With the objective of generating “shape-preserving” smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based ${\cal C}^1$-smooth univariate cubic $L_1$ splines. An $L_1$ spline minimizes the $L_1$ norm of the difference between the first-order derivative of the spline and the local divided difference of … Read more
Generating set search (GSS) is a family of direct search methods that encompasses generalized pattern search and related methods. We describe an algorithm for asynchronous linearly-constrained GSS, which has some complexities that make it different from both the asynchronous bound-constrained case as well as the synchronous linearly-constrained case. The algorithm has been implemented in the … Read more
We prove that if two smooth manifolds intersect transversally, then the method of alternating projections converges locally at a linear rate. We bound the speed of convergence in terms of the angle between the manifolds, which in turn we relate to the modulus of metric regularity for the intersection problem, a natural measure of conditioning. … Read more
We use the proximal point method with the phi-divergence given by phi(t) = t – log t – 1 for the minimization of quasiconvex functions subject to nonnegativity constraints. We establish that the sequence generated by our algorithm is well-defined in the sense that it exists and it is not cyclical. Without any assumption of … Read more
It has been known since the early 1970s that the Hessian matrices in quasi-Newton methods can be updated by variational means, in several different ways. The usual formulation of these variational problems uses a coordinate system, and the symmetry of the Hessian matrices are enforced as explicit constraints. As a result, the variational problems seem … Read more
Given a set of points $\cS = \{x^1,\ldots,x^m\} \subset \R^n$ and $\eps > 0$, we propose and analyze an algorithm for the problem of computing a $(1 + \eps)$-approximation to the the minimum volume axis-aligned ellipsoid enclosing $\cS$. We establish that our algorithm is polynomial for fixed $\eps$. In addition, the algorithm returns a small … Read more
We consider an implementation of the recursive multilevel trust-region algorithm proposed by Gratton, Sartenaer, Toint (2004), and provide significant numerical experience on multilevel test problems. A suitable choice of the algorithm’s parameters is identified on these problems, yielding a very satisfactory compromise between reliability and efficiency. The resulting default algorithm is then compared to alternative … Read more
In this paper we propose an extension of the proximal point method to solve minimization problems with quasiconvex objective functions on the Euclidean space and the nonnegative orthant. For the unconstrained minimization problem, assumming that the function is bounded from below and lower semicontinuous we prove that iterations fxkg given by 0 2 b@(f(:)+(k=2)jj:xk1jj2)(xk) are … Read more
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. Citation19 June 2006ArticleDownload View PDF
We discuss local convergence of Newton’s method to a singular solution $x^*$ of the nonlinear equations $F(x) = 0$, for $F:\R^n \rightarrow \R^n$. It is shown that an existing proof of Griewank, concerning linear convergence to a singular solution $x^*$ from a starlike domain around $x^*$ for $F$ twice Lipschitz continuously differentiable and $x^*$ satisfying … Read more