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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 ...
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 ...
A feasible solution that minimizes (or maximizes) the objective function is called an optimal solution. In mathematics, conventional optimization problems are usually stated in terms of minimization. A local minimum x* is defined as an element for which there exists some δ > 0 such that
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]
These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization , which aims to find (global) minima of the function f {\displaystyle f} .
Deterministic global optimization methods are typically used when locating the global solution is a necessity (i.e. when the only naturally occurring state described by a mathematical model is the global minimum of an optimization problem), when it is extremely difficult to find a feasible solution, or simply when the user desires to locate the ...
In convex analysis and variational analysis, a point (in the domain) at which some given function is minimized is typically sought, where is valued in the extended real number line [,] = {}. [1] Such a point, if it exists, is called a global minimum point of the function and its value at this point is called the global minimum (value) of the ...
These minimization problems arise especially in least squares curve fitting. The LMA interpolates between the Gauss–Newton algorithm (GNA) and the method of gradient descent. The LMA is more robust than the GNA, which means that in many cases it finds a solution even if it starts very far off the final minimum. For well-behaved functions and ...