Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex coneH of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are eq...

We consider variants of trust-region and cubic regularization methods for nonconvex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve ǫ-approximate second-order optimality which have shown to be tight. Our Hessian approximation condi...

We describe computational experience with automatic differentiation of mathematical programming problems expressed in the modeling language AMPL. Nonlinear expressions are translated to loop-free code, which makes it easy to compute gradients and Jacobians by backward automatic differentiation. The nonlinear expressions may be interpreted or, to gain some evaluation speed at the cost of increas...

To efficiently solve a large scale unconstrained minimization problem with a dense Hessian matrix, this paper proposes to use an incomplete Hessian matrix to define a new modified Newton method, called the incomplete Hessian Newton method (IHN). A theoretical analysis shows that IHN is convergent globally, and has a linear rate of convergence with a properly selected symmetric, positive definit...

Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001], we address the ill-posedness of the inverse scattering problem of electromagnetic waves due to an inhomogeneous medium by studying the Hessian of the data misfit. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton...

in this paper, we consider a class of connected oriented (with respect to z/p) closed g-manifolds with a non-empty finite fixed point set, each of which is g-equivariantly formal, where g = z/p and p is an odd prime. using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a g-manifold in terms of algebra. this makes it p...

This paper proposes a modification to the simultaneous per tu rba t ion stochastic approximation (SPSA) methods based on the comparisons made between the first o rder and the second order SPSA (1SPSA and 2SPSA) algori thms f rom the perspective of loss function Hessian. At finite iterations, the convergence rate depends on the matr ix conditioning of the loss function Hessian. It is shown that ...

Abstract We describe the first gradient methods on Riemannian manifolds to achieve accelerated rates in non-convex case. Under Lipschitz assumptions and Hessian of cost function, these find approximate first-order critical points faster than regular descent. A randomized version also finds second-order points. Both algorithms their analyses build extensively existing work Euclidean The basic op...

The score and hessian functions The score vector ∇L(β, x) and hessian matrix ∇ 2 L(β, x) used in the Newton-Raphson

in this paper, we obtain a necessary and sufficient condition for a conformal mapping between two weyl manifolds to preserve einstein tensor. then we prove that some basic curvature tensors of $w_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. also, we obtained the relation between the scalar curvatures of the weyl manifolds r...