## 1 Introduction
\begin{pmatrix}1& x_{1}&\cdots&a_{1n}\\ 1/x_{1}&1&\cdots&x_{k}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&1/x_{k}&\cdots&1\end{pmatrix} \]  
and \(\lambda_{max}(\mathbf{A}(\mathbf{x}))\) represents the maximum eigenvalue function of \(\mathbf{A}(\mathbf{x})\). In addition, an optimal solution \(\mathbf{x}^{*}=(x_{1}^{*},x_{2}^{*},\ldots,x_{k}^{*})\) to the problem (1) is known as an optimal completion of \(\mathbf{A}\). Such a problem was originally considered by [12,13]. However, the focus of early research was a specific term of the characteristic polynomial and not \(\lambda_{max}\), which instead, cannot be represented in a closed form. More recently, the minimization of the Perron eigenvalue (1) of incomplete PCMs has been object of further studies [14, 15, 16]. The reason for minimizing \(\lambda_{max}\), or equivalently, adopting [PERSON]'s inconsistency index as an objective function instead of alternative inconsistency indices, was discussed in a survey study [17] (p. 761). It was chosen because of its widespread use. An equal of even greater amount of interest was generated by the optimal completion of incomplete PCM by optimizing quantities other than \(\lambda_{max}\). For example, [PERSON] and [PERSON] [18] proposed a method that minimizes a global inconsistency index in order to compute the optimal values for the missing comparisons. [PERSON] et al. [19] provided a method based on the linearization process to address a consistent completion of incomplete PCMs by a specific actor. [PERSON] et al. [20] completed the incomplete PCMs by extending the geometric mean-induced bias matrix. [PERSON] et al. [21] proposed a DEMATEL-based completion method that estimates the missing values by deriving the total relation matrix from the direct relation matrix. [PERSON] [22] extended the geometric mean method (GMM) for an incomplete PCM, leading to the same results obtained using the logarithmic least square method (LLSM) for incomplete PCM by [PERSON] et al. [15], where the weight ratio \(w_{i}/w_{j}\) substitutes the missing comparisons in the incomplete PCM. In addition, other researchers have studied several completion methods for incomplete PCMs [23, 24, 25, 26]. The goal of this paper is to propose an efficient and scalable algorithm to solve the constrained problem, where the objective function is a maximum eigenvalue (Perron eigenvalue) function and the constraints are intervals. Due to the unavoidable uncertainty in expressing preferences, it often happens that a decision maker considers it more suitable to state their pairwise comparisons as intervals, rather than as a precise numerical value (see, e.g., [27, 28, 29, 30]). Interval judgments therefore indicate a range for the relative importance of the attributes giving a necessary flexibility to the preference assessment. For this reason, we consider it particularly important that the proposed algorithm, described in Section 3, is able to solve a constrained optimization problem, where variables are subject to interval constraints. All entries of each matrix in the paper consist of numerical values restricted to the interval \([1/9,9]\) in order to meet [PERSON]'s proposal and comply with AHP formulation. It must, however, be said that our choice is not binding and that this requirement can be relaxed by removing some constraints. The paper is organized as follows. Section 2 deals with basic definitions and illustrations regarding the purpose of the work. In Section 3, an algorithm is proposed to solve the constrained eigenvalue problem. In Section 4, illustrative examples are provided to demonstrate the simplex procedure and verify how the algorithm provides an optimal completion. In Section 5, numerical simulations are performed to validate the performance of the proposed algorithm. Finally, Section 6 concludes the paper.