Strong convexity gradient inequality ($\Vert \nabla f(x){\Vert }^2\ge 2\gamma (f(x)-f(x^{\ast }))$)

For a differentiable strongly convex function with modulus $\gamma >0$ and global minimizer $x^{\ast }$, the supporting inequality givesCompleting the square shows that the last two terms are at least $-\Vert \nabla f(x){\Vert }^2/(2\gamma )$, which proves the result. Combining this with the descent lemma converts squared gradient decrease into a contraction of the objective function gap.

Target: mathematical-optimization.bigb, under Line search.