SCOlib User Manual Version Version 1.0

## Evolutionary Algorithms

### Overview

Evolutionary search is an adaptive random search that maintains a collection of solutions that are ranked by their performance and uses a competition between these solutions to select solutions for further processing. Research on evolutionary search algorithms incorporates elements of both biological evolution and global optimization. These algorithms are inspired by biological evolutionary mechanisms and are often used to perform global optimization.

The exemplars of evolutionary search algorithms are genetic algorithms, evolutionary strategie and evolutionary programming [ BacHofSch91 , Fog95 , Gold89a ]. The design and motivation for these algorithms are different, but they incorporate the same basic adaptive

components [ BacHof91, HofBac90 ]. These methods use a collection of solutions (population of individuals) that are updated iteratively using selection mechanisms and genetic operators. The general process of each iteration (generation) is described

as follows:

- Initialize population (with uniformly generated solutions)
- Repeat
  - Evaluate solutions in the population
  - Perform competitive selection
  - Apply genetic operators
  - Perform local search (optional)
- Until convergence criteria satisfied

EA-1: Pseudo-algorithm for a genetic algorithm.

The selection mechanism performs a competition to select a subset of the solutions for further processing. The genetic operators are used to generate deviates from the selected individuals. Two types of genetic operators are commonly employed: mutation and recombination. Mutation uses a single individual to generate a deviate that is located in the local neighborhood of the individual. Recombination uses two individuals to generate another individual that is typically located in the smallest hypercube that contains them both. Local search is another genetic operator that is sometimes employed with evolutionary algorithms to refine solutions in their local neighborhood.

Using these genetic operators, evolutionary search algorithms perform a global search. Global convergence is not guaranteed for all evolutionary algorithms [Rud94], but experiments with these algorithms indicate that they often converge to regions of the search space that contain near-optimal solutions.

### Genetic Algorithms

For historical reasons, the development of evolutionary algorithms within SGOPT has focused on Genetic Algorithms (GAs). The GA was initially described using populations of binary strings in , which are evaluated by the objective function (fitness function) [Hol76,Gol89a,Mic96]. When searching spaces other than , the objective function decodes the binary string and performs the function evaluation.

Holland [Hol76] proposed a selection mechanism that stochastically selects individuals with probability

This selection mechanism is called proportional selection, since the number of copies of an individual will be in proportion to the its fraction of the population's total fitness. This method assumes the GA is minimizing and that the global minimum is greater than or equal to zero, but it can be easily modified to perform selection when maximizing a function, or when the global minimum is negative.

The binary GA proposed by Holland uses mutation and crossover operators. With binary strings, the mutation operator changes a single bit on a string, and it is typically used with low frequency. The crossover operator picks two points on the the binary representation and generates the new sample by taking all of the bits between these points from one parent and the remaining bits from the other parent. For example, if and the chosen points are and :

    Parent(1): 1111111111    Parent(2): 0000000000    Sample: 0011110000

Crossover is typically used with high frequency, so most of the individuals in each generation are generated using crossover.

The manner in which the parameters of the objective function are encoded on each string does not affect the mechanisms of the GA, though it can affect the GA's search dynamics. In particular, much research has been done examining how crossover composes and disrupts patterns in binary strings, based on their contribution to the total fitness of the individual [Gol89b,SpeDej91a,SpeDej91b,VosLie91]. This research has motivated the use of modified crossover operators that restrict the distribution of crossover points. For example, if the binary string is decoded into a vector of integers or floating point values, then crossover is often applied only between the integer or floating point values on the binary string [Dav91].

### Panmictic and Geographically Structured Genetic Algorithms

GAs can be distinguished by the manner in which the selection mechanism and genetic operators are applied to the population. Panmictic GAs employ selection mechanisms (like proportional selection) that use global information about the entire population to perform a global selection. In proportional selection the population's total fitness is used to perform selection. Panmictic GAs apply the crossover operator to pairs of individuals randomly taken from individuals selected from the entire population.

Geographically structured genetic algorithms (GSGAs) perform a structured selection in which individuals compete against a fixed subset of the population, and the genetic operators are applied to individuals selected from these subsets. The most common way of structuring the selection mechanism uses a toroidal two dimensional grid like the one in Figure EA-2 [AckLit93,ColJef91,McI92,SpiMan91]. Every element of the population is assigned to a location on the grid. The grid locations are not necessarily related to the individuals' solutions. They are often arbitrary designations used to perform selection. Thus, there are distinct notions of locality with respect to the population grid and the search space. When local search is performed with GSGAs, it is performed in the search space. When local selection is performed, it is performed in the population grid.

EA-2: The two dimensional grid used by GSGAs to define population subsets.

Two general methods of local selection have been used to perform selection in GSGAs: (1) fixed size neighborhoods have been used to define the set of neighboring individuals [DavYamNak93,GorWhi93], and (2) random walks have been used to stochastically sample the locations of neighboring individuals [ColJef91,McI92], Figure EA-2 illustrates the fixed size neighborhoods that could be used to perform selection. Proportional selection is applied to the solutions in each of these neighborhoods. Since one individual is assigned to each grid location, the selection procedure is used to select only as many individuals as are necessary to use the genetic operators. For example, two individuals will be selected if crossover is used. The new individual generated from a genetic operator is assigned to the grid location at which selection is performed.

The early motivation for GSGAs came from SIMD designs for GAs. McInerney [McI92] describes a SIMD GSGA and analyzes the effect of different methods of local selection. He shows how local selection encourages local regions of the 2D grid to form demes of very similar individuals, and argues that inter-deme competition enables GSGAs to perform search while maintaining diversity in the population. He observes that selection using random walks gave very good results in his experiments. He notes that this method enabled good solutions to diffuse through the population, while strongly encouraging the formation of demes.

Gordon and Whitley [GorWhi93] argue that the algorithmic nature of GSGAs may be of interest, independent from their implementation on a particular architecture. They experimentally compare GSGAs to panmictic GAs and observe that the GSGAs provide superior performance. This philosophy is echoed by Davidor, Yamada and Nakano [DavYamNak93] in their motivation for the ECO framework. The ECO framework provides a serial design for implementing a geographically structured GA.

Finally, we note that our definition of GSGAs includes GAs which structure the selection at a fine granularity. A number of GAs have been proposed whose competitive selection is intermediate between GSGAs and panmictic GAs. Muhlenbein [Muh91] makes a similar distinction and describes a GA which uses a set of independent subpopulations and structures the inter-population communication with a ladder structure. These subpopulations are typically small, so they perform a localized search of the function. For example, Whitley [WhiDomDas91] illustrates how a small population can perform a locallized search in the context of neural network optimization problems. Inter-population communication enables populations to combine disparate solutions and enables them to perform a global search.