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Suppose is a function taking as input a vector and outputting a scalar If all second partial derivatives of exist and are continuous over the domain of the function, then the Hessian matrix of is a square matrix, usually defined and arranged as follows:
or, by stating an equation for the coefficients using indices i and j,
The Hessian matrix is a symmetric matrix, since the hypothesis of continuity of the second derivatives implies that the order of differentiation does not matter (Schwarz's theorem).
The determinant of the Hessian matrix is called the Hessian determinant.
If the Hessian is positive-definite at then attains an isolated local minimum at If the Hessian is negative-definite at then attains an isolated local maximum at If the Hessian has both positive and negative eigenvalues, then is a saddle point for Otherwise the test is inconclusive. This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite.
For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point). However, more can be said from the point of view of Morse theory.
The second-derivative test for functions of one and two variables is simpler than the general case. In one variable, the Hessian contains exactly one second derivative; if it is positive, then is a local minimum, and if it is negative, then is a local maximum; if it is zero, then the test is inconclusive. In two variables, the determinant can be used, because the determinant is the product of the eigenvalues. If it is positive, then the eigenvalues are both positive, or both negative. If it is negative, then the two eigenvalues have different signs. If it is zero, then the second-derivative test is inconclusive.
Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost) minors (determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization--the case in which the number of constraints is zero. Specifically, the sufficient condition for a minimum is that all of these principal minors be positive, while the sufficient condition for a maximum is that the minors alternate in sign, with the minor being negative.
If the gradient (the vector of the partial derivatives) of a function is zero at some point then has a critical point (or stationary point) at The determinant of the Hessian at is called, in some contexts, a discriminant. If this determinant is zero then is called a degenerate critical point of or a non-Morse critical point of Otherwise it is non-degenerate, and called a Morse critical point of
The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The eigenvalues of the Hessian at that point are the principle curvatures of the function, and the eigenvectors are the principle directions of curvature. (See Gaussian curvature § Relation to principal curvatures.)
Use in optimization
Hessian matrices are used in large-scale optimization problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,
Such approximations may use the fact that an optimization algorithm uses the Hessian only as a linear operator and proceed by first noticing that the Hessian also appears in the local expansion of the gradient:
Letting for some scalar this gives
so if the gradient is already computed, the approximate Hessian can be computed by a linear (in the size of the gradient) number of scalar operations. (While simple to program, this approximation scheme is not numerically stable since has to be made small to prevent error due to the term, but decreasing it loses precision in the first term.)
A bordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given the function considered previously, but adding a constraint function such that the bordered Hessian is the Hessian of the Lagrange function
If there are, say, constraints then the zero in the upper-left corner is an block of zeros, and there are border rows at the top and border columns at the left.
The above rules stating that extrema are characterized (among critical points with a non-singular Hessian) by a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as if is any vector whose sole non-zero entry is its first.
The second derivative test consists here of sign restrictions of the determinants of a certain set of submatrices of the bordered Hessian. Intuitively, the constraints can be thought of as reducing the problem to one with free variables. (For example, the maximization of subject to the constraint can be reduced to the maximization of without constraint.)
Specifically, sign conditions are imposed on the sequence of leading principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, for which the first leading principal minors are neglected, the smallest minor consisting of the truncated first rows and columns, the next consisting of the truncated first rows and columns, and so on, with the last being the entire bordered Hessian; if is larger than then the smallest leading principal minor is the Hessian itself. There are thus minors to consider, each evaluated at the specific point being considered as a candidate maximum or minimum. A sufficient condition for a local maximum is that these minors alternate in sign with the smallest one having the sign of A sufficient condition for a local minimum is that all of these minors have the sign of (In the unconstrained case of these conditions coincide with the conditions for the unbordered Hessian to be negative definite or positive definite respectively).
Magnus, Jan R.; Neudecker, Heinz (1999). "The Second Differential". Matrix Differential Calculus : With Applications in Statistics and Econometrics (Revised ed.). New York: Wiley. pp. 99-115. ISBN0-471-98633-X.