I am given the following equality constrained convex QP.
$$\min_x \frac{1}{2}x'Hx+g'x$$ $$st. A'x+b=0$$
with $H\succ 0$.
I want to find the Lagrangian function for this problem and the first order necessary optimality conditions.
Attempt
I found the lagrangian to be:
$$L(x,\lambda)=\frac{1}{2}x'Hx+g'x-\lambda(A'x+b)$$
The first order necessary optimality condition I found to be: $$\nabla_x(\frac{1}{2}x'Hx+g'x)-\lambda\nabla(A'x+b)$$
I this correct, or am I at least on the right track? I am especially unsure about the first order necessary optimality condition....
I hope some of you can help me! The help would be very much appreciated! :)