560 research outputs found
Multipoint secant and interpolation methods with nonmonotone line search for solving systems of nonlinear equations
Multipoint secant and interpolation methods are effective tools for solving
systems of nonlinear equations. They use quasi-Newton updates for approximating
the Jacobian matrix. Owing to their ability to more completely utilize the
information about the Jacobian matrix gathered at the previous iterations,
these methods are especially efficient in the case of expensive functions. They
are known to be local and superlinearly convergent. We combine these methods
with the nonmonotone line search proposed by Li and Fukushima (2000), and study
global and superlinear convergence of this combination. Results of numerical
experiments are presented. They indicate that the multipoint secant and
interpolation methods tend to be more robust and efficient than Broyden's
method globalized in the same way
On the relationship between bilevel decomposition algorithms and direct interior-point methods
Engineers have been using bilevel decomposition algorithms to solve certain nonconvex large-scale optimization problems arising in engineering design projects. These algorithms transform the large-scale problem into a bilevel program with one upperlevel problem (the master problem) and several lower-level problems (the subproblems). Unfortunately, there is analytical and numerical evidence that some of these commonly used bilevel decomposition algorithms may fail to converge even when the starting point is very close to the minimizer. In this paper, we establish a relationship between a particular bilevel decomposition algorithm, which only performs one iteration of an interior-point method when solving the subproblems, and a direct interior-point method, which solves the problem in its original (integrated) form. Using this relationship, we formally prove that the bilevel decomposition algorithm converges locally at a superlinear rate. The relevance of our analysis is that it bridges the gap between the incipient local convergence theory of bilevel decomposition algorithms and the mature theory of direct interior-point methods
Local convergence of quasi-Newton methods under metric regularity
We consider quasi-Newton methods for generalized equations in Banach spaces under metric regularity and give a sufficient condition for q-linear convergence. Then we show that the well-known Broyden update satisfies this sufficient condition in Hilbert spaces. We also establish various modes of q-superlinear convergence of the Broyden update under strong metric subregularity, metric regularity and strong metric regularity. In particular, we show that the Broyden update applied to a generalized equation in Hilbert spaces satisfies the Dennis–Moré condition for q-superlinear convergence. Simple numerical examples illustrate the results.A. Belyakov was supported by the Austrian Science Foundation (FWF) under grant No P 24125-N13. A.L. Dontchev was supported by NSF Grant DMS 1008341 through the University of Michigan. M. López was supported by MINECO of Spain, Grant MTM2011-29064-C03-02
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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