Second-Order Sufficient Condition (SOSC)

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Suppose that for some feasible point $x^{\ast }\in {\mathbb{R}}^n$ there is a Lagrange multiplier vector ${\lambda }^{\ast }$ such that the KKT (Karush–Kuhn–Tucker) Conditions are satisfied. Suppose also that

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

Then $x^{\ast }$ is a strict local solution.