## 2 Problem Statement, Basic Notation, and Some Preliminary Results  
Given an integer \(p>1\), denote by \(\mathbb{R}^{p}_{+}\) the set of all \(p\) vectors with non-negative components (the non-negative orthant in \(\mathbb{R}^{p}\)); let \(\mathcal{S}^{p}\) and \(\mathcal{S}^{p}_{+}\) be the space of real symmetric \(p\times p\) matrices and the cone of symmetric positive semidefinite \(p\times p\) matrices, respectively; and let \(\mathcal{COP}^{p}\) stay for the cone of symmetric copositive \(p\times p\) matrices:  
\[\mathcal{COP}^{p}:=\{D\in\mathcal{S}^{p}:t^{\top}Dt\geq 0\ \forall t\in \mathbb{R}^{p}_{+}\}.\]  
Consider a linear copositive programming problem  
\[\min_{x\in\mathbb{R}^{n}}\ c^{\top}x,\ \ \ \text{s.t.}\ \mathcal{A}(x)\in \mathcal{COP}^{p}, \tag{1}\]  
where \(x=(x_{1},\ldots,x_{n})^{\top}\) is the vector of decision variables. The data of the problem above consist of the vector \(c\in\mathbb{R}^{n}\) and the constraints matrix function \(\mathcal{A}(x)\) in the form  
\[\mathcal{A}(x):=\sum_{j=1}^{n}A_{j}x_{j}+A_{0}, \tag{2}\]  
with given matrices \(A_{j}\in\mathcal{S}^{p},j=0,1,\ldots,n\). It is well known (see, e.g., [5]) that the copositive problem (1) is equivalent to the following convex SIP problem:  
\[\min_{x\in\mathbb{R}^{n}}c^{\top}x,\ \ \ \text{s.t.}\ t^{\top}\mathcal{A}(x)t\geq 0 \ \ \forall t\in T, \tag{3}\]  
with a \(p\)- dimensional compact index set \(T\). We define this set as a simplex  
\[T:=\{t\in\mathbb{R}^{p}_{+}:\mathbf{e}^{\top}t=1\}, \tag{4}\]  
where \(\mathbf{e}=(1,1,\ldots,1)^{\top}\in\mathbb{R}^{p}\). Denote by \(X\) the feasible set of the equivalent problems (1) and (3):  
\[X:=\{x\in\mathbb{R}^{n}:\mathcal{A}(x)\in\mathcal{COP}^{p}\}=\{x\in\mathbb{R}^ {n}:t^{\top}\mathcal{A}(x)t\geq 0\ \ \forall t\in T\}. \tag{5}\]  
Evidently, either the set \(X\) is convex or it is empty. Let us reformulate some standard definitions for the cone \(\mathcal{COP}^{p}\). A convex subset \(\mathbf{F}\) of the cone \(\mathcal{COP}^{p}\) is its _face_ if for any \(x\in\mathcal{COP}^{p}\) and \(y\in\mathcal{COP}^{p}\), the inclusion \(x+y\in\mathbf{F}\) implies \(x\in\mathbf{F},y\in\mathbf{F}\). Notice that any face of a cone is also a cone. A face \(\mathcal{F}\) of \(\mathcal{COP}^{p}\) is called _exposed_ if it can be presented as the intersection of \(\mathcal{COP}^{p}\) with a supporting hyperplane. Given the copositive problem \(\mathcal{COP}^{p}\), let \(\mathbf{F}_{min}\) be the smallest (by inclusion) face of \(\mathcal{COP}^{p}\) containing the image set \(\mathcal{D}\) of the constraints of this problem defined as follows:  
\[\mathcal{D}:=\{\mathcal{A}(x),\;x\in X\}. \tag{6}\]  
The face \(\mathbf{F}_{min}\) will be called here _the minimal face_ of the optimization problem (1). According to the commonly used definitions, the constraints of the copositive problem (1) satisfy _the Slater condition_ if  
\[\exists\pi\in\mathbb{R}^{n}\text{ such that }\mathcal{A}(\bar{x})\in\operatorname{ int}\mathcal{COP}^{p}=\{D\in\mathcal{S}^{p}:t^{\top}Dt>0\;\forall t\in \mathbb{R}^{p}_{+},t\
eq\mathbf{0}\}, \tag{7}\]  
where \(\operatorname{int}\mathcal{C}\) stays for the interior of a set \(\mathcal{C}\) and \(\mathbf{0}\) denotes the null vector in the corresponding finite dimensional space. As it was mentioned in the Introduction, in [17] we described a regularization algorithm REG-LCoP specially designed for problems in the form (1). In this paper we go further and develop new modifications of the algorithm REG-LCoP that are more explicit and have less additional requirements. To justify these modified algorithms, we need some preliminary results. In what follows, to simplify notation, for \(y=(y_{1},\ldots,y_{n})^{\top}\in\mathbb{R}^{n}\), \(y_{0}\in\mathbb{R}\), and \(\mu\in\mathbb{R}\), the vector \((y^{\top},y_{0})^{\top}\in\mathbb{R}^{n+1}\) will be denoted as \((y,y_{0})\in\mathbb{R}^{n+1}\), and the vector \((y^{\top},y_{0},\mu)^{\top}\in\mathbb{R}^{n+2}\) as \((y,y_{0},\mu)\in\mathbb{R}^{n+2}\).