Limited Perfect Monotonical Surrogates constructed using low-cost recursive linkage discovery with guaranteed output

Surrogates provide a cheap solution evaluation and offer significant leverage for optimizing computationally expensive problems. Usually, surrogates only approximate the original function. Recently, the perfect linear surrogates were proposed that ideally represent the original function. These surrogates do not mimic the original function. In fact, they are another (correct) representation of it and enable a wide range of possibilities, e.g., discovering the optimized function for problems where the direct transformation of the encoded solution into its evaluation is not available. However, many real-world problems can not be represented by linear models, making the aforementioned surrogates inapplicable. Therefore, we propose the Limited Monotonical Perfect Surrogate (LyMPuS), which overcomes this difficulty and enables the comparison of two solutions that differ by a single variable. Our proposition is suitable for limiting the cost of expensive local search procedures. The proposed surrogate is parameterless and can be trained on the fly without any separate surrogate-building step. It uses only the necessary fitness evaluations, and the already-paid costs are not wasted when the model is updated. Finally, it offers low-cost missing-linkage detection and low-cost linkage discovery, guaranteed to find a missing dependency in no more than $2\lceil\log_2(n)\rceil$ steps.

Obtaining Partition Crossover masks using Statistical Linkage Learning for solving noised optimization problems with hidden variable dependency structure

In optimization problems, some variable subsets may have a joint non-linear or non-monotonical influence on the function value. Therefore, knowledge of variable dependencies may be crucial for effective optimization, and many state-of-the-art optimizers leverage it to improve performance. However, some real-world problem instances may be the subject of noise of various origins. In such a case, variable dependencies relevant to optimization may be hard or impossible to tell using dependency checks sufficient for problems without noise, making highly effective operators, e.g., Partition Crossover (PX), useless. Therefore, we use Statistical Linkage Learning (SLL) to decompose problems with noise and propose a new SLL-dedicated mask construction algorithm. We prove that if the quality of the SLL-based decomposition is sufficiently high, the proposed clustering algorithm yields masks equivalent to PX masks for the noise-free instances. The experiments show that the optimizer using the proposed mechanisms remains equally effective despite the noise level and outperforms state-of-the-art optimizers for the problems with high noise.