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Reference.org
Smooth maximum
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a smooth maximum of an <a href="/facts/Indexed_family/aFqfzJN0">indexed family</a> <i>x</i>1, ..., <i>x</i><i>n</i> of numbers is a <a href="/facts/Smooth_approximation/mffL4ch2">smooth approximation</a> to the <a href="/facts/Maximum/ADlgnyzV">maximum</a> function max ( x 1 , … , x n ) , {\displaystyle \max(x_{1},\ldots ,x_{n}),} meaning a <a href="/facts/Parametric_family/YoEWw0I8">parametric family</a> of functions m α ( x 1 , … , x n ) {\displaystyle m_{\alpha }(x_{1},\ldots ,x_{n})} such that for every <i>α</i>, the function m α {\displaystyle m_{\alpha }} is smooth, and the family converges to the maximum function m α → max {\displaystyle m_{\alpha }\to \max } as α → ∞ {\displaystyle \alpha \to \infty } . The concept of smooth minimum is similarly defined. In many cases, a single family approximates both: maximum as the parameter goes to positive infinity, minimum as the parameter goes to negative infinity; in symbols, m α → max {\displaystyle m_{\alpha }\to \max } as α → ∞ {\displaystyle \alpha \to \infty } and m α → min {\displaystyle m_{\alpha }\to \min } as α → − ∞ {\displaystyle \alpha \to -\infty } . The term can also be used loosely for a specific smooth function that behaves similarly to a maximum, without necessarily being part of a parametrized family. </p>