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Given that $h_{i}(x) = 0$ and $g_j(x) \le 0$ are the constraints, observe the gradient of the Lagrangian:

$\nabla [f_0(x) + \sum_{i=1}^m\lambda_ih_i(x) + \sum_{j=1}^n\mu_jg_j(x)]$

Each of the partial derivatives with respect to the Lagrange multipliers on the equality constraints leaves $h_i(x) = 0$, which is just primal feasibility for said constraints. Is there any particular reason why feasibility of the equality constraints is considered a KKT condition anyways, or is it just for bookkeeping?

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