The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best uniform step size. But other, much faster methods usi...

The satissability (SAT) problem is a basic problem in computing theory. Presently, an active area of research on SAT problem is to design eecient optimization algorithms for nding a solution for a satissable CNF formula. A new formulation, the Universal SAT problem model, which transforms the SAT problem on Boolean space into an optimization problem on real space has been developed 31, 35, 34, ...

In recent years it has been made more and more clear that the critical issue in gradient methods is the choice of the step length, whereas using the gradient as search direction may lead to very effective algorithms, whose surprising behaviour has been only partially explained, mostly in terms of the spectrum of the Hessian matrix. On the other hand, the convergence of the classical Cauchy stee...

Among the compact homogeneous spaces, a very distinguished subclass is formed by the (generalized) real flag manifolds which by definition are the orbits of the isotropy representations of Riemannian symmetric spaces (sorbits). This class contains most compact symmetric spaces (e.g. all hermitian ones), all classical flag manifolds over real, complex and quaternionic vector spaces, all adjoint ...

For p 2 (1; 1) and n > 0 we consider the scalar doubly degenerate diiusion equation @ t s ? div(jrs n j p?2 rs n) = 0 in (0; 1) (1) with no{{ux boundary conditions. We argue that this evolution problem can be understood as steepest descent of the convex functional sign(m ? 1) Z s m ; provided m := n + p ? 2 p ? 1 > 0 ; (2) w. r. t. the Wasserstein metric of order p on the space of probability d...

We consider the special case of the restarted Arnoldi method for approximating the product of a function of a Hermitian matrix with a vector which results when the restart length is set to one. When applied to the solution of a linear system of equations, this approach coincides with the method of steepest descent. We show that the method is equivalent with an interpolation process in which the...