## 4 Regularization Algorithms Based on the Concept of Immobile Indices
### Regularization Algorithm RLCoP-1  
Based on the results of the previous section, we describe here in detail the iterations of the algorithm RLCoP-1. **Iteration # 0. ** Given the copositive problem in the form (1), consider the following _regular_ SIP problem:  
\[SIP_{0}^{1}:\quad\max_{(y,y_{0},\mu)\in\mathbb{R}^{n+2}}\ \mu,\ \text{s.t.}\ t^{ \top}\mathcal{B}(y,y_{0})t-\mu\geq 0\ \forall t\in T,\ y_{0}\geq 0,\]  
with the compact index set \(T\) defined in (4). If there exists a feasible solution \((\check{y},\check{y}_{0},\check{\mu})\) of the problem \((SIP_{0}^{1})\) with \(\check{\mu}>0\), then set \(m_{*}:=0\) and go to the _Final step_. Otherwise the vector \((y=\mathbf{0},y_{0}=0,\mu=0)\) is an optimal solution of this problem. By construction, in the problem \((SIP_{0}^{1})\), the index set \(T\) is compact, and the constraints satisfy the Slater condition. Hence, (see, e.g., [21]), it follows from the optimality conditions for the vector \((y=\mathbf{0},y_{0}=0,\mu=0)\) that there exist indices and numbers  
\[\tau(i)\in T,\ \ \gamma(i)>0\ \forall i\in I_{1},\ \ |I_{1}|\leq n+2, \tag{25}\]  
such that  
\[\sum_{i\in I_{1}}\gamma(i)(\tau(i))^{\top}A_{i}\tau(i)=0\ \forall j=1,...,n;\ \ \sum_{i\in I_{1}}\gamma(i)=1, \tag{26}\]  
\[\sum_{i\in I_{1}}\gamma(i)(\tau(i))^{\top}A_{0}\tau(i)=:\eta_{0}\leq 0. \tag{27}\]  
If \(\eta_{0}<0\), then STOP: from Proposition 4, it follows that \(X=\varnothing\). Suppose that \(\eta_{0}=0\). From the relations (26) and (27) (with \(\eta_{0}=0\)), one can conclude that \(I_{1}\
eq\varnothing\) and \(\tau(i)\in T_{im}(\mathcal{Z})\subset T\) for all \(i\in I_{1}\). Go to the next iteration with the set of indices \(W_{1}:=\{\tau(i),i\in I_{1}\}\). **Iteration # \(m\), \(m\geq 1\). ** By the beginning of the iteration, we have the index set \(W_{m}:=\{\tau(i),\ i\in I_{m}\}\), where  
\[\tau(i)\in T_{im}(\mathcal{Z})\ \ \forall i\in I_{m}.\]  
Consider a SIP problem  
\[\max_{(y,y_{0},\mu)\in\mathbb{R}^{n+2}}\mu,\] \[SIP_{m}^{1}:\quad\text{s.t.}\ t^{\top}\mathcal{B}(y,y_{0})t-\mu \geq  
where the set \(\Omega(W_{m})\) is constructed by the rules (12) and (13) with \(W=W_{m}\).  
Since \(W_{m}\) is a subset of the set of immobile indices of \(\mathcal{Z}\), according to Theorem 1, we conclude that \(\mathcal{Z}(W_{m})=\mathcal{Z}\), where the set \(\mathcal{Z}(W)\) is defined in (15).  
Notice that by the definition of the set \(\Omega(W_{m})\), it holds  
\[\rho(t,\text{conv}W_{m})\geq\sigma(W_{m})>0\ \ \forall t\in\Omega(W_{m}).\]  
If the problem \((SIP^{1}_{m})\) admits a feasible solution \((\bar{g},\bar{g}_{0},\bar{\mu})\) with \(\bar{\mu}>0\), then go to the _Final step_ with \(m_{*}:=m\) or else, the vector \((y=0,y_{0}=0,\mu=0)\) is an optimal solution of this problem. In the problem \((SIP^{1}_{m})\), the constraints \(\mathcal{B}(y,y_{0})\tau(i)\geq 0\ \forall\ i\in I_{m}\), are linear, the index set \(\Omega(W_{m})\) is compact, and the following Slater type condition is satisfied:  
\[\exists\ (\bar{y},\bar{y}_{0},\bar{\mu})\ \text{such that} \mathcal{B}(\bar{y},\bar{y}_{0})\tau(i)\geq 0\ \forall i\in I_{m};\] \[t^{\top}\mathcal{B}(\bar{y},\bar{y}_{0})t-\bar{\mu}>0\ \forall t\in\Omega(W_{m}),\ \bar{y}_{0}\geq 0.\]  
Due to these properties, the optimality of the vector mentioned above provides (see [22]) that there exist indices, numbers, and vectors  
\[\tau(i)\in\Omega(W_{m}),\gamma(i)>0,i\in\Delta I_{m},1\leq|\Delta I_{m}|\leq n +2;\lambda^{m}(i)\in\mathbb{R}^{p}_{+},i\in I_{m}, \tag{28}\]  
satisfying the following conditions:  
\[\tau(i)\in T,\ \ \gamma(i)>0\ \forall i\in I_{1},\ \ |I_{1}|\leq n+2, \tag{25}\]0  
\[\tau(i)\in T,\ \ \gamma(i)>0\ \forall i\in I_{1},\ \ |I_{1}|\leq n+2, \tag{25}\]1  
Moreover, by applying the _procedure DAM_ described in [20], to the data presented in (28), it is possible to ensure that the following conditions are met:  
\[\tau(i)\in T,\ \ \gamma(i)>0\ \forall i\in I_{1},\ \ |I_{1}|\leq n+2, \tag{25}\]2  
If \(\eta_{m}<0\), then STOP: it follows from Proposition 4 that \(X=\emptyset\). Suppose that \(\eta_{m}=0\).