## 1 Introduction
The facial reduction approach has been successfully applied to SDP and second-order cone programming problems, and to certain classes of optimization problem over symmetric (i.e., self-dual and homogeneous) and nice cones (see, for example, [13; 14; 15; 16]). At the same time, the question of the effectiveness of such an approach to other classes of problems remains open. This is because the known FRAs are more _conceptual_ than practical. In [17], using our previous results published in [18; 19], and others, we proposed a different approach to the regularization of linear CoP problems. This approach is based on the concept of _immobile indices_ (i.e., indices of the constraints that are active for all feasible solutions) originally introduced for SIP and SDP problems. We used this approach to develop two finite algorithms for the regularization of linear CoP problems: one method is called _one-step regularization_ and the another is the iterative algorithm REG-LCoP (regularization of linear copositive problems). The first method consists of one step, but certain difficulties may arise when applying it, since it is based on a priori possession of some additional information about some finite subset of immobile indices. Unlike the one-step regularization, the algorithm REG-LCoP does not require any additional information in advance. Comparing this algorithm with other algorithms, we showed that it is more constructive because its iterations (using the concept of immobile indices) can be more explicitly described than the iterations of other FRA-based algorithms (which use the concept of the minimal cone). As far as we know, the algorithm REG-LCoP is the first constructive regularization procedure developed specifically for linear copositive problems. This algorithm is important not only because it is based on previously unexplored properties of copositive problems, but also conceptually, since new algorithms and methods can be developed on its basis. In this paper, we propose three new algorithms that are modifications of REG-LCoP. Although these algorithms have structures similar to that of the REG-LCoP algorithm, they do not require either the feasibility of the original optimization problem or the initial knowledge of a set of immobile indices and can be considered as developments in the approach to the regularization of copositive problems, started in [17]. All the algorithms presented in the paper use the properties of the copositive problem established here and are based on alternative representations of its feasible set that are obtained using the concept of normalized immobile indices. The algorithms are finite and their iterations are explicitly described. The main differences between the modifications of the REG-LCoP algorithm presented here consist in the auxiliary problems that are solved in their iterations and in the properties of their solution sets. According to their specific needs, a user can choose the most appropriate version of the main algorithm. It is important to emphasize that the algorithms described in the paper are important not only from the point of view of applications, but also as they contribute to the theoretical study of the structural properties of copositive problems. The main steps of the algorithms are interpreted in the paper from the point of view of the facial geometry of the cone of copositive matrices. Therefore, the results of the paper allow a deeper understanding of the structure of the feasible set of a given copositive problem and can serve as a basis for new types of optimality conditions and various forms of dual problems that satisfy strong duality relations. The paper is organized as follows. Section 1 hosts the Introduction. In Section 2, we formulate the problem, and introduce basic notation and some preliminary results. The properties of the set of normalized immobile indices for the copositive problem under investigation are considered in Section 3. In Section 4, we describe and justify three versions of the REG-LCoP algorithm from [17].