Equality condition

The equality condition of a non-strict inequality is when all the variables are such that the sides of the inequality are equal. Equality conditions can be used in optimization problems. Many inequalities reach their equality conditions when the variables are equal, most notably the QM-GM-AM-HM inequality.

Uses in Optimization

In order to find the minimum or maximum of an expression, it is often useful to create an inequality equivalent to the expression. For example, in order to find the minimum of the square of a real number, (see Trivial Inequality) it is possible to construct the inequality $x^2\geq0$. The minimum, in this case, is the equality condition, $x=0$, so we can state that 0 is the minimum value of $x^2$.

Notable Inequalities and Their Equality Conditions

QM-GM-AM-HM- This inequality achieves equality when the numbers being averaged are equal.

Cauchy-Schwartz Inequality- This inequality achieves equality when the terms of the first set of numbers- say $a_1,a_2,...,a_i$, can be multiplied by a constant to get the second set of numbers- $b_1,b_2,...,b_i$. So $a_1=k*b_1$, and $a_i=k*b_i$ in general where $k$ is a constant.

Triangle Inequality- This inequality achieves equality when two of the sides of the triangle sum to the third side, or when the triangle is degenerate.

Trivial Inequality- This inequality achieves equality when the number being squared is equal to 0.

See Also

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