## 2 Related Work
### Searchability Deterioration of Pareto Dominance-Based Sorting  
As it was stated in the previous section, the Pareto dominance-based selection criterion does not provide enough selection pressure to guide the evolution process in the case of many-objective optimization [4; 9; 10]. Numerous attempts have been made to overcome this problem. We classify and schematically represent them in Figure 1.  
To the first group of approaches, we classify those that require specific **points** to guide the selection process. These can be knee-points [9; 15] and reference points either provided by a user or identified automatically, for example, the reference points in \(NSGA\)-\(III\)[16]. Apart from directing the selection process towards the true Pareto-frontier, these reference points also contribute to the diversity of the solution set as they are often widely distributed in the search space.  
Figure 1: Solutions for the searchability deterioration problem.  
The next group of methods employs **indicators** to select the best solutions. Among recent works in this area, we can mention [10; 17], which incorporate several ranking methods simultaneously, and [18] which used a novel convergence indicator. Computing the values of these indicators increases the execution time of the optimization process.  
Another group of approaches aims to solve the root of the problem by **reducing the dimensionality**. This can be achieved, for example, by using a transfer matrix [19] or by considering single-objective optimization problems via scalarization [12]. Such methods simplify complex relationships between objectives. This can lead to under-coverage of the search space.  
There was also a suggestion to **ensemble** several methods. In the approach from [20], one population is evolved using conventional non-dominated sorting, and another is evolved using an approximate non-dominated sorting procedure. This combination is supposed to improve both diversity and convergence at the same time.  
An alternative approach proposed in [21] performs sorting not with the aim to select the best individuals but to remove the most unfitted ones (**problem redefinition**). It was implemented within the the framework of _NSGA-III_ and is based on the idea of the niche-preservation operation of this algorithm.  
The methods from the last group attempt to **re-define** the way the **dominance** is calculated. According to [11], we can define two main subgroups here: _value_-based dominance and _number_-based dominance. Value-based dominance methods modify the Pareto dominance by changing the objective values of the solutions when comparing them, for example, \(\epsilon\)-dominance [22]. Number-based methods try to compare a solution to another one by counting the number of objectives where it is better than, the same as, or worse than another; \((1-k)\)-dominance [23] and \(L\)-optimality [24] are prominent examples. Alternative methods also include the fuzzification of Pareto-dominance, see [25; 26], and the definition of dominance within a subgroup of a population, for example, \(\theta\)-dominance, which was used to improve the convergence of _NSGA-III_ in [27].  
The method studied in this work, \(k\)-Pareto optimality, falls within the subgroup of number-based approaches. As other methods from this group, it is straightforward and easy to implement. It does not require additional constructions such as reference points or indicators. In addition, to the best of our knowledge, it is the only procedure with theoretical guarantees of diversity, see [14]. As the current paper is a proof of concept, we perform empirical comparison of the newly proposed method with the well-established approaches, _NSGA-II_ and _NSGA-III_. In future work, we will aim to compare its performance with that of other value- and number-based approaches.