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.

Subfunction Structure Matters: A New Perspective on Local Optima Networks

Local optima networks (LONs) capture fitness landscape information. They are typically constructed in a black-box manner; information about the problem structure is not utilised. This also applies to the analysis of LONs: knowledge about the problem, such as interaction between variables, is not considered. We challenge this status-quo with an alternative approach: we consider how LON analysis can be improved by incorporating subfunction-based information - this can either be known a-priori or learned during search. To this end, LONs are constructed for several benchmark pseudo-boolean problems using three approaches: firstly, the standard algorithm; a second algorithm which uses deterministic grey-box crossover; and a third algorithm which selects perturbations based on learned information about variable interactions. Metrics related to subfunction changes in a LON are proposed and compared with metrics from previous literature which capture other aspects of a LON. Incorporating problem structure in LON construction and analysing it can bring enriched insight into optimisation dynamics. Such information may be crucial to understanding the difficulty of solving a given problem with state-of-the-art linkage learning optimisers. In light of the results, we suggest incorporation of problem structure as an alternative paradigm in landscape analysis for problems with known or suspected subfunction structure.

Seeking and leveraging alternative variable dependency concepts in gray-box-elusive bimodal land-use allocation problems

Solving land-use allocation problems can help us to deal with some of the most urgent global environmental issues. Since these problems are NP-hard, effective optimizers are needed to handle them. The knowledge about variable dependencies allows for proposing such tools. However, in this work, we consider a real-world multi-objective problem for which standard variable dependency discovery techniques are inapplicable. Therefore, using linkage-based variation operators is unreachable. To address this issue, we propose a definition of problem-dedicated variable dependency. On this base, we propose obtaining masks of dependent variables. Using them, we construct three novel crossover operators. The results concerning real-world test cases show that introducing our propositions into two well-known optimizers (NSGA-II, MOEA/D) dedicated to multi-objective optimization significantly improves their effectiveness.

On Revealing the Hidden Problem Structure in Real-World and Theoretical Problems Using Walsh Coefficient Influence

Gray-box optimization employs Walsh decomposition to obtain non-linear variable dependencies and utilize them to propose masks of variables that have a joint non-linear influence on fitness value. These masks significantly improve the effectiveness of variation operators. In some problems, all variables are non-linearly dependent, making the aforementioned masks useless. We analyze the features of the real-world instances of such problems and show that many of their dependencies may have noise-like origins. Such noise-caused dependencies are irrelevant to the optimization process and can be ignored. To identify them, we propose extending the use of Walsh decomposition by measuring variable dependency strength that allows the construction of the weighted dynamic Variable Interaction Graph (wdVIG). wdVIGs adjust the dependency strength to mixed individuals. They allow the filtering of irrelevant dependencies and re-enable using dependency-based masks by variation operators. We verify the wdVIG potential on a large benchmark suite. For problems with noise, the wdVIG masks can improve the optimizer's effectiveness. If all dependencies are relevant for the optimization, i.e., the problem is not noised, the influence of wdVIG masks is similar to that of state-of-the-art structures of this kind.

Moving between high-quality optima using multi-satisfiability characteristics in hard-to-solve Max3Sat instances

Gray-box optimization proposes effective and efficient optimizers of general use. To this end, it leverages information about variable dependencies and the subfunction-based problem representation. These approaches were already shown effective by enabling \textit{tunnelling} between local optima even if these moves require the modification of many dependent variables. Tunnelling is useful in solving the maximum satisfiability problem (MaxSat), which can be reformulated to Max3Sat. Since many real-world problems can be brought to solving the MaxSat/Max3Sat instances, it is important to solve them effectively and efficiently. Therefore, we focus on Max3Sat instances for which tunnelling fails to introduce improving moves between locally optimal high-quality solutions and the region of globally optimal solutions. We analyze the features of such instances on the ground of phase transitions. Based on these observations, we propose manipulating clause-satisfiability characteristics that allow connecting high-quality solutions distant in the solution space. We utilize multi-satisfiability characteristics in the optimizer built from typical gray-box mechanisms. The experimental study shows that the proposed optimizer can solve those Max3Sat instances that are out of the grasp of state-of-the-art gray-box optimizers. At the same time, it remains effective for instances that have already been successfully solved by gray-box.