I am reading a paper. I have a problems with understanding their lemma

Lemma: The nonzero elements of every vector $u \in U(x)$ are given by a vector $\bar{u} \in E^k, \bar{k} \leqslant k$, where $\bar{u}=-\left(\bar{H}^{\prime} \bar{H}\right)^{-1} \bar{H}^{\prime} g(x, \bar{r}) \geqslant 0$. The $m \times \bar{k}$ matrix $\bar{H}=\bar{H}(x)$ consists of $\bar{k}$ linearly independent columns of $H(x)$ selected from $\nabla h_j(x)$ for $j \in J$.

Let me recall some useful notations

We define an $m \times k$ matrix $H(x)$ whose $jth$ column is $\nabla h_j(x)$ $$H(x)=\left[\begin{array}{lll} \nabla h_1(x) & \nabla h_2(x) \ldots \nabla h_k(x) \end{array}\right]$$

$g(x,r)$ is a mapping of $E^m$ into itself $$g(x, r)=\left[\begin{array}{c} r_1 \nabla_1 \varphi_1(x) \\ r_2 \nabla_2 \varphi_2(x) \\ \vdots \\ r_n \nabla_n \varphi_n(x) \end{array}\right]$$

We define the mapping $f(x,u,r)$ of $E^{m+k} \to E^m$ for each fixed $\bar{r}>0$ as follows $$f(x, u, \bar{r})=g(x, \bar{r})+H(x) u$$

where $u \in U(x) \subset E^k$ such that $$U(x)=\left\{u \mid\|f(x, u, \bar{r})\|=\min _{\substack{v_j \geqslant 0, j \in J \\ v_j=0, j \neq J}}\|f(x, v, \bar{r})\|\right\}$$

$$J=J(x)=\left\{j \mid h_j(x) \leqslant 0\right\}$$

Thank you very much for your help. I appreciate it a lot



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