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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.
Positive-definite and positive-semidefinite real matrices are at the basis of convex optimization, since, given a function of several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point , then the function is convex near p, and, conversely, if the ...
This update maintains the symmetry and positive definiteness of the Hessian matrix. Given a function f ( x ) {\displaystyle f(x)} , its gradient ( ∇ f {\displaystyle \nabla f} ), and positive-definite Hessian matrix B {\displaystyle B} , the Taylor series is
One can, for example, modify the Hessian by adding a correction matrix so as to make ″ + positive definite. One approach is to diagonalize the Hessian and choose B k {\displaystyle B_{k}} so that f ″ ( x k ) + B k {\displaystyle f''(x_{k})+B_{k}} has the same eigenvectors as the Hessian, but with each negative eigenvalue replaced by ϵ > 0 ...
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M,
This update maintains the symmetry of the matrix but does not guarantee that the update be positive definite. The sequence of Hessian approximations generated by the SR1 method converges to the true Hessian under mild conditions, in theory; in practice, the approximate Hessians generated by the SR1 method show faster progress towards the true ...
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.