HessianKriterien

HessianKriterien, often called the Hessian criterion, refers to a set of criteria used in multivariable calculus and optimization to classify critical points of smooth functions by analyzing the Hessian matrix. The Hessian is the matrix of second-order partial derivatives and is symmetric when the function is twice continuously differentiable. If the gradient is zero at a point x*, this point is a candidate for a local extremum, and the definiteness of the Hessian at x* determines the type of extremum.

The core idea is definiteness. If the Hessian at x* is positive definite, the point is a strict local minimum. If it is negative definite, the point is a strict local maximum. If the Hessian is indefinite, the point is a saddle point. If the Hessian is positive semidefinite or negative semidefinite, the test is inconclusive, and higher-order derivatives or alternative methods may be required to reach a conclusion.

Practically, to check definiteness one can examine eigenvalues of the Hessian or use Sylvester’s criterion, which involves leading principal minors. In the two-variable case, with f_xx, f_xy, and f_yy as second partial derivatives, define D = f_xx f_yy − (f_xy)^2. If D > 0 and f_xx > 0, there is a local minimum; if D > 0 and f_xx < 0, there is a local maximum; if D < 0, the point is a saddle; if D = 0, the test is inconclusive.

Beyond classification, Hessian criteria relate to convexity: if the Hessian is positive semidefinite for all points in a domain, the function is convex; if it is positive definite everywhere, it is strictly convex. The Hessian criterion is a local tool; global conclusions often require additional structure of the function.