A popular trend in evolutionary computation is to adapt numerical algorithms to combinatorial optimization problems. For instance, this is the case of a variety of Particle Swarm Optimization and Differential Evolution implementations for both binary and permutation-based optimization problems. In this paper, after highlighting the main drawbacks of the approaches in literature, we provide an algebraic framework which allows to derive fully discrete variants of a large class of numerical evolutionary algorithms to tackle many combinatorial problems. The strong mathematical foundations upon which the framework is built allow to redefine numerical evolutionary operators in such a way that their original movements in the continuous space are simulated in the discrete space. Algebraic implementations of Differential Evolution and Particle Swarm Optimization are then proposed. Experiments have been held to compare the algebraic algorithms to the most popular schemes in literature and to the state-of-the-art results for the tackled problems. Experimental results clearly show that algebraic algorithms outperform the competitors and are competitive with the state-of-the-art results.

进化计算的一个流行趋势是使数值算法适应组合优化问题。例如，针对基于二进制和基于置换的优化问题，有许多粒子群优化和差分演化实现的情况就是这种情况。在本文中，在强调了文献中方法的主要缺点之后，我们提供了一个代数框架，该框架允许派生出大型数值进化算法的完全离散变体，以解决许多组合问题。建立框架的强大数学基础允许重新定义数值演化算子，从而在离散空间中模拟其在连续空间中的原始运动。然后提出了差分进化和粒子群优化的代数实现。已经进行了实验，以将代数算法与文献中最流行的方案进行比较，并与解决问题的最新结果进行比较。实验结果清楚地表明，代数算法的性能优于竞争对手，并且与最新的结果竞争。