On complexity of optimized crossover for binary representations

We consider the computational complexity of producing the best possible offspring in a crossover, given two solutions of the parents. The crossover operators are studied on the class of Boolean linear programming problems, where the Boolean vector of variables is used as the solution representation. By means of efficient reductions of the optimized gene transmitting crossover problems (OGTC) we show the polynomial solvability of the OGTC for the maximum weight set packing problem, the minimum weight set partition problem and for one of the versions of the simple plant location problem. We study a connection between the OGTC for linear Boolean programming problem and the maximum weight independent set problem on 2-colorable hypergraph and prove the NP-hardness of several special cases of the OGTC problem in Boolean linear programming.Comment: Dagstuhl Seminar 06061 "Theory of Evolutionary Algorithms", 200

The Factor-Portfolios Approach to Asset Management using Genetic Algorithms

We present an investment process that: (i) decomposes securities into risk factors; (ii) allows for the construction of portfolios of assets that would selectively expose the manager to desired risk factors; (iii) perform a risk allocation between these portfolios, allowing for tracking error restrictions in the optimization process and (iv) give the flexibility to manage dinamically the transfer coeffficient (TC). The contribution of this article is to present an investment process that allows the asset manager to limit risk exposure to macro-factors - including expectations on correlation dynamics - whilst allowing for selective exposure to risk factors using mimicking portfolios that emulate the behaviour of given specific. An Artificial Intelligence (AI) optimisation technique is used for risk-budget allocation to factor-portfolios.Active Management, Portfolio Optimization, Genetic Algorithms, Propensities. Classification JEL: G11; G14; G32.

Iterated function systems and shape representation

We propose the use of iterated function systems as an isomorphic shape representation scheme for use in a machine vision environment. A concise description of the basic theory and salient characteristics of iterated function systems is presented and from this we develop a formal framework within which to embed a representation scheme. Concentrating on the problem of obtaining automatically generated two-dimensional encodings we describe implementations of two solutions. The first is based on a deterministic algorithm and makes simplifying assumptions which limit its range of applicability. The second employs a novel formulation of a genetic algorithm and is intended to function with general data input. Keywords: Machine Vision, Shape Representation, Iterated Function Systems, Genetic Algorithms

An Analysis of the Genetic Algorithm and Abstract Search Space Visualisation

The Genetic Algorithm (Holland, 1975) is a powerful search technique based upon the principles of Darwinian evolution. In its simplest form the GA consists of three main operators - crossover, mutation and selection. The principal theoretical treatment of the Genetic Algorithm (GA) is provided by the Schema Theorem and building block hypothesis (Holland, 1975). The building block hypothesis describes the GA search process as the combination, sampling and recombination of fragments of solutions known as building blocks. The crossover operator is responsible for the combination of building blocks, whilst the selection operator allocates increasing numbers of samples to good building blocks. Thus the GA constructs the optimal (or near-optimal) solution from those fragments of solutions which are, in some sense, optimal. The first part of this thesis documents the development of a technique for the isolation of building blocks from the populations of the GA. This technique is shown to extract exactly those building blocks of interest - those which are sampled most regularly by the GA. These building blocks are used to empirically investigate the validity of the building block hypothesis. It is shown that good building blocks do not combine to form significantly better solution fragments than those resulting from the addition of randomly generated building blocks to good building blocks. This results casts some doubt onto the value of the building block hypothesis as an account of the GA search process (at least for the functions used during these experiments). The second part of this thesis describes an alternative account of the action of crossover. This account is an approximation of the geometric effect of crossover upon the population of samples maintained by the GA. It is shown that, for a simple function, this description of the crossover operator is sufficiently accurate to warrant further investigation. A pair of performance models for the GA upon this function are derived and shown to be accurate for a wide range of crossover schemes. Finally, the GA search process is described in terms of this account of the crossover operator and parallels are drawn with the search process of the simulated annealing algorithm (Kirkpatrick et al, 1983). The third and final part of this thesis describes a technique for the visualisation of high dimensional surfaces, such as are defined by functions of many parameters. This technique is compared to the statistical technique of projection pursuit regression (Friedman & Tukey, 1974) and is shown to compare favourably both in terms of computational expense and quantitative accuracy upon a wide range of test functions. A fundamental flaw of this technique is that it may produce poor visualisations when applied to functions with a small high frequency (or order) components