Search results
Results from the WOW.Com Content Network
The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point x {\displaystyle x} is a local maximum, local minimum, or a saddle point, as follows:
The following test can be applied at any critical point a for which the Hessian matrix is invertible: If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a. If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a.
A twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix of second ... matrix is a convex function ...
A multivariate polynomial is SOS-convex (or sum of squares convex) if its Hessian matrix H can be factored as H(x) = S T (x)S(x) where S is a matrix (possibly rectangular) which entries are polynomials in x. [1] In other words, the Hessian matrix is a SOS matrix polynomial.
There also exist various quasi-Newton methods, where an approximation for the Hessian (or its inverse directly) is built up from changes in the gradient. If the Hessian is close to a non-invertible matrix, the inverted Hessian can be numerically unstable and the solution may diverge. In this case, certain workarounds have been tried in the past ...
When is a convex quadratic function with positive-definite Hessian , one would expect the matrices generated by a quasi-Newton method to converge to the inverse Hessian =. This is indeed the case for the class of quasi-Newton methods based on least-change updates.
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector , where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
Since a quadratic has a diagonal Hessian matrix, this superposition essentially adds a number to all diagonal elements of the original Hessian, such that the resulting Hessian is positive-semidefinite. Thus, the resulting relaxation is a convex function.