Find the minimum of the following quadratic function
$$ f_{\alpha}(x) := \frac{1}{2} x^T H x + c^T x + \frac{\alpha}{2}(b^Tx)^2 $$
where matrix $H$ is symmetric and positive definite, and $\alpha \geq 0$.
I try to set gradient to zero and get
$$Hx+c+\alpha(b^Tx)b = 0$$
Now I have troubles to get $x$, because it is in the matrix multiplication and in the scalar product. Here some help would be useful.
The whole thing is actually a penalty approximation of the restricted problem to minimize
$$\text{minimize} \quad f(x)=\frac{1}{2}x^THx+c^Tx \quad \text{ subject to } x^T c = 0$$
I may show that the sequence of minima of $f_{\alpha}(x)$ converges to the minimum $f(x)$ with growing $\alpha$.
Hint.
$$ \frac 12(b^T x)^2 = \frac 12(x^T b)(b^T x)\Rightarrow \partial_x\frac 12(b^T x)^2 = b(b^T x) $$
$$ \left(H+\alpha b b^T\right)x + c = 0\Rightarrow x = -\left(H+\alpha bb^T\right)^{-1}c $$