3,962 research outputs found
On memory gradient method with trust region for unconstrained optimization
In this paper we present a new memory gradient method with trust region for unconstrained optimization problems. The method combines line search method and trust region method to generate new iterative points at each iteration and therefore has both advantages of line search method and trust region method. It sufficiently uses the previous multi-step iterative information at each iteration and avoids the storage and computation of matrices associated with the Hessian of objective functions, so that it is suitable to solve large scale optimization problems. We also design an implementable version of this method and analyze its global convergence under weak conditions. This idea enables us to design some quick convergent, effective, and robust algorithms since it uses more information from previous iterative steps. Numerical experiments show that the new method is effective, stable and robust in practical computation, compared with other similar methods.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45437/1/11075_2005_Article_9008.pd
Limited-memory BFGS Systems with Diagonal Updates
In this paper, we investigate a formula to solve systems of the form (B +
{\sigma}I)x = y, where B is a limited-memory BFGS quasi-Newton matrix and
{\sigma} is a positive constant. These types of systems arise naturally in
large-scale optimization such as trust-region methods as well as
doubly-augmented Lagrangian methods. We show that provided a simple condition
holds on B_0 and \sigma, the system (B + \sigma I)x = y can be solved via a
recursion formula that requies only vector inner products. This formula has
complexity M^2n, where M is the number of L-BFGS updates and n >> M is the
dimension of x
Optimization Methods for Inverse Problems
Optimization plays an important role in solving many inverse problems.
Indeed, the task of inversion often either involves or is fully cast as a
solution of an optimization problem. In this light, the mere non-linear,
non-convex, and large-scale nature of many of these inversions gives rise to
some very challenging optimization problems. The inverse problem community has
long been developing various techniques for solving such optimization tasks.
However, other, seemingly disjoint communities, such as that of machine
learning, have developed, almost in parallel, interesting alternative methods
which might have stayed under the radar of the inverse problem community. In
this survey, we aim to change that. In doing so, we first discuss current
state-of-the-art optimization methods widely used in inverse problems. We then
survey recent related advances in addressing similar challenges in problems
faced by the machine learning community, and discuss their potential advantages
for solving inverse problems. By highlighting the similarities among the
optimization challenges faced by the inverse problem and the machine learning
communities, we hope that this survey can serve as a bridge in bringing
together these two communities and encourage cross fertilization of ideas.Comment: 13 page
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