arXiv subjects

Aban Ansari-Önnestam

Publications and source records attributed to Aban Ansari-Önnestam.

3 recordsLinked to original sources

arXiv2026

On Modified Newton Methods for Unconstrained Optimization Based on Lanczos Tridiagonalization

For a strongly convex unconstrained minimization problem, Newton's method with a line search offers rapid convergence to a minimizer by incorporating second-order information. However, for a general unconstrained problem the Hessian need not be positive definite, which can lead to the inability to find a search direction or to search directions that fail to produce descent. Modified Newton methods are designed to handle this situation. This paper considers modifications to a Newton-conjugate-gradient approach based on Lanczos tridiagonalization. Existing methods are extended using dynamic modification schemes that provide a descent direction while keeping the modification bounded. A Lanczos-based scheme and a conjugate-gradient (CG)-based scheme are formulated and analyzed. These methods are mathematically equivalent in exact arithmetic, but their practical behavior differs in finite precision. Finally, it is shown how the convergence and computational cost are affected by varying the amount of additional curvature introduced and by allowing the early termination of the inner iterations.

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arXiv2026

Finite Newton-Step Generation via Subspace Quasi-Newton Methods for Quadratic Problems

We study the curvature information sufficient for finite Newton-step generation by quasi-Newton methods on strictly convex quadratic problems. In exact arithmetic, the memoryless BFGS quasi-Newton method terminates finitely on such a problem when exact line search is used, although its quasi-Newton approximation incorporates curvature information associated with only one direction. We show that exact line search can be replaced by curvature information associated with one additional direction. Information associated with at most two directions suffices to generate the Newton step after finitely many iterations, independently of the preceding step lengths. More precisely, knowledge of the action of the Hessian on a suitably chosen subspace of dimension at most two is sufficient to generate a step that decomposes into a subspace Newton step and a component parallel to the next conjugate direction. Applying this construction recursively generates the full Newton step after finitely many iterations, after which a unit step reaches the minimizer. The required Hessian actions can also be recovered from gradient differences, yielding a first-order formulation. The resulting algorithm is intended as a constructive device for identifying sufficient curvature information, rather than as a practical alternative to the method of conjugate gradients.

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arXiv2025

Adaptive Gradient Descent on Riemannian Manifolds with Nonnegative Curvature

In this paper, we present an adaptive gradient descent method for geodesically convex optimization on a Riemannian manifold with nonnegative sectional curvature. The method automatically adapts to the local geometry of the function and does not use additional expensive computations other than calculation of the derivative of the Riemannian exponential. We prove the convergence of the method under the assumption of geodesic completeness. The performance of the method is illustrated by experiments on the sphere, the manifold of symmetric positive definite matrices equipped with the Bures-Wasserstein metric.

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