sieving based secant method to address
the computational challenges of solving sparse optimization problems with
least-squares constraints. A level-set method has been introduced in [X. Li,
D.F. Sun, and K.-C. Toh, SIAM J. Optim., 28 (2018), pp. 1842--1866] that solves
these problems by using the bisection method to find a root of a univariate
nonsmooth equation for some , where
is the value function computed by a solution of the
corresponding regularized least-squares optimization problem. When the
objective function in the constrained problem is a polyhedral gauge function,
we prove that (i) for any positive integer , is piecewise
in an open interval containing the solution to the equation
; (ii) the Clarke Jacobian of is
always positive. These results allow us to establish the essential ingredients
of the fast convergence rates of the secant method. Moreover, an adaptive
sieving technique is incorporated into the secant method to effectively reduce
the dimension of the level-set subproblems for computing the value of
. The high efficiency of the proposed algorithm is demonstrated
by extensive numerical results
Physiological Characterization of Cut-to-Cut Yield Variations of Alfalfa Genotypes under Controlled Greenhouse Conditions
In a temperate region, alfalfa (Medicago sativa) crops are usually harvested 3-6 times per annum. The biomass yields of first and second cuts in the spring are generally the high-est. However, in subsequent cuts the biomass yields decline, with the final 1 or 2 cuts producing the lowest yields (Wang et al. 2009). This seasonal reduction in alfalfa biomass yields could be associated with prevailing changes in environmental factors such as rainfall and heat stress or due to biological characteristics of alfalfa crop itself. In this study, alfalfa was grown under controlled greenhouse conditions with suitable temperature, light, water and nutrient supply to determine the driving force in cut-to-cut biomass yield variations among alfalfa genotypes
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