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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 that, for all x in X within distance ε of x ∗ with x ≠ x ∗, we ...
The constrained extrema of f are critical points of the Lagrangian , but they are not necessarily local extrema of (see § Example 2 below). One may reformulate the Lagrangian as a Hamiltonian , in which case the solutions are local minima for the Hamiltonian.
Fermat's theorem is central to the calculus method of determining maxima and minima: in one dimension, one can find extrema by simply computing the stationary points (by computing the zeros of the derivative), the non-differentiable points, and the boundary points, and then investigating this set to determine the extrema.
[e] The extremum [] is called a local maximum if everywhere in an arbitrarily small neighborhood of , and a local minimum if there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema , depending on whether the first derivatives of the continuous functions are respectively ...
The optimization of portfolios is an example of multi-objective optimization in economics. Since the 1970s, economists have modeled dynamic decisions over time using control theory. [14] For example, dynamic search models are used to study labor-market behavior. [15] A crucial distinction is between deterministic and stochastic models. [16]
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]
The extrema must occur at the pass and stop band edges and at either ω=0 or ω=π or both. The derivative of a polynomial of degree L is a polynomial of degree L−1, which can be zero at most at L−1 places. [3] So the maximum number of local extrema is the L−1 local extrema plus the 4 band edges, giving a total of L+3 extrema.
In the geometry of complex algebraic curves, a local parameter for a curve C at a smooth point P is a meromorphic function on C that has a simple zero at P. This concept can be generalized to curves defined over fields other than (or schemes), because the local ring at a smooth point P of an algebraic curve C (defined over an algebraically closed field) is always a discrete valuation ring. [1]