Enhancing partition crossover with articulation points analysis

Partition Crossover is a recombination operator for pseudo-Boolean optimization with the ability to explore an exponential number of solutions in linear or square time. It decomposes the objective function as a sum of subfunctions, each one depending on a different set of variables. The decomposition makes it possible to select the best parent for each subfunction independently, and the operator provides the best out of $2^q$ solutions, where $q$ is the number of subfunctions in the decomposition. These subfunctions are defined over the connected components of the recombination graph: a subgraph of the objective function variable interaction graph containing only the differing variables in the two parents. In this paper, we advance further and propose a new way to increase the number of linearly independent subfunctions by analyzing the articulation points of the recombination graph. These points correspond to variables that, once flipped, increase the number of connected components. The presence of a connected component with an articulation point increases the number of explored solutions by a factor of, at least, 4. We evaluate the new operator using Iterated Local Search combined with Partition Crossover to solve NK Landscapes and MAX-SAT.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Funding was provided by the Fulbright program, the Spanish Ministry of Education, Culture and Sport (CAS12/00274), the Spanish Ministry of Economy and Competitiveness and FEDER (TIN2014-57341-R and TIN2017-88213-R), the Air Force Office of Scientific Research, (FA9550-11-1-0088), the Leverhulme Trust (RPG-2015-395), the FAPESP (2015/06462-1) and CNPq (304400/2014-9)

Quasi-Optimal Recombination Operator

The output of an optimal recombination operator for two parent solutions is a solution with the best possible value for the objective function among all the solutions fulfilling the gene transmission property: the value of any variable in the offspring must be inherited from one of the parents. This set of solutions coincides with the largest dynastic potential for the two parent solutions of any recombination operator with the gene transmission property. In general, exploring the full dynastic potential is computationally costly, but if the variables of the objective function have a low number of non-linear interactions among them, the exploration can be done in $O(4^{\beta}(n+m)+n^2)$ time, for problems with $n$ variables, $m$ subfunctions and $\beta$ a constant. In this paper, we propose a quasi-optimal recombination operator, called Dynastic Potential Crossover (DPX), that runs in $O(4^{\beta}(n+m)+n^2)$ time in any case and is able to explore the full dynastic potential for low-epistasis combinatorial problems. We compare this operator, both theoretically and experimentally, with two recently defined efficient recombination operators: Partition Crossover (PX) and Articulation Points Partition Crossover (APX). The empirical comparison uses NKQ Landscapes and MAX-SAT instances.Universidad de Málaga. Campus de Excelencia International Andalucía Tech. Ministerio de Economía y Competitividad y FEDER (proyecto TIN2017-88213-R

Dynastic Potential Crossover Operator

An optimal recombination operator for two parent solutions provides the best solution among those that take the value for each variable from one of the parents (gene transmission property). If the solutions are bit strings, the offspring of an optimal recombination operator is optimal in the smallest hyperplane containing the two parent solutions. Exploring this hyperplane is computationally costly, in general, requiring exponential time in the worst case. However, when the variable interaction graph of the objective function is sparse, exploration can be done in polynomial time. In this paper, we present a recombination operator, called Dynastic Potential Crossover (DPX), that runs in polynomial time and behaves like an optimal recombination operator for low-epistasis combinatorial problems. We compare this operator, both theoretically and experimentally, with traditional crossover operators, like uniform crossover and network crossover, and with two recently defined efficient recombination operators: partition crossover and articulation points partition crossover. The empirical comparison uses NKQ Landscapes and MAX-SAT instances. DPX outperforms the other crossover operators in terms of quality of the offspring and provides better results included in a trajectory and a population-based metaheuristic, but it requires more time and memory to compute the offspring.This research is partially funded by the Universidad de M\'alaga, Consejería de Economía y Conocimiento de la Junta de Andalucía and FEDER under grant number UMA18-FEDERJA-003 (PRECOG); under grant PID 2020-116727RB-I00 (HUmove) funded by MCIN/AEI/10.13039/501100011033; and TAILOR ICT-48 Network (No 952215) funded by EU Horizon 2020 research and innovation programme. The work is also partially supported in Brazil by São Paulo Research Foundation (FAPESP), under grants 2021/09720-2 and 2019/07665-4, and National Council for Scientific and Technological Development (CNPq), under grant 305755/2018-8

Modularity and Innovation in Complex Systems

The problem of designing, coordinating, and managing complex systems has been central to the management and organizations literature. Recent writings have tended to offer modularity as, at least, a partial solution to this design problem. However, little attention has been paid to the problem of identifying what constitutes an appropriate modularization of a complex system. We develop a formal simulation model that allows us to carefully examine the dynamics of innovation and performance in complex systems. The model points to the trade-off between the destabilizing effects of overly refined modularization and the modest levels of search and a premature fixation on inferior designs that can result from excessive levels of integration. The analysis highlights an asymmetry in this trade-off, with excessively refined modules leading to cycling behavior and a lack of performance improvement. We discuss the implications of these arguments for product and organization design.

Partition crossover for continuous optimization: EPX

Partition crossover (PX) is an efficient recombination operator for gray-box optimization. PX is applied in problems where the objective function can be written as a sum of subfunctions fl(.). In PX, the variable interaction graph (VIG) is decomposed by removing vertices with common variables. Parent variables are inherited together during recombination if they are part of the same connected recombining component of the decomposed VIG. A new way of generating the recombination graph is proposed here. The VIG is decomposed by removing edges associated with subfunctions fl(.) that have similar evaluation for combinations of variables inherited from the parents. By doing so, the partial evaluations of fl(.) are taken into account when decomposing the VIG. This allows the use of partition crossover in continuous optimization. Results of experiments where local optima are recombined indicate that more recombining components are found. When the proposed epsilon-PX (ePX) is compared with other recombination operators in Genetic Algorithms and Differential Evolution, better performance is obtained when the epistasis degree is low

Advances and applications in high-dimensional heuristic optimization

“Applicable to most real-world decision scenarios, multiobjective optimization is an area of multicriteria decision-making that seeks to simultaneously optimize two or more conflicting objectives. In contrast to single-objective scenarios, nontrivial multiobjective optimization problems are characterized by a set of Pareto optimal solutions wherein no solution unanimously optimizes all objectives. Evolutionary algorithms have emerged as a standard approach to determine a set of these Pareto optimal solutions, from which a decision-maker can select a vetted alternative. While easy to implement and having demonstrated great efficacy, these evolutionary approaches have been criticized for their runtime complexity when dealing with many alternatives or a high number of objectives, effectively limiting the range of scenarios to which they may be applied. This research introduces mechanisms to improve the runtime complexity of many multiobjective evolutionary algorithms, achieving state-of-the-art performance, as compared to many prominent methods from the literature. Further, the investigations here presented demonstrate the capability of multiobjective evolutionary algorithms in a complex, large-scale optimization scenario. Showcasing the approach’s ability to intelligently generate well-performing solutions to a meaningful optimization problem. These investigations advance the concept of multiobjective evolutionary algorithms by addressing a key limitation and demonstrating their efficacy in a challenging real-world scenario. Through enhanced computational efficiency and exhibited specialized application, the utility of this powerful heuristic strategy is made more robust and evident”--Abstract, page iv