Exact computation of the expectation curves for uniform crossover

Chicano, F., Whitley D., & Alba E. (2012). Exact computation of the expectation curves for uniform crossover. (Soule, T., & Moore J. H., Ed.).Genetic and Evolutionary Computation Conference, GECCO'12, Philadelphia, PA, USA, July 7-11, 2012. 1301–1308.Uniform crossover is a popular operator used in genetic algorithms to combine two tentative solutions of a problem represented as binary strings. We use the Walsh decomposition of pseudo-Boolean functions and properties of Krawtchouk matrices to exactly compute the expected value for the fitness of a child generated by uniform crossover from two parent solutions. We prove that this expectation is a polynomial in , the probability of selecting the best-parent bit. We provide efficient algorithms to compute this polynomial for ONEMAX and MAX-kSAT problems, but the results also hold for domains such as NK-Landscapes.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech. Spanish Ministry of Science and Innovation and FEDER under contract TIN2011-28194 (the roadME project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project)

Exact computation of the expectation surfaces for uniform crossover along with bit-flip mutation

Theoretical Computer Science 545, 2014, pp.76-93,Uniform crossover and bit-flip mutation are two popular operators used in genetic algorithms to generate new solutions in an iteration of the algorithm when the solutions are represented by binary strings. We use the Walsh decomposition of pseudo-Boolean functions and properties of Krawtchouk matrices to exactly compute the expected value for the fitness of a child generated by uniform crossover followed by bit-flip mutation from two parent solutions. We prove that this expectation is a polynomial in ρ, the probability of selecting the best-parent bit in the crossover, and μ, the probability of flipping a bit in the mutation. We provide efficient algorithms to compute this polynomial for Onemax and MAX-SAT problems, but the results also hold for other problems such as NK-Landscapes. We also analyze the features of the expectation surfaces.Spanish Ministry of Science and Innovation and FEDER under contract TIN2011-28194 (the roadME project). Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-11-1-0088