Presenter : Tsung-Yu Ho

Review Mixing Problem Categories Crossover as a Mixer (√) Crossover as a Innovator (√) Crossover as a Disrupter All in one Crossover as a Disrupter Conclusion

Goldberg, Thierens, and Deb, 1993 “Toward a Better Understanding of Mixing in Genetic Algorithms.” Control Map How well “fixed crossover operators” solve ? GA-easy problem GA-hard Problem Before design a better operator Studies on existed crossover operators.

Four Crossover Categories (Different Model Problem) Crossover as a Mixer Crossover as an Innovator (BB) Crossover as a Disrupter (BB) All in one model (Innovator and Disrupter) What kind of crossover operators can solve problem well? Fixed operators. Other operators.

Robbin’s equilibrium (1918) & Geiringer’s Theorem (1944) Equilibrium distribution (Biology) Predicted equilibrium in GA Linkage disequilibrium (Christiansen, 1989) Rate of convergence to equilibrium (Rabani, 1998) Relaxation Time Uniform crossover : O( ln l ) One-Point crossover : O( l ln l ) Pr Ű gel-Bennett (2001) prove the result.

GA-Easy Problem Goldberg, Thierens, and Deb, 1993 Mixing time Control Map S and Pc Fixed crossover operators are good. GA-Hard Problem BB mixing. Two BB mixing m BB nixing Sweet-spot shrinks exponentially. Population size grows exponential.

Focus on Schemata Disruption Motivation from schema theorem. Studies suggest schema theory should be obeyed. Satisfy schema theorem does not guarantee BB mixing. Schemata Disruption models are useful to compare different crossover operators.

Syswerda (1989) Schema survive for analysis of fixed crossover operators. Spears and De Jong (1991) Multi-point crossover. De Jong and Spears (1992) The effect of crossover operators on population size. Goldberg and Sastry (2001) Satisfy Schema theory issue.

Syswerda, “Uniform crossover in genetic algorithms.”, 1989 Analyzed schema survival rates. Use empirical result to suggest that uniform crossover outperformed one-point crossover and two-point crossover in the most case.

Spears and De Jong, “An Analysis of Multi-Point Crossover.” 1991 Analyzed multi-point crossover to compare with uniform crossover. Uniform has higher schema disruption rate. They suggest the disruption has a positive role in balancing the exploration and exploitation.

Sampling disrupter is important for population diversity when population become homogeneous. Crossover Productivity The property of crossover for diversity. Crossover Productivity is easy to measure.

De Jong and Spears(1992) The effect of crossover operators on population size Small population sizes Uniform performs better. Large population sizes Two-point performs better. But, they did not give any analytical framework.

Goldberg and Sastry, “A Practical Schema Theorem for Genetic Algorithms Design and Tuning.”, Control Map Selection Pressure. Crossover Probability. Show obeying schema theorem can not guarantee BB mixing.

Schema Theorem (Holland, 1978; De Jong, 1975) Proportionate selection and one-point crossover. Simplified Practical Schema Theorem Goldberg & Deb, 1991

Large selection pressure (Sp -> ∞) BB growth ensure even if crossover is fully disruptive Small crossover probability ( Pc -> 0) BB growth is ensured for any selection pressure Easiest way to obey schema theorem Does not guarantee mixing.

8-bit deceptive trap function Single building block Global optimum : Local optimum :

Mixer Models Uniform Crossover is suggested. Disrupter Models n-point crossover is suggested. Innovation Models Fixed Crossover operators are not enough. All in one Models Does not help in GA design.