1,487 research outputs found
On the global convergence of the inexact semi-smooth Newton method for absolute value equation
In this paper, we investigate global convergence properties of the inexact
nonsmooth Newton method for solving the system of absolute value equations
(AVE). Global -linear convergence is established under suitable assumptions.
Moreover, we present some numerical experiments designed to investigate the
practical viability of the proposed scheme
Inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations
In this paper, we propose a new method that combines the inexact Newton
method with a procedure to obtain a feasible inexact projection for solving
constrained smooth and nonsmooth equations. The local convergence theorems are
established under the assumption of smoothness or semismoothness of the
function that defines the equation and its regularity at the solution. In
particular, we show that a sequence generated by the method converges to a
solution with linear, superlinear, or quadratic rate, under suitable
conditions. Moreover, some numerical experiments are reported to illustrate the
practical behavior of the proposed method
Level-set methods for convex optimization
Convex optimization problems arising in applications often have favorable
objective functions and complicated constraints, thereby precluding first-order
methods from being immediately applicable. We describe an approach that
exchanges the roles of the objective and constraint functions, and instead
approximately solves a sequence of parametric level-set problems. A
zero-finding procedure, based on inexact function evaluations and possibly
inexact derivative information, leads to an efficient solution scheme for the
original problem. We describe the theoretical and practical properties of this
approach for a broad range of problems, including low-rank semidefinite
optimization, sparse optimization, and generalized linear models for inference.Comment: 38 page
A semi-smooth Newton method for projection equations and linear complementarity problems with respect to the second order cone
In this paper a special semi-smooth equation associated to the second order
cone is studied. It is shown that, under mild assumptions, the semi-smooth
Newton method applied to this equation is well-defined and the generated
sequence is globally and Q-linearly convergent to a solution. As an
application, the obtained results are used to study the linear second order
cone complementarity problem, with special emphasis on the particular case of
positive definite matrices. Moreover, some computational experiments designed
to investigate the practical viability of the method are presented.Comment: 18 page
Inexact Newton Methods for Stochastic Nonconvex Optimization with Applications to Neural Network Training
We study stochastic inexact Newton methods and consider their application in
nonconvex settings. Building on the work of [R. Bollapragada, R. H. Byrd, and
J. Nocedal, IMA Journal of Numerical
Analysis, 39 (2018), pp. 545--578] we derive bounds for convergence rates in
expected value for stochastic low rank Newton methods, and stochastic inexact
Newton Krylov methods. These bounds quantify the errors incurred in subsampling
the Hessian and gradient, as well as in approximating the Newton linear solve,
and in choosing regularization and step length parameters. We deploy these
methods in training convolutional autoencoders for the MNIST and CIFAR10 data
sets. Numerical results demonstrate that, relative to first order methods,
these stochastic inexact Newton methods often converge faster, are more
cost-effective, and generalize better
An inexact Douglas-Rachford splitting method for solving absolute value equations
The last two decades witnessed the increasing of the interests on the
absolute value equations (AVE) of finding such that
, where and . In
this paper, we pay our attention on designing efficient algorithms. To this
end, we reformulate AVE to a generalized linear complementarity problem (GLCP),
which, among the equivalent forms, is the most economical one in the sense that
it does not increase the dimension of the variables. For solving the GLCP, we
propose an inexact Douglas-Rachford splitting method which can adopt a relative
error tolerance. As a consequence, in the inner iteration processes, we can
employ the LSQR method ([C.C. Paige and M.A. Saunders, ACM Trans. Mathe. Softw.
(TOMS), 8 (1982), pp. 43--71]) to find a qualified approximate solution for
each subproblem, which makes the cost per iteration very low. We prove the
convergence of the algorithm and establish its global linear rate of
convergence. Comparing results with the popular algorithms such as the exact
generalized Newton method [O.L. Mangasarian, Optim. Lett., 1 (2007), pp. 3--8],
the inexact semi-smooth Newton method [J.Y.B. Cruz, O.P. Ferreira and L.F.
Prudente, Comput. Optim. Appl., 65 (2016), pp. 93--108] and the exact SOR-like
method [Y.-F. Ke and C.-F. Ma, Appl. Math. Comput., 311 (2017), pp. 195--202]
are reported, which indicate that the proposed algorithm is very promising.
Moreover, our method also extends the range of numerically solvable of the AVE;
that is, it can deal with not only the case that , the commonly
used in those existing literature, but also the case where .Comment: 25 pages, 3 figures, 3 table
Physarum Dynamics and Optimal Transport for Basis Pursuit
We study the connections between Physarum Dynamics and Dynamic Monge
Kantorovich (DMK) Optimal Transport algorithms for the solution of Basis
Pursuit problems. We show the equivalence between these two models and unveil
their dynamic character by showing existence and uniqueness of the solution for
all times and constructing a Lyapunov functional with negative Lie-derivative
that drives the large-time convergence. We propose a discretization of the
equation by means of a combination of implicit time-stepping and Newton method
yielding an efficient and robust method for the solution of general basis
pursuit problems. Several numerical experiments run on literature benchmark
problems are used to show the accuracy, efficiency, and robustness of the
proposed method
A class of inexact modified Newton-type iteration methods for solving the generalized absolute value equations
In Wang et al. (J. Optim. Theory Appl., \textbf{181}: 216--230, 2019), a
class of effective modified Newton-tpye (MN) iteration methods are proposed for
solving the generalized absolute value equations (GAVE) and it has been found
that the MN iteration method involves the classical Picard iteration method as
a special case. In the present paper, it will be claimed that a
Douglas-Rachford splitting method for AVE is also a special case of the MN
method. In addition, a class of inexact MN (IMN) iteration methods are
developed to solve GAVE. Linear convergence of the IMN method is established
and some specific sufficient conditions are presented for symmetric positive
definite coefficient matrix. Numerical results are given to demonstrate the
efficiency of the IMN iteration method.Comment: 14 pages, 4 table
An Optimal Block Diagonal Preconditioner for Heterogeneous Saddle Point Problems in Phase Separation
The phase separation processes are typically modeled by Cahn-Hilliard
equations. This equation was originally introduced to model phase separation in
binary alloys, where phase stands for concentration of different components in
alloy. When the binary alloy under preparation is subjected to a rapid
reduction in temperature below a critical temperature, it has been
experimentally observed that the concentration changes from a mixed state to a
visibly distinct spatially separated two phase for binary alloy. This rapid
reduction in the temperature, the so-called "deep quench limit", is modeled
effectively by obstacle potential. The discretization of Cahn-Hilliard equation
with obstacle potential leads to a block {\em non-linear} system,
where the block has a non-linear and non-smooth term. Recently a
globally convergent Newton Schur method was proposed for the non-linear Schur
complement corresponding to this non-linear system. The proposed method is
similar to an inexact active set method in the sense that the active sets are
first approximately identified by solving a quadratic obstacle problem
corresponding to the block of the block system, and later
solving a reduced linear system by annihilating the rows and columns
corresponding to identified active sets. For solving the quadratic obstacle
problem, various optimal multigrid like methods have been proposed. In this
paper, we study a non-standard norm that is equivalent to applying a block
diagonal preconditioner to the reduced linear systems. Numerical experiments
confirm the optimality of the solver and convergence independent of problem
parameters on sufficiently fine mesh.Comment: 2 figure
Implementation of the ADMM to Parabolic Optimal Control Problems with Control Constraints and Beyond
Parabolic optimal control problems with control constraints are generally
challenging, from either theoretical analysis or algorithmic design
perspectives. Conceptually, the well-known alternating direction method of
multipliers (ADMM) can be directly applied to such a problem. An attractive
advantage of this direct ADMM application is that the control constraint can be
untied from the parabolic PDE constraint; these two inherently different
constraints thus can be treated individually in iterations. At each iteration
of the ADMM, the main computation is for solving an unconstrained parabolic
optimal control problem. Because of its high dimensionality after
discretization, the unconstrained parabolic optimal control problem at each
iteration can be solved only inexactly by implementing certain numerical scheme
internally and thus a two-layer nested iterative scheme is required. It then
becomes important to find an easily implementable and efficient inexactness
criterion to execute the internal iterations, and to prove the overall
convergence rigorously for the resulting two-layer nested iterative scheme. To
implement the ADMM efficiently, we propose an inexactness criterion that is
independent of the mesh size of the involved discretization, and it can be
executed automatically with no need to set empirically perceived constant
accuracy a prior. The inexactness criterion turns out to allow us to solve the
resulting unconstrained optimal control problems to medium or even low accuracy
and thus saves computation significantly, yet convergence of the overall
two-layer nested iterative scheme can be still guaranteed rigorously.
Efficiency of this ADMM implementation is promisingly validated by preliminary
numerical results. Our methodology can also be extended to a range of optimal
control problems constrained by other linear PDEs such as elliptic equations
and hyperbolic equations
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