Optimization with Gradient and Hessian Information Calculated Using Hyper-Dual Numbers Jeffrey A. Fike and Juan J. Alonso Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, U.S.A. Sietse Jongsma and Edwin van der Weide Department of Engineering Technology, University of Twente, the Netherlands 29th AIAA Applied Aerodynamics Conference Honolulu, Hawaii … The Hessian matrix has a number of important applications in a variety of di erent elds, such as optimzation, image processing and statistics.

A Hessian matrix or simply a Hessian is a matrix of all the second-order partial derivatives of a function $ f:\R^n\to\R $.For example, given the function $ f(x,y)=4xy+y^2 $ The resulting Hessian is $ \bold{H}(f)=\begin{bmatrix}0&4+2y\\4+2y&2\end{bmatrix} $ The Hessian matrix will be symmetric if the partial derivatives of the function are continuous.. Mathematically, this matrix is defined to hold the information about every possible second order derivative (here shown in the 2D case) 3: In point-free notation, we denote by the Hessian matrix function, and we define it as: Interpretation as second derivative . Hessian-Laplace Feature Detector and Haar Descriptor for Image Matching by Akshay Bhatia Thesis submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulflllment of the requirements For the M.A.Sc.
He restricts the inde nite metric 1=d(d 1)@2f=@x i@x j to the hypersurface M:= ff= 1gin the index cone, where it is a Riemannian metric, which he calls the Hodge metric.

For the Hessian, this implies the stationary point is a saddle point. A function f: R → R f:\mathbb{R}\to\mathbb{R} f: R → R whose second order partial derivatives are well defined in it's domain so we can have the Hessian matrix of f f f. Note that the Hessian matrix here is always symmetric. Wilson considers the \index cone," the open subset where the Hessian matrix of f is Lorentzian (that is, of signature (1;)) and f is positive. If any of the leading principal minors is zero, then a separate analysis (to investigate whether the matrix could be positive semi-definite or negative semi-definite) is needed.

1 Hessian of trace of some matrix functions Let F: R n!R n be some matrix function and ˚: R n!R be its trace ˚(X) = tr F(X): (1) We are interested in the Hessian of ˚: H˚(X) = @2˚(X) @vec(X)@vec(X)T: (2) In the derivation, the following matrix serves as a handy tool to transform between vec(A) and vec(AT). The Hessian matrix of is a -matrix-valued function with domain a subset of the domain of , defined as follows: the Hessian matrix at any point in the domain is the Jacobian matrix of the gradient vector of at the point.


The Hessian matrix.

Ge-ometrically, the Hessian matrix de- scribes the local curvature of scalar functions f: RP!R, and is for this reason perhaps mostly known in the eld of optimization [5].

Usually Hessian in two variables are easy and interesting to look for. the matrix is indefinite.


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