3,680 research outputs found
Newton-MR: Inexact Newton Method With Minimum Residual Sub-problem Solver
We consider a variant of inexact Newton Method, called Newton-MR, in which
the least-squares sub-problems are solved approximately using Minimum Residual
method. By construction, Newton-MR can be readily applied for unconstrained
optimization of a class of non-convex problems known as invex, which subsumes
convexity as a sub-class. For invex optimization, instead of the classical
Lipschitz continuity assumptions on gradient and Hessian, Newton-MR's global
convergence can be guaranteed under a weaker notion of joint regularity of
Hessian and gradient. We also obtain Newton-MR's problem-independent local
convergence to the set of minima. We show that fast local/global convergence
can be guaranteed under a novel inexactness condition, which, to our knowledge,
is much weaker than the prior related works. Numerical results demonstrate the
performance of Newton-MR as compared with several other Newton-type
alternatives on a few machine learning problems.Comment: 35 page
On affine scaling inexact dogleg methods for bound-constrained nonlinear systems
Within the framework of affine scaling trust-region methods for bound constrained problems, we discuss the use of a inexact dogleg method as a tool for simultaneously handling the trust-region and the bound constraints while seeking for an approximate minimizer of the model. Focusing on bound-constrained systems of nonlinear equations, an inexact affine scaling method for large scale problems, employing the inexact dogleg procedure, is described. Global convergence results are established without any Lipschitz assumption on the Jacobian matrix, and locally fast convergence is shown under standard assumptions. Convergence analysis is performed without specifying the scaling matrix used to handle the bounds, and a rather general class of scaling matrices is allowed in actual algorithms. Numerical results showing the performance of the method are also given
Efficient Solution of Large-Scale Algebraic Riccati Equations Associated with Index-2 DAEs via the Inexact Low-Rank Newton-ADI Method
This paper extends the algorithm of Benner, Heinkenschloss, Saak, and
Weichelt: An inexact low-rank Newton-ADI method for large-scale algebraic
Riccati equations, Applied Numerical Mathematics Vol.~108 (2016), pp.~125--142,
doi:10.1016/j.apnum.2016.05.006 to Riccati equations associated with Hessenberg
index-2 Differential Algebratic Equation (DAE) systems. Such DAE systems arise,
e.g., from semi-discretized, linearized (around steady state) Navier-Stokes
equations. The solution of the associated Riccati equation is important, e.g.,
to compute feedback laws that stabilize the Navier-Stokes equations. Challenges
in the numerical solution of the Riccati equation arise from the large-scale of
the underlying systems and the algebraic constraint in the DAE system. These
challenges are met by a careful extension of the inexact low-rank Newton-ADI
method to the case of DAE systems. A main ingredient in the extension to the
DAE case is the projection onto the manifold described by the algebraic
constraints. In the algorithm, the equations are never explicitly projected,
but the projection is only applied as needed. Numerical experience indicates
that the algorithmic choices for the control of inexactness and line-search can
help avoid subproblems with matrices that are only marginally stable. The
performance of the algorithm is illustrated on a large-scale Riccati equation
associated with the stabilization of Navier-Stokes flow around a cylinder.Comment: 21 pages, 2 figures, 4 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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