MANIFOLD : the Rosenbrock ' s Function Case

Steepest descent method on a Riemannian manifold: the convex case

In this paper we are interested in the asymptotic behavior of the trajectories of the famous steepest descent evolution equation on Riemannian manifolds. It writes ẋ (t) + gradφ (x (t)) = 0. It is shown how the convexity of the objective function φ helps in establishing the convergence as time goes to infinity of the trajectories towards points that minimize φ. Some numerical illustrations are ...

The Nehari Manifold for a Class of Indefinite Weight Semilinear Elliptic Equations

A Modified Algorithm of Steepest Descent Method for Solving Unconstrained Nonlinear Optimization Problems

The steepest descent method (SDM), which can be traced back to Cauchy (1847), is the simplest gradient method for unconstrained optimization problem. The SDM is effective for well-posed and low-dimensional nonlinear optimization problems without constraints; however, for a large-dimensional system, it converges very slowly. Therefore, a modified steepest decent method (MSDM) is developed to dea...

A note about the complexity of minimizing Nesterov's smooth Chebyshev-Rosenbrock function

This short note considers and resolves the apparent contradiction between known worst-case complexity results for first and second-order methods for solving unconstrained smooth nonconvex optimization problems and a recent note by Jarre (2011) implying a very large lower bound on the number of iterations required to reach the solution’s neighbourhood for a specific problem with variable dimension.

On Rosenbrock models over a commutative ring

Rosenbrock’s notion of system equivalence is general in nature; it is a kind of equivalence which in algebra is often termed as stable. We have shown recently that Fuhrmann’s notion of system equivalence can be viewed as a homotopy equivalence, and as such is also general in nature. This note deals with a generalization of the theory of system equivalences from the field case to the commutative...