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The definition of local minimum point can also proceed similarly. In both the global and local cases, the concept of a strict extremum can be defined. For example, x ∗ is a strict global maximum point if for all x in X with x ≠ x ∗ , we have f ( x ∗ ) > f ( x ) , and x ∗ is a strict local maximum point if there exists some ε > 0 such ...
In mathematics, a mathematical ... Perhaps the best-known example of the idea of locality lies in the concept of local minimum (or local maximum), which is a point in ...
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables.
A turning point of a differentiable function is a point at which the derivative has an isolated zero and changes sign at the point. [2] A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). A turning point is thus a stationary point, but not all stationary points are turning points.
A critical point (where the function is differentiable) may be either a local maximum, a local minimum or a saddle point. If the function is at least twice continuously differentiable the different cases may be distinguished by considering the eigenvalues of the Hessian matrix of second derivatives.
If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is called a "saddle point", and if none of these cases hold (i.e., some of the eigenvalues are zero) then the test is considered to be inconclusive.
In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point).
Global optimization is distinguished from local optimization by its focus on finding the minimum or maximum over the given set, as opposed to finding local minima or maxima. Finding an arbitrary local minimum is relatively straightforward by using classical local optimization methods. Finding the global minimum of a function is far more ...