R2-EMOA: Focused Multiobjective Search Using R2-Indicator-Based Selection

short paperInternational audienceAn indicator-based evolutionary multiobjective optimization algorithm (EMOA) is introduced which incorporates the contribution to the unary R2-indicator as the secondary selection criterion. First experiments indicate that the R2-EMOA accurately approximates the Pareto front of the considered continuous multiobjective optimization problems. Furthermore, decision makers' preferences can be included by adjusting the weight vector distributions of the indicator which results in a focused search behavior

Hybridizing Non-dominated Sorting Algorithms: Divide-and-Conquer Meets Best Order Sort

Many production-grade algorithms benefit from combining an asymptotically efficient algorithm for solving big problem instances, by splitting them into smaller ones, and an asymptotically inefficient algorithm with a very small implementation constant for solving small subproblems. A well-known example is stable sorting, where mergesort is often combined with insertion sort to achieve a constant but noticeable speed-up. We apply this idea to non-dominated sorting. Namely, we combine the divide-and-conquer algorithm, which has the currently best known asymptotic runtime of $O(N (\log N)^{M - 1})$, with the Best Order Sort algorithm, which has the runtime of $O(N^2 M)$ but demonstrates the best practical performance out of quadratic algorithms. Empirical evaluation shows that the hybrid's running time is typically not worse than of both original algorithms, while for large numbers of points it outperforms them by at least 20%. For smaller numbers of objectives, the speedup can be as large as four times.Comment: A two-page abstract of this paper will appear in the proceedings companion of the 2017 Genetic and Evolutionary Computation Conference (GECCO 2017

Dual guidance in evolutionary multi-objective optimization by localization

In this paper, we propose a framework using local models for multi-objective optimization to guide the search heuristic in both the decision and objective spaces. The localization is built using a limited number of adaptive spheres in the decision space. These spheres are usually guided, using some direction information, in the decision space towards the areas with non-dominated solutions. We use a second mechanism to adjust the spheres to specialize on different parts of the Pareto front using the guided dominance technique in the objective space. With this dual guidance, we can easily guide spheres towards different parts of the Pareto front while also exploring the decision space efficiently

Preference Articulation by Means of the R2 Indicator

International audienceIn multi-objective optimization, set-based performance indicators have become the state of the art for assessing the quality of Pareto front approximations. As a consequence, they are also more and more used within the design of multi-objective optimization algorithms. The R2 and the Hypervolume (HV) indicator represent two popular examples. In order to understand the behavior and the approximations preferred by these indicators and algorithms, a comprehensive knowledge of the indicator's properties is required. Whereas this knowledge is available for the HV, we presented a first approach in this direction for the R2 indicator just recently. In this paper, we build upon this knowledge and enhance the considerations with respect to the integration of preferences into the R2 indicator. More specifically, we analyze the effect of the reference point, the domain of the weights, and the distribution of weight vectors on the optimization of μ solutions with respect to the R2 indicator. By means of theoretical findings and empirical evidence, we show the potentials of these three possibilities using the optimal distribution of μ solutions for exemplary setups

A Study of Archiving Strategies in Multi-Objective PSO for Molecular Docking

Molecular docking is a complex optimization problem aimed at predicting the position of a ligand molecule in the active site of a receptor with the lowest binding energy. This problem can be formulated as a bi-objective optimization problem by minimizing the binding energy and the Root Mean Square Deviation (RMSD) difference in the coordinates of ligands. In this context, the SMPSO multi-objective swarm-intelligence algorithm has shown a remarkable performance. SMPSO is characterized by having an external archive used to store the non-dominated solutions and also as the basis of the leader selection strategy. In this paper, we analyze several SMPSO variants based on different archiving strategies in the scope of a benchmark of molecular docking instances. Our study reveals that the SMPSOhv, which uses an hypervolume contribution based archive, shows the overall best performance.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

An Improvement Study of the Decomposition-based Algorithm Global WASF-GA for Evolutionary Multiobjective Optimization

The convergence and the diversity of the decompositionbased evolutionary algorithm Global WASF-GA (GWASF-GA) relies on a set of weight vectors that determine the search directions for new non-dominated solutions in the objective space. Although using weight vectors whose search directions are widely distributed may lead to a well-diversified approximation of the Pareto front (PF), this may not be enough to obtain a good approximation for complicated PFs (discontinuous, non-convex, etc.). Thus, we propose to dynamically adjust the weight vectors once GWASF-GA has been run for a certain number of generations. This adjustment is aimed at re-calculating some of the weight vectors, so that search directions pointing to overcrowded regions of the PF are redirected toward parts with a lack of solutions that may be hard to be approximated. We test different parameters settings of the dynamic adjustment in optimization problems with three, five, and six objectives, concluding that GWASF-GA performs better when adjusting the weight vectors dynamically than without applying the adjustment.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A Bug in the Multiobjective Optimizer IBEA: Salutary Lessons for Code Release and a Performance Re-Assessment

International audienceThe Indicator-Based Evolutionary Algorithm (IBEA) is one of the first indicator-based multiobjective optimization algorithms and due to its wide availability in several algorithm packages is often used as a reference algorithm when benchmarking multiobjective optimizers. The original publication on IBEA proposes to use two specific variants: one based on the ε-indicator and one based on the hypervolume. Several experimental studies concluded that, surprisingly, the IBEA variant with the ε-indicator performs better than the one with the hypervolume—even if the (unary) hypervolume indicator itself is the quality measure used in the performance assessment. Recently, a small bug has been found in the hypervolume variant of IBEA with large implications on its performance. Here, we not only explain the bug in detail and correct it, but also present the (improved) results of the corrected version. Moreover, and probably even more important for the scientific community, we point out that this bug has been transferred to other than the original software package, discuss how this obscured the bug, and argue in favor of some simple, even obvious guidelines how the optimization community should deal with algorithm source codes, documentation, and the (natural) existence of bugs in the future

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

Bringing Order to Special Cases of Klee's Measure Problem

Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm has runtime O*(n^{d/2}) [Overmars,Yap'91]. There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex), and k-Grounded (where the projection onto the first k dimensions is a Hypervolume instance). In this paper we bring some order to these special cases by providing reductions among them. In addition to the trivial inclusions, we establish Hypervolume as the easiest of these special cases, and show that the runtimes of Unitcube-KMP and Cube-KMP are polynomially related. More importantly, we show that any algorithm for one of the special cases with runtime T(n,d) implies an algorithm for the general case with runtime T(n,2d), yielding the first non-trivial relation between KMP and its special cases. This allows to transfer W[1]-hardness of KMP to all special cases, proving that no n^{o(d)} algorithm exists for any of the special cases under reasonable complexity theoretic assumptions. Furthermore, assuming that there is no improved algorithm for the general case of KMP (no algorithm with runtime O(n^{d/2 - eps})) this reduction shows that there is no algorithm with runtime O(n^{floor(d/2)/2 - eps}) for any of the special cases. Under the same assumption we show a tight lower bound for a recent algorithm for 2-Grounded [Yildiz,Suri'12].Comment: 17 page