We introduce the map $$\begin{aligned} J : \mathbb { R } ^ { N } & \rightarrow \mathbb { R } \\ x & \mapsto \frac { 1 } { 2 } \langle x , A x \rangle - \langle b , x \rangle \end{aligned}$$ where $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, $b \in \mathbb { R } ^ { N }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ and $x _ { 0 } + H _ { k }$ denotes the subset of points of $\mathbb { R } ^ { N }$ of the form $x _ { 0 } + x$ where $x$ ranges over $H _ { k }$. Show that $J$ admits a unique minimizer on the subset $x _ { 0 } + H _ { k }$, for any $k \in \mathbb { N }$.
We introduce the map
$$\begin{aligned} J : \mathbb { R } ^ { N } & \rightarrow \mathbb { R } \\ x & \mapsto \frac { 1 } { 2 } \langle x , A x \rangle - \langle b , x \rangle \end{aligned}$$
where $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, $b \in \mathbb { R } ^ { N }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$,
$$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$
and $x _ { 0 } + H _ { k }$ denotes the subset of points of $\mathbb { R } ^ { N }$ of the form $x _ { 0 } + x$ where $x$ ranges over $H _ { k }$.
Show that $J$ admits a unique minimizer on the subset $x _ { 0 } + H _ { k }$, for any $k \in \mathbb { N }$.