An Analysis on Selection for High-Resolution Approximations in Many-Objective Optimization

This work studies the behavior of three elitist multi- and many-objective evolutionary algorithms generating a high-resolution approximation of the Pareto optimal set. Several search-assessment indicators are defined to trace the dynamics of survival selection and measure the ability to simultaneously keep optimal solutions and discover new ones under different population sizes, set as a fraction of the size of the Pareto optimal set.Comment: apperas in Parallel Problem Solving from Nature - PPSN XIII, Ljubljana : Slovenia (2014

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

A multi-objective window optimisation problem

We present an optimisation problem which seeks to locate the Pareto-optimal front of building window and shading designs minimising two objectives: projected energy use of the operational building and its construction cost. This problem is of particular interest because it has many variable interactions and each function evaluation is relatively timeconsuming. It also makes use of a freely-available building simulation program EnergyPlus which may be used in many other building design optimisation problems. We describe the problem and report the results of experiments comparing the performance of a number of existing multi-objective evolutionary algorithms applied to it. We conclude that this represents a promising real-world application area

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

An Hypervolume based constraint handling technique for multi-objective optimization problems

Formulation of structural optimization problems usually leads to the individuation of one or more objective functions to be minimized under different constraints. Many multi-objective evolutionary algorithms are approached by a Pareto-compliant ranking method, where no a priori information on the problem is needed and the concept of non-dominated solutions is used. In this paper a constraint handling technique based on the concept of hypervolume indicator is presented. Initially proposed to compare different multi-objective algorithms hypervolume indicator is the only single set quality measure to reflects the dominance of solution’s sets. The constraint handling technique proposed use an extension of stochastic ranking approach for single-objective optimization problem to multi-objective ones. The extension proposed use the hypervolume indicator to compares different solutions and is tested on a structural constrained multi-objective problems. Results show the suitability of the proposed approach

Set-based Multiobjective Fitness Landscapes: A Preliminary Study

Fitness landscape analysis aims to understand the geometry of a given optimization problem in order to design more efficient search algorithms. However, there is a very little knowledge on the landscape of multiobjective problems. In this work, following a recent proposal by Zitzler et al. (2010), we consider multiobjective optimization as a set problem. Then, we give a general definition of set-based multiobjective fitness landscapes. An experimental set-based fitness landscape analysis is conducted on the multiobjective NK-landscapes with objective correlation. The aim is to adapt and to enhance the comprehensive design of set-based multiobjective search approaches, motivated by an a priori analysis of the corresponding set problem properties