The KKT conditions are usually defined as follows

\begin{align} \nabla_x \cal L(x^*) &= \nabla f(x^*) - \sum_{i\in\cal I}\mu_i g(x^*) - \sum_{i\in\cal E}\lambda_i h(x^*) = 0 \\ g(x^*) &\leq 0 \\ h(x^*) &= 0 \\ \mu_i &\geq 0 \\ \mu_i g(x^*) &= 0 \end{align}Wikipedia

However, I've encountered another set of equations which claim to also be the KKT conditions

\begin{align} \frac{\partial \cal L}{\partial x_i}(x^*) &\leq 0 \\ x_i^*\frac{\partial \cal L}{\partial x_i}(x^*) &= 0 \\ g_j(x^*) &\leq 0 \\ x_i^* &\geq 0 \\ \lambda_j &\geq 0 \\ \lambda_j g_j(x^*) &= 0 \end{align} Some paper

However, this features an inequality for $\nabla_x\cal L(x^*)$ instead of an equality, so is this just a special case or is the equality enforced by conditions in other equations?


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