Abstract
The algorithms of multi-objective optimisation had a relative growth in the last years. Thereby, it's requires some way of comparing the results of these. In this sense, performance measures play a key role. In general, it's considered some properties of these algorithms such as capacity, convergence, diversity or convergence-diversity. There are some known measures such as generational distance (GD), inverted generational distance (IGD), hypervolume (HV), Spread($\Delta$), Averaged Hausdorff distance ($\Delta_p$), R2-indicator, among others. In this paper, we focuses on proposing a new indicator to measure convergence based on the traditional formula for Shannon entropy. The main features about this measure are: 1) It does not require tho know the true Pareto set and 2) Medium computational cost when compared with Hypervolume.
Highlights
Nowadays, the evolutionary algorithms (EAs) are used to obtain approximate solutions of multiobjective optimisation problems (MOP) and these EAs are called multi-objective evolutionary algorithms (MOEAs)
The evolutionary algorithms (EAs) are used to obtain approximate solutions of multiobjective optimisation problems (MOP) and these EAs are called multi-objective evolutionary algorithms (MOEAs). Some of these algorithms are very well-known among the community such that NSGA-II (See [1]), SPEA-II (See [2]), MO-PSO (See [3]) and MO-CMA-ES (See [4])
To avoid the phenomenon caused by Pareto relation, some researchers indicates others way of comparative the elements (See [7]) or change into Non-Pareto-based MOEAs, such as indicator-based and aggregation-based approaches (See [8, 9])
Summary
The evolutionary algorithms (EAs) are used to obtain approximate solutions of multiobjective optimisation problems (MOP) and these EAs are called multi-objective evolutionary algorithms (MOEAs). Another one based on this measure was propose so called Hausdorff Measure (see [12]) which combines the IGD and GD and takes its maximum These indicator is efficient to obtain informations about closeness of the output of some algorithm with the True Pareto set. A classical work (See [16]) established a relationship between the points of PS and gradient informations from the problem (2.1) That connection it is known by Karush-Kuhn-Tucker (KKT) conditions for Pareto optimality that we define as follow: Theorem 1 (KKT Condition [16]). This theorem will be fundamental to this paper because we will use this fact to formulate our proposal
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