From Direct to Directional Variable Dependencies—Nonsymmetrical Dependencies Discovery in Real-World and Theoretical Problems

The knowledge about variable interactions is frequently employed in state-of-the-art research concerning genetic algorithms (GAs). Whether these interactions are known a priori (gray-box optimization) or are discovered by the optimizer (black-box optimization), they are used for many purposes, including proposing more effective mixing operators. Frequently, the quality of the problem structure decomposition is decisive to the optimizers’ effectiveness. However, in gray- and black-box optimization, the dependency between the variables is assumed to be symmetric. This work identifies and defines the nonsymmetrical (directional) variable dependencies. We show that these dependencies may exist (together with symmetrical) in the considered real-world problem, in which we must optimize subsequent variable groups (one after the other) in the appropriate optimization order that is not known by the optimizer. To improve GA’s effectiveness in solving the problem of such features, we propose a new linkage learning (LL) technique that can discover symmetrical and nonsymmetrical dependencies (in binary and nonbinary discrete domains) and distinguish them from each other. We show that telling these two types of dependencies from each other may significantly increase the optimizer’s effectiveness in solving real-world and theoretical problems with nonsymmetrical dependencies. Finally, we show that using the proposed LL technique does not deteriorate the effectiveness of the state-of-the-art optimizer in solving typical benchmarks containing only symmetrical dependencies.