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Finding global maxima and minima is the goal of mathematical optimization. If a function is continuous on a closed interval, then by the extreme value theorem, global maxima and minima exist. Furthermore, a global maximum (or minimum) either must be a local maximum (or minimum) in the interior of the domain, or must lie on the boundary of the ...
In which case, a property is said to be local if it can be detected from the local subgroups. Global and local properties formed a significant portion of the early work on the classification of finite simple groups, which was carried out during the 1960s.
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 ...
In a convex problem, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.
So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one. Is there an efficient way to find the global maximum/minimum? Take for example the sine integral. It has an infinite ...
Finding the extrema of functionals is similar to finding the maxima and minima of functions. The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero). The extrema of functionals may be obtained by finding functions for which the functional derivative is equal to zero.
[5] The LMA is used in many software applications for solving generic curve-fitting problems. By using the Gauss–Newton algorithm it often converges faster than first-order methods. [6] However, like other iterative optimization algorithms, the LMA finds only a local minimum, which is not necessarily the global minimum.
If the second derivative is null, the critical point is generally an inflection point, but may also be an undulation point, which may be a local minimum or a local maximum. For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point.