A generating set search method exploiting curvature and sparsity

Generating Set Search method are one of the few alternatives for optimising high fidelity functions with numerical noise. These methods are usually only efficient when the number of variables is relatively small. This paper presents a modification to an existing Generating Set Search method, which makes it aware of the sparsity structure of the Hessian. … Read more

Finding optimal algorithmic parameters using a mesh adaptive direct search

The objectives of this paper are twofold; we first demonstrate the flexibility of the mesh adaptive direct search (MADS) in identifying locally optimal algorithmic parameters. This is done by devising a general framework for parameter tuning. The framework makes provision for surrogate objectives. Parameters are sought so as to minimize some measure of performance of … Read more

A shifted Steihaug-Toint method for computing a trust-region step.

Trust-region methods are very convenient in connection with the Newton method for unconstrained optimization. The More-Sorensen direct method and the Steihaug-Toint iterative method are most commonly used for solving trust-region subproblems. We propose a method which combines both of these approaches. Using the small-size Lanczos matrix, we apply the More-Sorensen method to a small-size trust-region … Read more

Sensitivity of trust-region algorithms on their parameters

In this paper, we examine the sensitivity of trust-region algorithms on the parameters related to the step acceptance and update of the trust region. We show, in the context of unconstrained programming, that the numerical efficiency of these algorithms can easily be improved by choosing appropriate parameters. Recommanded ranges of values for these parameters are … Read more

Recursive Trust-Region Methods for Multilevel Nonlinear Optimization (Part I): Global Convergence and Complexity

A class of trust-region methods is presented for solving unconstrained nonlinear and possibly nonconvex discretized optimization problems, like those arising in systems governed by partial differential equations. The algorithms in this class make use of the discretization level as a mean of speeding up the computation of the step. This use is recursive, leading to … Read more

Additional properties of shifted valiable metric methods.

Some supplements to shifted variable metric or quasi-Newton methods for unconstrained minimization are given, including new limited-memory methods. Global convergence of these methods can be established for convex sufficiently smooth functions. Some encouraging numerical experience is reported. CitationReport No. V899-03, Institute of Computer Scienc, Czech Academy of Sciences, Prague, December 2003 (revised May 2004).ArticleDownload View … Read more

On the Relationship Between Convergence Rates of Discrete and Continuous Dynamical Systems

Considering iterative sequences that arise when the approximate solution $x_k$ to a numerical problem is updated by $x_{k+1} = x_k+v(x_k)$, where $v$ is a vector field, we derive necessary and sufficient conditions for such discrete methods to converge to a stationary point of $v$ at different Q-rates in terms of the differential properties of $v$ … Read more

Semidefinite Approximations for Global Unconstrained Polynomial Optimization

We consider here the problem of minimizing a polynomial function on $\oR^n$. The problem is known to be hard even for degree $4$. Therefore approximation algorithms are of interest. Lasserre \cite{lasserre:2001} and Parrilo \cite{Pa02a} have proposed approximating the minimum of the original problem using a hierarchy of lower bounds obtained via semidefinite programming relaxations. We … Read more

A filter-trust-region method for unconstrained optimization

A new filter-trust-region algorithm for solving unconstrained nonlinear optimization problems is introduced. Based on the filter technique introduced by Fletcher and Leyffer, it extends an existing technique of Gould, Leyffer and Toint (SIAM J. Optim., to appear 2004) for nonlinear equations and nonlinear least-squares to the fully general unconstrained optimization problem. The new algorithm is … Read more

Optimal Direct Determination of Sparse Jacobian Matrices

It is well known that a sparse Jacobian matrix can be determined with fewer function evaluations or automatic differentiation \emph{passes} than the number of independent variables of the underlying function. In this paper we show that by grouping together rows into blocks one can reduce this number further. We propose a graph coloring technique for … Read more