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

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

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

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

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

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

Maternal obesity has little effect on the immediate offspring but impacts on the next generation

Maternal obesity during pregnancy has been linked to an increased risk of obesity and cardiometabolic disease in the offspring, a phenomenon attributed to developmental programming. Programming effects may be transmissible across generations through both maternal and paternal inheritance, although the mechanisms remain unclear. Using a mouse model, we explored the effects of moderate maternal diet-induced obesity (DIO) on weight gain and glucose-insulin homeostasis in first-generation (F1) and second-generation offspring. DIO was associated with insulin resistance, hyperglycemia and dyslipidemia before pregnancy. Birth weight was reduced in female offspring of DIO mothers (by 6%, P = .039), and DIO offspring were heavier than controls at weaning (males by 47%, females by 27%), however there were no differences in glucose tolerance, plasma lipids, or hepatic gene expression at 6 months. Despite the relative lack of effects in the F1, we found clear fetal growth restriction and persistent metabolic changes in otherwise unmanipulated second-generation offspring with effects on birth weight, insulin levels, and hepatic gene expression that were transmitted through both maternal and paternal lines. This suggests that the consequences of the current dietary obesity epidemic may also have an impact on the descendants of obese individuals, even when the phenotype of the F1 appears largely unaffected

Amantadine and levodopa in the treatment of Parkinson's disease

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/117040/1/cpt197213128.pd