Many optimization algorithms represent solutions as permutations. However, despite their apparent simplicity, permutations pose significant challenges—especially for Global Random Search (GRS) algorithms—due to the mutual-exclusivity constraint. This constraint complicates both the learning and sampling of probability distributions over the permutation space, often leading to computationally expensive procedures. A promising alternative involves transforming permutation-encoded solutions into integer vectors using bijective functions on the symmetric group Sn, resulting in what are known as inversion vectors. While inversion vectors have been studied for centuries, a unified and formal framework encompassing all their codifications has been lacking. In this paper, we introduce precise definitions and a unified notation for various types of inversion vector codifications. We establish bijective transformations between them, providing a formal characterization of their relationships and properties. Leveraging this theoretical foundation, we analyze and explain the behavior of GRS algorithms across different permutation problems when using different inversion vector representations.

Advances in large models, reinforcement learning, and open-endedness have accelerated progress toward autonomous agents that can learn and interact in the real world. To achieve this, flexible tools are needed to create rich, yet computationally efficient, environments. While scalable 2D environments fail to address key real-world challenges like 3D navigation and spatial reasoning, more complex 3D environments are computationally expensive and lack features like customizability and multi-agent support. This paper introduces Craftium, a highly customizable and easy-to-use platform for building rich 3D single- and multi-agent environments. We showcase environments of different complexity and nature: from single- and multi-agent tasks to vast worlds with many creatures and biomes, and customizable procedural task generators. Benchmarking shows that Craftium significantly reduces the computational cost of alternatives of similar richness, achieving +2K steps per second more than Minecraft-based frameworks.

This work introduces a growable and modular neural network architecture that naturally avoids catastrophic forgetting and interference in continual reinforcement learning. The structure of each module allows the selective combination of previous policies along with its internal policy accelerating the learning process on the current task. Unlike previous growing neural network approaches, we show that the number of parameters of the proposed approach grows linearly with respect to the number of tasks, and does not sacrifice plasticity to scale. Experiments conducted in benchmark continuous control and visual problems reveal that the proposed approach achieves greater knowledge transfer and performance than alternative methods.

Probability-based algorithms have proven to be a solid alternative for approaching optimization problems. Nevertheless, in many cases, using probabilistic models that efficiently exploit the characteristics of the problem involves large computational overheads, and therefore, lower complexity models such as those that are univariate are usually employed within approximation algorithms. With the motivation to address such an issue, in this article, we aim to introduce an iterative optimization framework that employs generative models to efficiently estimate the parameters of probability models for optimization problems. This allows the use of complex probabilistic models (or those that are appropriate for each problem) in a way that is feasible to apply them iteratively. Specifically, the framework is composed of three elements: a generative model, a probability model whose probability rule is differentiable, and a loss function. The possibility of modifying any of the three elements of the framework offers the flexibility to design algorithms that best adapt to the problem at hand. Experiments conducted on two case studies reveal that the presented approach has strong performance in terms of objective value and execution time when compared to other probability-based algorithms. Moreover, the experimental analysis demonstrates that the convergence of the algorithms is controllable by adjusting the components of the framework. For the sake of reproducibility, the source code, results, scripts, figures, and other material related to the manuscript are available at https://github.com/mikelma/nnco_lib .

Since their introduction, Estimation of Distribution Algorithms (EDAs) have proved to be very competitive algorithms to solve many optimization problems. However, despite recent developments, in the case of permutation-based combinatorial optimization problems, there are still many aspects that deserve further research. One of them is the influence of the codification employed to represent the solutions on the overall performance of the algorithm. When considering classical EDAs, optimizing permutation problems is challenging, and specific mechanisms are needed to hold the restrictions associated with the permutation nature of solutions.In this paper, in addition to the permutation-vector codification, we investigate alternative representations to describe solutions of permutation problems in the context of EDAs. In order to evaluate their influence, we adopted a classical EDA and conducted an experimental study on two different permutation problems and representations for codifying solutions. The results revealed a narrow relationship between the type of combinatorial problem optimized and the selected representation used to codify its solutions. Moreover, the results point out that choosing the appropriate representation to codify solutions of the given permutation problem is critical for the performance of the algorithm.