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...

Quadratic programs (QP) with an indefinite Hessian matrix arise naturally in some direct optimal control methods, e.g. as subproblems in a sequential quadratic programming (SQP) scheme. Typically, the Hessian is approximated with a positive definite matrix to ensure having a unique solution; such a procedure is called regularization. We present a novel regularization method tailored for QPs wit...

The problem of recovering a parameter function based on measurements of solutions of a system of partial diierential equations in several space variables leads to a number of computational challenges. Upon discretization of a regularized formulation a large, sparse constrained optimization problem is obtained. Typically in the literature , the constraints are eliminated and the resulting uncons...

Improvements to the optimal control Newton-Raphson GRAPE method for exact Hessian calculations are shown with examples of auxiliary matrix and ESCALADE methods. For an ensemble two-level systems realistic MRI conditions, new accelerated between 200 600 times faster ESCALADE, respectively.

In this paper, we propose the block-diagonal matrix to approximate the Hessian matrix in the Levenberg Mar-quardt method in the training of neural networks. Two weight updating strategies, namely asynchronous and synchronous updating methods were investigated. Asyn-chronous method updates weights of one block at a time while synchronous method updates all weights at the same time. Variations of...

In the social sciences, researchers typically assume the accuracy of generalized linear models by using an asymptotic normal approximation to the likelihood function or, occasionally, by using the full posterior distribution. Thus, for standard maximum likelihood analyses, only point estimates and the variance at the maximum are normally seen as necessary. For Bayesian posterior analysis, the m...

In this paper, a new ear anatomy feature edge extraction method based on Hessian matrix is proposed. Stable edge is obtained from principal curvature image across scale space. Firstly, the side face image that includes an ear is filtered and forms Gaussian pyramid. Secondly, the 2D gray image in the pyramid was regarded as a surface, maximum and minimum principal curvature and their direction w...

The problem of approximating the Pareto front a multiobjective optimization can be reformulated as finding set that maximizes hypervolume indicator. This paper establishes analytical expression Hessian matrix mapping from (fixed size) collection n points in d-dimensional decision space (or m dimensional objective space) to scalar indicator value. To define matrix, input is vectorized, and deriv...

An explicit formula based on matrix algebra to approximate the diagonal entries of a Hessian with any number sample points is introduced. When derivative-free technique called generalized centered simplex gradient used gradient, then can be computed for only one additional function evaluation. error bound introduced and provides information form set that should matrix. If built in specific mann...