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In mathematical analysis, the maximum and minimum [a] of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum , [ b ] they may be defined either within a given range (the local or relative extrema) or on the entire domain (the global or absolute extrema) of a function.
The Lagrange multiplier theorem states that at any local maximum (or minimum) of the function evaluated under the equality constraints, if constraint qualification applies (explained below), then the gradient of the function (at that point) can be expressed as a linear combination of the gradients of the constraints (at that point), with the ...
For example, given a function defined on the interval [,] and a degree bound , a minimax polynomial approximation algorithm will find a polynomial of degree at most to minimize max a ≤ x ≤ b | f ( x ) − p ( x ) | . {\displaystyle \max _{a\leq x\leq b}|f(x)-p(x)|.} [ 3 ]
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
If the knowledge of this correlation function is not perfectly available, a popular minimax robust optimization approach [6] is to define a set characterizing the uncertainty about the correlation function, and then pursuing a minimax optimization over the uncertainty set and the estimator respectively. Similar minimax optimizations can be ...
A continuous function () on the closed interval [,] showing the absolute max (red) and the absolute min (blue).. In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed and bounded interval [,], then must attain a maximum and a minimum, each at least once.
However, the normalised sinc function (blue) has arg min of {−1.43, 1.43}, approximately, because their global minima occur at x = ±1.43, even though the minimum value is the same. [ 1 ] In mathematics , the arguments of the maxima (abbreviated arg max or argmax ) and arguments of the minima (abbreviated arg min or argmin ) are the input ...
In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces. It can be viewed as the starting point of many results of similar nature.