Extremal Optimization - The Technique

The Technique

EO was designed as a local search algorithm for combinatorial optimization problems. Unlike genetic algorithms, which work with a population of candidate solutions, EO evolves a single solution and makes local modifications to the worst components. This requires that a suitable representation be selected which permits individual solution components to be assigned a quality measure ("fitness"). This differs from holistic approaches such as ant colony optimization and evolutionary computation that assign equal-fitness to all components of a solution based upon their collective evaluation against an objective function. The algorithm is initialized with an initial solution, which can be constructed randomly, or derived from another search process.

The technique is a fine-grained search, and superficially resembles a hill climbing (local search) technique. A more detailed examination reveals some interesting principles, which may have applicability and even some similarity to broader population-based approaches (evolutionary computation and artificial immune system). The governing principle behind this algorithm is that of improvement through selectively removing low-quality components and replacing them with a randomly selected component. This is obviously at odds with genetic algorithms, the quintessential evolutionary computation algorithm that selects good solutions in an attempt to make better solutions.

The resulting dynamics of this simple principle is firstly a robust hill climbing search behaviour, and secondly a diversity mechanism that resembles that of multiple-restart search. Graphing holistic solution quality over time (algorithm iterations) shows periods of improvement followed by quality crashes (avalanche) very much in the manner as described by punctuated equilibrium. It is these crashes or dramatic jumps in the search space that permit the algorithm to escape local optima and differentiate this approach from other local search procedures. Although such punctuated-equilibrium behaviour can be "designed" or "hard-coded", it should be stressed that this is an emergent effect of the negative-component-selection principle fundamental to the algorithm.

EO has primarily been applied to combinatorial problems such as graph partitioning and the travelling salesman problem, as well as problems from statistical physics such as spin glasses.