Recently several methods were proposed for sparse optimization which make careful use of second-order information [11, 30, 17, 3] to improve local convergence rates. These methods construct a composite quadratic approximation using Hessian information, optimize this approximation using a first-order method, such as coordinate descent and employ a line search to ensure sufficient descent. Here … Read more
We provide a formula for variational quasi-Newton updates with multiple weighted secant equations. The derivation of the formula leads to a Sylvester equation in the correction matrix. Examples are given. CitationReport naXys-09-2013, Namur Centre for Complex Systems, Unibersity of Namur, Namur (Belgium)ArticleDownload View PDF
It is well known that the conjugate gradient method and a quasi-Newton method, using any well-defined update matrix from the one-parameter Broyden family of updates, produce identical iterates on a quadratic problem with positive-definite Hessian. This equivalence does not hold for any quasi-Newton method. We define precisely the conditions on the update matrix in the … Read more
We present a low-memory approach for the best-state estimate (data assimilation) of hidden Markov models where model error is considered. In particular, our findings apply for the 4D- Var framework. The novelty of our approach resides in the fact that the storage needed by our estimation framework, while including model error, is dramatically reduced from … Read more
The quasi-Newton strategy presented in this paper preserves one of the most important features of the stabilized Sequential Quadratic Programming method, the local convergence without constraint qualifications assumptions. It is known that the primal-dual sequence converges quadratically assuming only the second-order sufficient condition. In this work, we show that if the matrices are updated by … Read more
The properties of multilevel optimization problems defined on a hierarchy of discretization grids can be used to define approximate secant equations, which describe the second-order behaviour of the objective function. Following earlier work by Gratton and Toint (2009), we introduce a quasi-Newton method (with a linesearch) and a nonlinear conjugate gradient method that both take … Read more
We investigate the BFGS algorithm with an inexact line search when applied to nonsmooth functions, not necessarily convex. We define a suitable line search and show that it generates a sequence of nested intervals containing points satisfying the Armijo and weak Wolfe conditions, assuming only absolute continuity. We also prove that the line search terminates … Read more
We investigate the behavior of the BFGS algorithm with an exact line search on nonsmooth functions. We show that it may fail on a simple polyhedral example, but that it apparently always succeeds on the Euclidean norm function, spiraling into the origin with a Q-linear rate of convergence; we prove this in the case of … Read more
New approximate secant equations are shown to result from the knowledge of (problem dependent) invariant subspace information, which in turn suggests improvements in quasi-Newton methods for unconstrained minimization. It is also shown that this type of information may often be extracted from the multigrid structure of discretized infinite dimensional problems. A new limited-memory BFGS using … Read more
It is shown that a compact storage implementation of a quasi-Newton method based on the adjoint Broyden update reduces in the affine case exactly to the well established GMRES procedure. Generally, storage and linear algebra effort per step are small multiples of n k, where n is the number of variables and k the number … Read more