## 2 Problem Statement, Basic Notation, and Some Preliminary Results
Introduce a matrix function \(\mathcal{B}(y,y_{0}):=\sum_{j=1}^{n}A_{j}y_{j}+A_{0}y_{0}\) and consider the set  
\[ \begin{split}\mathcal{Z}:=&\{(y,y_{0})\in\mathbb{R }^{n+1}:\mathcal{B}(y,y_{0})\in\mathcal{COP}^{p},\;y_{0}\geq 0\}\\ =&\{(y,y_{0})\in\mathbb{R}^{n+1}:t^{\top}B(y,y_{0})t \geq 0\;\forall t\in T,\;y_{0}\geq 0\}.\end{split}  \tag{8}\]  
Notice that \(\mathcal{Z}\
eq\emptyset\). Given the sets \(X\
eq\emptyset\) and \(\mathcal{Z}\) defined in (5) and (8), respectively, following [19], we define here the corresponding _sets of (normalized) immobile indices_\(T_{im}(X)\) and \(T_{im}(\mathcal{Z})\):  
\[T_{im}(X):=\{t\in T:t^{\top}\mathcal{A}(x)t=0\;\;\forall x\in X\}, \tag{9}\]  
\[T_{im}(\mathcal{Z}):=\{t\in T:t^{\top}\mathcal{B}(y,y_{0})t=0\;\;\forall(y,y_{ 0})\in\mathcal{Z}\}. \tag{10}\]  
Generally speaking, the set of immobile indices of a given optimization problem is defined by the indices of constraints which are active for all feasible solutions. Based on Lemma 1 and Proposition 1 in [19], let us formulate the following lemma. **Lemma 1**. : _Given the linear copositive problem (1), suppose that \(X\
eq\emptyset\). Then_  
1. _the Slater condition (_7_) is equivalent to the emptiness of the set of normalized immobile indices_ \(T_{im}(X)\)_,_
2. _The set_ \(T_{im}(X)\) _is either empty or can be presented as a union of a finite number of convex closed bounded polyhedra._  
For a vector \(t=(t_{k},k\in P)^{\top}\in\mathbb{R}^{p}_{+}\) with \(P:=\{1,2,\ldots,p\}\), define the index set  
\[P_{+}(t):=\{k\in P:t_{k}>0\}.\]Given a set \(\mathcal{C}\subset\mathbb{R}^{p}\) and a vector \(l\in\mathbb{R}^{p}\), denote by \(\rho(l,\mathcal{C})\) the distance between this set and the vector, \(\rho(l,\mathcal{C}):=\min\limits_{\tau\in\mathcal{C}}\sum\limits_{k\in p}|l_{k} -\tau_{k}|\), and by \(\mathrm{conv}\mathcal{C}\), the convex hull of the set \(\mathcal{C}\). Here and in what follows, for a vector \(\tau\in\mathbb{R}^{p}\), we denote by \(\tau_{k}\), \(k\in P\), its components. Consider a finite non-empty subset \(W\) of the simplex \(T\) defined in (4). Without loss of generality, we suppose that the elements of the set \(W\) are indexed as follows:  
\[W=\{\tau(i),i\in I\}\subset T,\ 0<|I|<\infty, \tag{11}\]  
where \(I\) is an index set. For the set \(W\), define the number  
\[\sigma(W):=\min\{\tau_{k}(i),\ \ k\in P_{+}(\tau(i)),\ i\in I\}>0, \tag{12}\]  
and the sets  
\[\Omega(W) :=\{t\in T:\rho(t,\mathrm{conv}W)\geq\sigma(W)\}, \tag{13}\] \[\mathcal{X}(W) :=\{x\in\mathbb{R}^{n}:\mathcal{A}(x)\tau(i)\geq 0\ \forall i\in I;\ \ t^{\top}\mathcal{A}(x)t\geq 0 \ \forall t\in\Omega(W)\},\] (14) \[\mathcal{Z}(W) :=\{(y,y_{0})\in\mathbb{R}^{n+1}:\mathcal{B}(y,y_{0})\tau(i) \geq 0\ \forall i\in I;\] (15) \[\ t^{\top}\mathcal{B}(y,y_{0})t\geq 0\ \forall t\in\Omega(W),y_{0}\geq 0\}.\]  
Suppose that the constraints of problem (1) do not satisfy the Slater condition; i.e., the set of normalized immobile indices \(T_{im}(X)\) of this problem is not empty. By following the proof of Theorem 5 in [18], we can prove the next theorem. Theorem 1: _Consider problem (1) with the feasible set \(X\
eq\emptyset\) given in (5) and the set \(\mathcal{Z}\) given in (8). Then_  
1. _For any set_ \(W\subset T_{im}(X)\) _in the form (_11_), the equality_ \(X=\mathcal{X}(W)\) _holds true, where the set_ \(\mathcal{X}(W)\) _is defined in (_14_)._
2. _For any set_ \(W\subset T_{im}(\mathcal{Z})\) _in the form (_11_), the equality_ \(\mathcal{Z}=\mathcal{Z}(W)\) _holds true, where the set_ \(\mathcal{Z}(W)\) _is defined in (_15_)._  
This theorem gives us alternative representations of the related sets \(X\) and \(\mathcal{Z}\) on the basis of a given finite set (11). This theorem and the results of the next section form a basis for new regularization algorithms that will be described below.