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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 $\alpha$.

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  • $\begingroup$ Yes, α≥0 this is is required by the algorithm $\endgroup$ – Nikolskyy May 19 at 16:21
  • $\begingroup$ Since $b^Tx$ is a scalar, it can right-multiply vector $b$ instead. Hence, $$Hx + c + \alpha (b^Tx) b = Hx + c + \alpha b b^T x = (H + \alpha b b^T) x + c = 0$$ where matrix $H + \alpha b b^T$ is clearly symmetric and positive definite. Keep in mind that there is seldom a good reason to invert a matrix. Linear systems are solved using Gaussian elimination. $\endgroup$ – Rodrigo de Azevedo May 19 at 16:28
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Hint.

$$ \frac 12(b^T x)^2 = \frac 12(x^T b)(b^T x)\Rightarrow \partial_x\frac 12(b^T x)^2 = b(b^T x) $$

$$ \left(H+\alpha b b^T\right)x + c = 0\Rightarrow x = -\left(H+\alpha bb^T\right)^{-1}c $$

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  • $\begingroup$ thank you very much for help! $\endgroup$ – Nikolskyy May 19 at 16:22

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