Increasing the density of available pareto optimal solutions

The set of available multi-objective optimization algorithms continues to grow. This fact can be partially attributed to their widespread use and applicability. However this increase also suggests several issues remain to be addressed satisfactorily. One such issue is the diversity and the number of solutions available to the decision maker (DM). Even for algorithms very well suited for a particular problem, it is difficult - mainly due to the computational cost - to use a population large enough to ensure the likelihood of obtaining a solution close to the DMs preferences. In this paper we present a novel methodology that produces additional Pareto optimal solutions from a Pareto optimal set obtained at the end run of any multi-objective optimization algorithm. This method, which we refer to as Pareto estimation, is tested against a set of 2 and 3-objective test problems and a 3-objective portfolio optimization problem to illustrate its’ utility for a real-world problem

Methods for many-objective optimization: an analysis

Decomposition-based methods are often cited as the solution to problems related with many-objective optimization. Decomposition-based methods employ a scalarizing function to reduce a many-objective problem into a set of single objective problems, which upon solution yields a good approximation of the set of optimal solutions. This set is commonly referred to as Pareto front. In this work we explore the implications of using decomposition-based methods over Pareto-based methods from a probabilistic point of view. Namely, we investigate whether there is an advantage of using a decomposition-based method, for example using the Chebyshev scalarizing function, over Paretobased methods

Generalized decomposition and cross entropy methods for many-objective optimization

Decomposition-based algorithms for multi-objective optimization problems have increased in popularity in the past decade. Although their convergence to the Pareto optimal front (PF) is in several instances superior to that of Pareto-based algorithms, the problem of selecting a way to distribute or guide these solutions in a high-dimensional space has not been explored. In this work, we introduce a novel concept which we call generalized decomposition. Generalized decomposition provides a framework with which the decision maker (DM) can guide the underlying evolutionary algorithm toward specific regions of interest or the entire Pareto front with the desired distribution of Pareto optimal solutions. Additionally, it is shown that generalized decomposition simplifies many-objective problems by unifying the three performance objectives of multi-objective evolutionary algorithms – convergence to the PF, evenly distributed Pareto optimal solutions and coverage of the entire front – to only one, that of convergence. A framework, established on generalized decomposition, and an estimation of distribution algorithm (EDA) based on low-order statistics, namely the cross-entropy method (CE), is created to illustrate the benefits of the proposed concept for many objective problems. This choice of EDA also enables the test of the hypothesis that low-order statistics based EDAs can have comparable performance to more elaborate EDAs

On the Impact of Multiobjective Scalarizing Functions

Recently, there has been a renewed interest in decomposition-based approaches for evolutionary multiobjective optimization. However, the impact of the choice of the underlying scalarizing function(s) is still far from being well understood. In this paper, we investigate the behavior of different scalarizing functions and their parameters. We thereby abstract firstly from any specific algorithm and only consider the difficulty of the single scalarized problems in terms of the search ability of a (1+lambda)-EA on biobjective NK-landscapes. Secondly, combining the outcomes of independent single-objective runs allows for more general statements on set-based performance measures. Finally, we investigate the correlation between the opening angle of the scalarizing function's underlying contour lines and the position of the final solution in the objective space. Our analysis is of fundamental nature and sheds more light on the key characteristics of multiobjective scalarizing functions.Comment: appears in Parallel Problem Solving from Nature - PPSN XIII, Ljubljana : Slovenia (2014

Vortex flows in the solar atmoshpere: automated identification and statistical analysis

Vortices on the photosphere are fundamentally important as these coherent flows have the potential to form coherent magnetic field structures in the solar atmosphere, e.g., twisted magnetic flux tubes. These flows have traditionally been identified by tracking magnetic bright points (BPs) using primarily visual inspection. This approach has the shortcoming that it introduces bias into the statistical analyses. In this work we fully automate the process of vortex identification using an established method from hydrodynamics for the study of eddies in turbulent flows. For the first time, we apply this to detect intergranular photospheric intensity vortices. Using this automated approach, we find that the expected lifetime of intensity vortices is much shorter (≈17 s) compared with previously observed magnetic BP swirls. We suggest that at any time there are 1.48 × 106 such small-scale intensity vortices covering about 2.8% of the total surface of the solar photosphere. Lastly, we compare our results with previous works and speculate what this could imply with regards to estimating the global energy flux due magnetic tornadoes in the solar atmosphere with future higher resolution instrumentation

Improved Sampling of Decision Space for Pareto Estimation

Pareto Estimation (PE) is a novel method for increasing the density of Pareto optimal solutions across the entire Pareto Front or in a specific region of interest. PE identifies the inverse mapping of Pareto optimal solutions, namely, from objective space to decision space. This identification can be performed using a number of modeling techniques, how- ever, for the sake of simplicity in this work we use a radial basis neural network. In any modeling method, the quality of the resulting model depends heavily on the training samples used. The original version of PE uses the result- ing set of Pareto optimal solutions from any multi-objective optimization algorithm and then utilizes this set to identify the aforementioned mapping. However, we argue that this selection may not always be the best possible and propose an alternative scheme to improve the resulting set of Pareto optimal solutions in order to produce higher quality samples for the identification scheme in PE. The proposed approach is integrated with MAEA-gD, and the resulting solutions are used with PE. The results show that the proposed method shows promise, in that there is measurable improvement in the quality of the estimated PE in terms of the coverage and density

Liger : a cross-platform open-source integrated optimization and decision-making environment

Real-world optimization problems involving multiple conflicting objectives are commonly best solved using multi-objective optimization as this provides decision-makers with a family of trade-off solutions. However, the complexity of using multi-objective optimization algorithms often impedes the optimization process. Knowing which optimization algorithm is the most suitable for the given problem, or even which setup parameters to pick, requires someone to be an optimization specialist. The lack of supporting software that is readily available, easy to use and transparent can lead to increased design times and increased cost. To address these challenges, Liger is presented. Liger has been designed for ease of use in industry by non-specialists in optimization. The user interacts with Liger via a visual programming language to create an optimization workflow, enabling the user to solve an optimization problem. Liger contains a novel optimization library known as Tigon. The library utilizes the concept of design patterns to enable the composition of optimization algorithms by making use of simple reusable operator nodes. The library offers a varied range of multi-objective evolutionary algorithms which cover different paradigms in evolutionary computation; and supports a wide variety of problem types, including support for using more than one programming language at a time to implement the optimization model. Additionally, Liger functionality can be easily extended by plugins that provide access to state-of-the-art visualization tools and are responsible for managing the graphical user interface. Lastly, new user-driven interactive capabilities are shown to facilitate the decision-making process and are demonstrated on a control engineering optimization problem

Resonant Absorption of Axisymmetric Modes in Twisted Magnetic Flux Tubes

It has been shown recently that magnetic twist and axisymmetric MHD modes are ubiquitous in the solar atmosphere, and therefore the study of resonant absorption for these modes has become a pressing issue because it can have important consequences for heating magnetic flux tubes in the solar atmosphere and the observed damping. In this investigation, for the first time, we calculate the damping rate for axisymmetric MHD waves in weakly twisted magnetic flux tubes. Our aim is to investigate the impact of resonant damping of these modes for solar atmospheric conditions. This analytical study is based on an idealized configuration of a straight magnetic flux tube with a weak magnetic twist inside as well as outside the tube. By implementing the conservation laws derived by Sakurai et al. and the analytic solutions for weakly twisted flux tubes obtained recently by Giagkiozis et al. we derive a dispersion relation for resonantly damped axisymmetric modes in the spectrum of the Alfvén continuum. We also obtain an insightful analytical expression for the damping rate in the long wavelength limit. Furthermore, it is shown that both the longitudinal magnetic field and the density, which are allowed to vary continuously in the inhomogeneous layer, have a significant impact on the damping time. Given the conditions in the solar atmosphere, resonantly damped axisymmetric modes are highly likely to be ubiquitous and play an important role in energy dissipation. We also suggest that, given the character of these waves, it is likely that they have already been observed in the guise of Alfvén waves

Mapping structural diversity in networks sharing a given degree distribution and global clustering: Adaptive resolution grid search evolution with Diophantine equation-based mutations

Methods that generate networks sharing a given degree distribution and global clustering can induce changes in structural properties other than that controlled for. Diversity in structural properties, in turn, can affect the outcomes of dynamical processes operating on those networks. Since exhaustive sampling is not possible, we propose a novel evolutionary framework for mapping this structural diversity. The three main features of this framework are: (a) subgraph-based encoding of networks, (b) exact mutations based on solving systems of Diophantine equations, and (c) heuristic diversity-driven mechanism to drive resolution changes in the MapElite algorithm.We show that our framework can elicit networks with diversity in their higher-order structure and that this diversity affects the behaviour of the complex contagion model. Through a comparison with state of the art clustered network generation methods, we demonstrate that our approach can uncover a comparably diverse range of networks without needing computationally unfeasible mixing times. Further, we suggest that the subgraph-based encoding provides greater confidence in the diversity of higher-order network structure for low numbers of samples and is the basis for explaining our results with complex contagion model. We believe that this framework could be applied to other complex landscapes that cannot be practically mapped via exhaustive sampling