This article may be found at Adaptive Meta-Lamarckian http://www.soton.ac.uk/~cedc/posters.html Learning in Hybrid Genetic Algorithms

Correspondence to A variety of constrained and unconstrained nonlinear local [email protected], or Y.S. Ong, C search methods were employed Computational Engineering and in the study. Nine hybrid GA- Design Centre, LSs are presented here: GA-BC,E Search traces (average of 20 runs) for University of Southampton, GA-CO, GA-DS, GA-HO, GA- minimizing 10D Griewank function Highfield, Southampton, using the Adaptive Meta-Lamarckian FL, GA-LA, GA-NM, GA-PD Strategies, i.e., Traces GA-S1 and GA- SO17 1BJ, U.K. and GA-PO. These D S2. Tel. +44-(0)23-8059-3392, abbreviations have the Fax +44-(0)23-8059-3230. following meanings: The convergence trends for GA: Standard GA; C the various adaptive GA-BC: GA with Bit Climbing strategies of Meta-Lamarckian Algorithm by Davis; learning when applied on the GA-CO: GA with Complex Method of M.J.Box as implemented by Griewank function is as Schwefel; shown in the figure. It is seen GA-DS: GA with Davies, Swann that both adaptive strategies, and Campey Search with Gram- GA-S1 and GA-S2, were able Schmidt orthogonalization as to correctly select the most C implemented by Schwefel; appropriate local search GA-HO: GA with Hooke and Jeeves method for the Griewank Direct Search by Siddall; GA-FL: GA with Fletcher’s 1972 function, thus displaying E method by Siddall; performances close to GA-DS. GA-LA: GA with Repeated A Stochastic Approach, Biased Lagrangian Interpolation as The other great advantage of D Roulette Wheel – GA-S2 implemented by Schwefel; the adaptive Meta- GA-NM: GA with Simplex Method Lamarckian GA-LS strategies We present strategies for by Nelder and Meade; is that further improvement hybrid Genetic Algorithm- Experimental GA-PD: GA with Powell’s Direct LocalC Searches (GA-LS) control Studies Search Method as produced by of search performance further AERE Harwell; search may be attained when that decide, at runtime, which The basic steps of the hybrid GA-PO: GA with Powell’s Direct human designer knowledge is local method from a pool of GA-LS search with Meta- different local methods, is Search Method [27] as incorporated into the search. Lamarckian Learning are implemented by Schwefel; Trace GA-S2A illustrates the chosen to locally improve the outlined below: next chromosome. The use of case where the DS method is multiple local methods during BEGIN biased with twice the chances a hybrid GA-LS search in the Initialize: Generate an initial GA population. of being selected, as spirit of Lamarckian learning While (Stopping condition are not satisfied) compared to the other local is termed Meta-Lamarckian Evaluation of All Individuals in the Population methods in the same pool. Learning. Two adaptive For each individual in the population Trace GA-S2B is obtained strategies are studied on the • Select LS using the Meta-Lamarckian Learning when the designer chooses to Strategy employed and proceed with local improvement. use six local methods (PO, Griewank test function. The • Reward/Update fitness of selected local search method. proposed approach is shown • Replace the genotype in the population with the locally improved solution. NM, CO, BC, PD and DS) as to yield robust and improved End For the pool to perform a search design search performance. Apply standard GA operators to create a new population; i.e., on the Griewank function. It Selection, Mutation and Crossover. is seen that superior search Adaptive Meta- End While performances are obtained END over the most appropriate Lamarckian Learning local method, GA-DS. Algorithm for Hybrid GA-LS With Inspired by the research Meta-Lamarckian Learning GA-B: This represents the estimated works on different kinds of performance one might expect effort in social evolution, two Experimental to get when the selection of a adaptive strategies of Meta- fixed LS from the entire pool for GA hybrid is made randomly; Lamarckian learning in Results On GA-AV: This is an average of the hybrid GA-LS are structured Griewank performances for the entire pool to promote cooperation and of nine fixed LS hybrids on a competition among the Benchmark Test problem. different LSs, working Problem The convergence trends of the together to accomplish the various local methods in GA Griewank function is a shared optimization goal. hybrid are shown in the figure. It multimodal function with Search traces (average of 20 runs) for is evident how the choice of the minimizing 10D Griewank function many local minima and a local search method employed using the Adaptive Meta-Lamarckian Two Cooperation global minimum located at greatly effect the efficiency of Strategies with the incorporation of (0,...,0). It has a very rugged Human Designer Knowledge, i.e., and Competition problem searches. The most landscape and is given by: Traces GA-S2A and GA-S2B. Adaptive Meta- appropriate local search method for the Griewank function is Lamarckian shown to be GA-DS hybrid. Strategies

A Heuristic Approach, Sub-problem One-dimensional slice of the Search traces (average of 20 runs) for Decomposition – GA-S1 Griewank function for [-200, 200]10. minimizing 10D Griewank function using the nine local search methods.