I am using a black-box solver to solve the following non-convex QCQP to global optimality.
$$ \min_x x^TQ_0x + c^T x \\ s.t. \quad x^TQ_1x+c_1^Tx=b_1 \\ Ax=b \\ l\leq x\leq u $$
where $Q_0$ is indefinite diagonal matrix and $Q_1$ is positive semidefinite diagonal matrix. I have obtained a globally optimal solution of the the above program, say $x^*$. How can I calculate the optimal dual multipliers associated with the constraints? I am thinking of using the KKT conditions, will that be right? The Lagrange dual I am referring to is formulated as $$ \max_{\lambda,\nu,\omega,\mu} \min_{x \in \mathbb R^n} L(x,\lambda,\nu,\omega,\mu) \\ \lambda \geq 0, \omega \leq 0, \mu \geq 0 \\ L(x,\lambda,\nu,\omega,\mu) = x^TQ_0x+c_0^Tx+\sum_{k=1}^m \lambda_k(x^TQ_kx+c_k^Tx-b_k)+ \sum_{p=1}^q \nu_p(a_p^Tx-b)+\omega^T(x-l)+\mu^T(x-u) $$ Upon fixing $x$ to $x^*$ the Lagrange function becomes : $$ (x^*)^TQ_0x^*+c_0^Tx^*+\omega^T(x^*-l)+\mu^T(x^*-u) $$ The first order conditions are $\nabla{L_x}|_{x=x^*}=0$. The constraints are all satisfied since $x^*$ is an optimal solution I also think the only possible non-zero multipliers are $\lambda_k$ and $\nu_p$ (from complimentary slackness). Are these conditions sufficient to find the optimal multiplier vector? What is the second order condition here? $Q_0 \succeq 0$? Is second order condition necessary to guarantee global optimality?