Accelerated line-search and trust-region methods

In numerical optimization, line-search and trust-region methods are two important classes of descent schemes, with well-understood global convergence properties. Here we consider “accelerated” versions of these methods, where the conventional iterate is allowed to be replaced by any point that produces at least as much decrease in the cost function as a fixed fraction of … Read more

H2-optimal model reduction of MIMO systems

We consider the problem of approximating a $p\times m$ rational transfer function $H(s)$ of high degree by another $p\times m$ rational transfer function $\hat{H}(s)$ of much smaller degree. We derive the gradients of the $H_2$-norm of the approximation error and show how stationary points can be described via tangential interpolation. Citation Technical report UCL-INMA-2007.034, Department … Read more

An implicit trust-region method on Riemannian manifolds

We propose and analyze an “implicit” trust-region method in the general setting of Riemannian manifolds. The method is implicit in that the trust-region is defined as a superlevel set of the ratio of the actual over predicted decrease in the objective function. Since this method potentially requires the evaluation of the objective function at each … Read more

Convergence analysis of Riemannian trust-region methods

A general scheme for trust-region methods on Riemannian manifolds is proposed and analyzed. Among the various approaches available to (approximately) solve the trust-region subproblems, particular attention is paid to the truncated conjugate-gradient technique. The method is illustrated on problems from numerical linear algebra. Citation 19 June 2006 Article Download View Convergence analysis of Riemannian trust-region … Read more

Constraint Reduction for Linear Programs with Many Inequality Constraints

Consider solving a linear program in standard form, where the constraint matrix $A$ is $m \times n$, with $n \gg m \gg 1$. Such problems arise, for example, as the result of finely discretizing a semi-infinite program. The cost per iteration of typical primal-dual interior-point methods on such problems is $O(m^2n)$. We propose to reduce … Read more

Newton-KKT Interior-Point Methods for Indefinite Quadratic Programming

Two interior-point algorithms are proposed and analyzed, for the (local) solution of (possibly) indefinite quadratic programming problems. They are of the Newton-KKT variety in that (much like in the case of primal-dual algorithms for linear programming) search directions for the `primal´ variables and the Karush-Kuhn-Tucker (KKT) multiplier estimates are components of the Newton (or quasi-Newton) … Read more