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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 ...
The maximum of a subset of a preordered set is an element of which is greater than or equal to any other element of , and the minimum of is again defined dually. In the particular case of a partially ordered set , while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements.
Solar minima and maxima are the two extremes of the Sun's 11-year and 400-year activity cycle. [1] At a maximum, the Sun is peppered with sunspots, solar flares erupt, and the Sun hurls billion-ton clouds of electrified gas into space. Sky watchers may see more auroras, and space agencies must monitor radiation storms for astronaut protection.
The sample maximum and minimum are the least robust statistics: they are maximally sensitive to outliers.. This can either be an advantage or a drawback: if extreme values are real (not measurement errors), and of real consequence, as in applications of extreme value theory such as building dikes or financial loss, then outliers (as reflected in sample extrema) are important.
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 ...
Start (Maximum) Spotless days [10] Solar cycle 10–11 1860 – Feb 406 Solar cycle 11–12 1870 – Aug 1028 Solar cycle 12–13 1883 – Dec 736 Solar cycle 13–14 1894 – Jan 934 Solar cycle 14–15 1906 – Feb 1023 Solar cycle 15–16 1917 – Aug 534 Solar cycle 16–17 1928 – Apr 568 Solar cycle 17–18 1937 – Apr 269
Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.
So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one. Is there an efficient way to find the global maximum/minimum? Take for example the sine integral. It has an infinite ...