## 1 Introduction  
Increasingly complex decisions are being made everyday and especially in positions of high responsibility, they must be \"defensible\" in front of stakeholders. In this context, decision analysis can be seen as a set of formal tools which can help decision makers articulate their decisions in a more transparent and justifiable way. Due to its nature, in decision analysis, much of the information required by various models is represented by subjective judgments--very often representing preferences--expressed by one or more experts on a reference set. In complex problems where the reference set can hardly be considered in its entirety, a divide and conquer approach can be helpful, so that judgments are expressed on pairs of alternatives or attributes and can then be combined to reach a global result (see, e.g., [1]). In this way, an originally complex problem was decomposed in many smaller and more tractable subproblems. Such judgments on pairs are called pairwise comparisons and are widely used in a number of multiple-criteria decision-making methods as, for instance, the analytic hierarchy process (AHP) by [PERSON] [2, 3]. In addition to having been employed in the AHP, the technique of pairwise comparisons and some of its variants are currently used in other multi-criteria decision-making techniques as, for instance, multi-attribute utility theory [4]; the best-worst method [5]; ELECTRE [6]; PROMETHEE [7]; MACBETH [8]; and PAPRIKA [9]. Hence, it is hard to underestimate the importance of pairwise comparisons in multi-criteria decision analysis. A pairwise comparison matrix (PCM) is utilized to obtain a priority vector over a set of alternatives for a given criterion or quantify the priorities of the criteria. There are several methods for deriving priority vectors when all elements of the PCM are already known (see, e.g., [10]). In contrast to the elegance of the approach based on PCMs, it must be said that it is often impractical (and sometimes even undesirable) to ask an expert to express their judgments on each possible pair of alternatives. Hence, incomplete pairwise comparison matrices [11] appeared due to decision makers' limited knowledge about some alternatives or criteria, lack of time because of a large number of alternatives, lack of data, or simply to avoid to overload the decision maker asking a significant amount of questions. In decision-making contexts, it is often important to infer the values of the missing comparisons starting from the knowledge of the existing ones. This procedure can be automatic or, even better, the inferred values can act as simple suggestions, and not impositions, to help and guide the expert during the elicitation process. In this paper, we considered an optimal completion algorithm for incomplete PCMs. The first step in this direction consists of considering the missing entries as variables. Let \(\mathbb{R}_{+}^{k}\) be the positive orthant of the \(k\)-dimensional Euclidean space and \(\mathbb{R}_{+}^{n\times n}\) be the set of \(n\times n\) positive matrices. Then, for a given incomplete matrix \(\mathbf{A}\), let \(\mathbf{x}=(x_{1},x_{2},\ldots,x_{k})\in\mathbb{R}_{+}^{k}\) be a vector of missing comparisons expressed as variables \(x_{1},x_{2},\ldots,x_{k}\). At this point, it is reasonable to assume that the missing entries are estimated so that they fit as much as possible with the existing ones. That is, all the entries, known and estimates, minimize the global inconsistency of the matrix \(\mathbf{A}\). In other words, the estimated missing entries must be as coherent as possible with the already elicited judgments. If we consider [PERSON]'s \(CR\) index [2,3], the optimization problem is the following:  
\[\min_{\mathbf{x}>\mathbf{0}}\ \ \lambda_{max}(\mathbf{A}(\mathbf{x})) \tag{1}\]  
where \(\mathbf{A}(\mathbf{x})\) is an incomplete PCM that contains a total of \(2k\) missing entries that can be written as  
\[\mathbf{A}(\mathbf{x})=\mathbf{A}(x_{1},x_{2},\ldots,x_{k})=