Second-Order Necessary Conditions (SONC)

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Suppose that $x^{\ast }$ is a local solution, and that the Linear Independence Constraint Qualification conditions is satisified. Let ${\lambda }^{\ast }$ be the Lagrange multiplier vector for which the KKT (Karush–Kuhn–Tucker) Conditions are satisifed. Then

$$w^T{\nabla }_{xx}^2\mathcal{L}(x^{\ast },{\lambda }^{\ast })w\ge 0,\forall w\in \mathcal{C}(x^{\ast },{\lambda }^{\ast })$$

Theorem 2.3

If $x^{\ast }$ is a local minimizer of $f$ and ${\nabla }^{2}f$ exists and is continuous in an open neighborhood of $x^{\ast }$, then $\nabla f(x^{\ast })=0$ and ${\nabla }^{2}f(x^{\ast })$ is positive semidefinite.