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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 ...
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 a function whose functional value is the smallest (resp., largest) within an immediate neighborhood of points. [1]
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.
Consequently, the set of global minimisers of a convex function is a convex set: - convex. Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum. [9]
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). [1]
When the function is convex, all local minima are also global minima, so in this case gradient descent can converge to the global solution. This process is illustrated in the adjacent picture. Here, F {\displaystyle F} is assumed to be defined on the plane, and that its graph has a bowl shape.
has exactly one minimum for = (at (,,)) and exactly two minima for —the global minimum at (,,...,) and a local minimum near ^ = (,, …,). This result is obtained by setting the gradient of the function equal to zero, noticing that the resulting equation is a rational function of x {\displaystyle x} .