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Define a vector $z:=\begin{bmatrix} z_1^\top & z_2^\top \end{bmatrix}^\top$.

Can this optimization problem \begin{equation*} \min_z z_1^\top H z_1 + f^\top z_1 - \delta(z_2) \end{equation*} be solved? Where matrix $H$ is a positive definite matrix, $f$ is a constant vector and $\delta(\cdot)$ denotes the Dirac delta function.

The point I am confused is that according to the definition of the Dirac delta function, $\delta(z_2)=\infty$ when $z_2=0$, which makes the value of objective funtion equals to -$\infty$.

Does this optimization is equivellent to solve \begin{equation*} \min_{z_1} z_1^\top H z_1 + f^\top z_1 \end{equation*} with $z_2=0$?

Stephen Ge
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  • This is a strange problem, how did it come about? There is no minimum as you have observed, the problem should be written as an $\inf$, not a $\min$. And the value if $-\infty$ as you have noted. Since the rest of the value is finite, the infimisers are $(z_1,z_2) = (t,0)$ for arbitrary $t$. – copper.hat Feb 19 '16 at 16:34
  • Yes, I feel strange too, I will check the formulation again, I think if there is a positive sign with the delta function rather than negative sign, then it will be fine – Stephen Ge Feb 19 '16 at 17:04
  • @ Stephen : $\min_{x,y} f(x) + g(y)$ without any constraint on $(x,y)$ is equivalent to minimize separately $f(x)$ and $g(y)$. and do you know how to solve $\min_z z^T H z + f^T z$ ? – reuns Feb 19 '16 at 17:22

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