= Strong convexity gradient inequality
{title2=$\|\nabla f(x)\|^2\geq2\gamma(f(x)-f(x^*))$}
For a <differentiable> <strongly convex function> with modulus $\gamma>0$ and <global minimizer> $x^*$, the supporting inequality gives
$$
f(x^*)\geq f(x)+\nabla f(x)^T(x^*-x)+\frac\gamma2\|x^*-x\|^2.
$$
Completing the square shows that the last two terms are at least $-\|\nabla f(x)\|^2/(2\gamma)$, which proves the result. Combining this with the <descent lemma> converts squared <gradient> decrease into a contraction of the <objective function> gap.
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Target: mathematical-optimization.bigb, under Line search.
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