Optimizing One Million Variable NK Landscapes by Hybridizing Deterministic Recombination and Local Search

In gray-box optimization, the search algorithms have access to the variable interaction graph (VIG) of the optimization problem. For Mk Landscapes (and NK Landscapes) we can use the VIG to identify an improving solution in the Hamming neighborhood in constant time. In addition, using the VIG, deterministic Partition Crossover is able to explore an exponential number of solutions in a time that is linear in the size of the problem. Both methods have been used in isolation in previous search algorithms. We present two new gray-box algorithms that combine Partition Crossover with highly efficient local search. The best algorithms are able to locate the global optimum on Adjacent NK Landscape instances with one million variables. The algorithms are compared with a state-of-the-art algorithm for pseudo-Boolean optimization: Gray-Box Parameterless Population Pyramid. The results show that the best algorithm is always one combining Partition Crossover and highly efficient local search. But the results also illustrate that the best optimizer differs on Adjacent and Random NK Landscapes.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

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)

Optimal Recombination in Genetic Algorithms

This paper surveys results on complexity of the optimal recombination problem (ORP), which consists in finding the best possible offspring as a result of a recombination operator in a genetic algorithm, given two parent solutions. We consider efficient reductions of the ORPs, allowing to establish polynomial solvability or NP-hardness of the ORPs, as well as direct proofs of hardness results

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

Unbiased Black-Box Complexities of Jump Functions

We analyze the unbiased black-box complexity of jump functions with small, medium, and large sizes of the fitness plateau surrounding the optimal solution. Among other results, we show that when the jump size is $(1/2 - \varepsilon)n$, that is, only a small constant fraction of the fitness values is visible, then the unbiased black-box complexities for arities $3$ and higher are of the same order as those for the simple \textsc{OneMax} function. Even for the extreme jump function, in which all but the two fitness values $n/2$ and $n$ are blanked out, polynomial-time mutation-based (i.e., unary unbiased) black-box optimization algorithms exist. This is quite surprising given that for the extreme jump function almost the whole search space (all but a $\Theta(n^{-1/2})$ fraction) is a plateau of constant fitness. To prove these results, we introduce new tools for the analysis of unbiased black-box complexities, for example, selecting the new parent individual not by comparing the fitnesses of the competing search points, but also by taking into account the (empirical) expected fitnesses of their offspring.Comment: This paper is based on results presented in the conference versions [GECCO 2011] and [GECCO 2014