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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 have f(x ∗) > f(x). Note that a point is a strict global maximum point if and only if ...
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 ...
Fermat and Lagrange found calculus-based formulae for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum. The term " linear programming " for certain optimization cases was due to George B. Dantzig , although much of the theory had been introduced by Leonid Kantorovich in 1939.
In Pareto efficiency, a Pareto optimum is a maximal element with respect to the partial order of Pareto improvement, and the set of maximal elements is called the Pareto frontier. In decision theory , an admissible decision rule is a maximal element with respect to the partial order of dominating decision rule .
The maximum sustainable yield is usually higher than the optimum sustainable yield and maximum economic yield. MSY is extensively used for fisheries management . Unlike the logistic ( Schaefer ) model, [ 1 ] MSY has been refined in most modern fisheries models and occurs at around 30% of the unexploited population size.
In the design of experiments, optimal experimental designs (or optimum designs [2]) are a class of experimental designs that are optimal with respect to some statistical criterion. The creation of this field of statistics has been credited to Danish statistician Kirstine Smith .
The optimum quantity is the same as the optimum quantity in the first diagram. If the firm is a monopolist, the marginal revenue curve would have a negative slope as shown in the next graph, because it would be based on the downward-sloping market demand curve.
The continuity of in the maximum theorem is the result of combining two independent theorems together. Theorem 1 . [ 9 ] [ 10 ] [ 11 ] If f {\displaystyle f} is upper semicontinuous and C {\displaystyle C} is upper hemicontinuous, nonempty and compact-valued, then f ∗ {\displaystyle f^{*}} is upper semicontinuous.