Life on the Edge: Characterising the Edges of Mutually Non-dominating Sets

Copyright © 2014 Massachusetts Institute of TechnologyMulti-objective optimisation yields an estimated Pareto front of mutually nondominating solutions, but with more than three objectives understanding the relationships between solutions is challenging. Natural solutions to use as landmarks are those lying near to the edges of the mutually non-dominating set. We propose four definitions of edge points for many-objective mutually non-dominating sets and examine the relations between them. The first defines edge points to be those that extend the range of the attainment surface. This is shown to be equivalent to finding points which are not dominated on projection onto subsets of the objectives. If the objectives are to be minimised, a further definition considers points which are not dominated under maximisation when projected onto objective subsets. A final definition looks for edges via alternative projections of the set. We examine the relations between these definitions and their efficacy in many dimensions for synthetic concave- and convex shaped sets, and on solutions to a prototypical many-objective optimisation problem, showing how they can reveal information about the structure of the estimated Pareto front. We show that the “controlling dominance area of solutions” modification of the dominance relation can be effectively used to locate edges and interior points of high-dimensional mutually non-dominating sets

Visualising many-objective populations

Copyright © 2012 ACM14th International Conference on Genetic and Evolutionary Computation (GECCO 2012), Philadelphia, USA, 7-11 July 2012Optimisation problems often comprise a large set of objectives, and visualising the set of solutions to a problem can help with understanding them, assisting a decision maker. If the set of objectives is larger than three, visualising solutions to the problem is a difficult task. Techniques for visualising high-dimensional data are often difficult to interpret. Conversely, discarding objectives so that the solutions can be visualised in two or three spatial dimensions results in a loss of potentially important information. We demonstrate four methods for visualising many-objective populations, two of which use the complete set of objectives to present solutions in a clear and intuitive fashion and two that compress the objectives of a population into two dimensions whilst minimising the information that is lost. All of the techniques are illustrated on populations of solutions to optimisation test problems

Edges of Mutually Non-dominating Sets

Copyright © 2013 ACM. This is the accepted, peer-reviewed version of the work. It is posted here for your personal use. Not for redistribution. The definitive Version of Record was published in Proceedings of the 15th annual conference on Genetic and Evolutionary Computation (GECCO ’13), pp. 607-614, http://dx.doi.org/10.1145/2463372.246345215th annual conference on Genetic and Evolutionary Computation (GECCO ’13), Amsterdam, The Netherlands, 6-10 July 2013Notes: Won the Best Paper Award in the EMO trackMulti-objective optimisation yields an estimated Pareto front of mutually non-dominating solutions, but with more than three objectives understanding the relationships between solutions is challenging. Natural solutions to use as landmarks are those lying near to the edges of the mutually non-dominating set. We propose four definitions of edge points for many-objective mutually non-dominating sets and examine the relations between them. The first defines edge points to be those that extend the range of the attainment surface. This is shown to be equivalent to finding points which are not dominated on projection onto subsets of the objectives. If the objectives are to be minimised, a further definition considers points which are not dominated under maximisation when projected onto objective subsets. A final definition looks for edges via alternative projections of the set. We examine the relations between these definitions and their efficacy for synthetic concave- and convex-shaped sets, and on solutions to a prototypical many-objective optimisation problem, showing how they can reveal information about the structure of the estimated Pareto front

The Bayesian Decision Tree Technique with a Sweeping Strategy

The uncertainty of classification outcomes is of crucial importance for many safety critical applications including, for example, medical diagnostics. In such applications the uncertainty of classification can be reliably estimated within a Bayesian model averaging technique that allows the use of prior information. Decision Tree (DT) classification models used within such a technique gives experts additional information by making this classification scheme observable. The use of the Markov Chain Monte Carlo (MCMC) methodology of stochastic sampling makes the Bayesian DT technique feasible to perform. However, in practice, the MCMC technique may become stuck in a particular DT which is far away from a region with a maximal posterior. Sampling such DTs causes bias in the posterior estimates, and as a result the evaluation of classification uncertainty may be incorrect. In a particular case, the negative effect of such sampling may be reduced by giving additional prior information on the shape of DTs. In this paper we describe a new approach based on sweeping the DTs without additional priors on the favorite shape of DTs. The performances of Bayesian DT techniques with the standard and sweeping strategies are compared on a synthetic data as well as on real datasets. Quantitatively evaluating the uncertainty in terms of entropy of class posterior probabilities, we found that the sweeping strategy is superior to the standard strategy

Visualising Mutually Non-dominating Solution Sets in Many-objective Optimisation

Copyright © 2013 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works.As many-objective optimization algorithms mature, the problem owner is faced with visualizing and understanding a set of mutually nondominating solutions in a high dimensional space. We review existing methods and present new techniques to address this problem. We address a common problem with the well-known heatmap visualization, since the often arbitrary ordering of rows and columns renders the heatmap unclear, by using spectral seriation to rearrange the solutions and objectives and thus enhance the clarity of the heatmap. A multiobjective evolutionary optimizer is used to further enhance the simultaneous visualization of solutions in objective and parameter space. Two methods for visualizing multiobjective solutions in the plane are introduced. First, we use RadViz and exploit interpretations of barycentric coordinates for convex polygons and simplices to map a mutually nondominating set to the interior of a regular convex polygon in the plane, providing an intuitive representation of the solutions and objectives. Second, we introduce a new measure of the similarity of solutions—the dominance distance—which captures the order relations between solutions. This metric provides an embedding in Euclidean space, which is shown to yield coherent visualizations in two dimensions. The methods are illustrated on standard test problems and data from a benchmark many-objective problem

Rank-based dimension reduction for many-criteria populations

Copyright © 2011 ACM13th annual conference on Genetic and Evolutionary Computation (GECCO '11), Dublin, Ireland, 12-16 July 2011Interpreting individuals described by a set of criteria can be a difficult task when the number of criteria is large. Such individuals can be ranked, for instance in terms of their average rank across criteria as well as by each distinct criterion. We therefore investigate criteria selection methods which aim to preserve the average rank of individuals but with fewer criteria. Our experiments show that these methods perform effectively, identifying and removing redundancies within the data, and that they are best incorporated into a multi-objective algorithm

Visualisation and ordering of many-objective populations

Copyright © 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.We introduce novel methods of visualising and ordering multi-and many-objective populations. We compare individuals by the probability that one will beat another in a tournament on a randomly selected objective. This defines a weighted directed graph representing the population. We introduce a novel graphical representation of the many objective population based on Pareto shells. We examine leagues, Pareto shells, preference ordering, average rank, outflow, the stationary distribution and the power index for ordering the population finding that the average rank is equivalent to outflow and that these together with the power index are generally superior. Finally, we show how to seriate objectives to enhance the interpretability of heatmap visualisations

A Toolkit for Generating Scalable Stochastic Multiobjective Test Problems

Real-world optimization problems typically include uncertainties over various aspects of the problem formulation. Some existing algorithms are designed to cope with stochastic multiobjective optimization problems, but in order to benchmark them, a proper framework still needs to be established. This paper presents a novel toolkit that generates scalable, stochastic, multiobjective optimization problems. A stochastic problem is generated by transforming the objective vectors of a given deterministic test problem into random vectors. All random objective vectors are bounded by the feasible objective space, defined by the deterministic problem. Therefore, the global solution for the deterministic problem can also serve as a reference for the stochastic problem. A simple parametric distribution for the random objective vector is defined in a radial coordinate system, allowing for direct control over the dual challenges of convergence towards the true Pareto front and diversity across the front. An example for a stochastic test problem, generated by the toolkit, is provided

Learning assignment order in an ant colony optimiser for the university course timetabling problem

This is the author accepted manuscript. The final version is available from ACM via the DOI in this recordPrevious studies have employed Ant Colony Optimisation to solve the University Course Timetabling task — which requires the order of lecture assignments to be defined for its construction graph. Various heuristic or random ordering techniques have been proposed in the literature, but uncertainty remains regarding the best approach for this. We investigate the effect that permuting assignment order has on the quality of timetable produced. As part of this we develop a novel MAX-MIN Ant System including dynamic constraint handling and partial function evaluations. We also explore algorithm variants with and without Local Search and employ a form of transfer learning to identify appropriate permutations. We find that between smaller problems in the International Timetabling Competition 2007 benchmark, timetabling performance can be improved using such an approach. However we find that we lose this performance gain when attempting to transfer to considerably larger problems — indicating that similar structures are required when using a ‘learnt’ permutation approach in such a framework.Engineering and Physical Sciences Research Council (EPSRC

On Evidence Weighted Mixture Classification

2005 Joint Annual Meeting of the Interface and the Classification Society of North America, St. Louis, Missouri, 8-12 June 2005Calculation of the marginal likelihood or evidence is a problem central to model selection and model averaging in a Bayesian framework. Many sampling methods, especially (Reversible Jump) Markov chain Monte Carlo techniques, have been devised to avoid explicit calculation of the evidence, but they are limited to models with a common parameterisation. It is desirable to extend model averaging to models with disparate architectures and parameterisations. In this paper we present a straightforward general computational scheme for calculating the evidence, applicable to any model for which samples can be drawn from the posterior distribution of parameters conditioned on the data. The scheme is demonstrated on a simple feature subset selection example