Assume the primal problem of: $$max_x \mu^Tx$$ $$st. x^T\Sigma x\leq \sigma^2$$ $$ Ax=b$$$$ Cx\geq d$$
What is the KKT optimality conditions for this?
So far this is what I got: $$L(x,\lambda,w,v)= -\mu^Tx+\lambda^T(x^{T}\Sigma x-\sigma^2))-w^T(cx-d)+v^T(b-Ax)$$ $$x^{*T}\Sigma x^*\leq\sigma^2$$ $$-Cx^*\leq -d$$ $$Ax^*=b$$ $$\lambda^*\geq 0$$ $$w^*\geq 0$$ $$\lambda^T(x^{*T}\sigma x^*-\sigma^2)=0$$ $$w^T(Cx^*-d)=0$$ $$\nabla (-\mu^Tx^*)+\lambda^T(\nabla(x^{*T}\Sigma x^*-\sigma^2))-w^T(\nabla(cx^*-d))+v^T (\nabla(b-Ax^*))=0$$ Is this correct? How should I solve the last equation?
Thanks for your help in advance!