Multi-objective optimization problems arise frequently in applications but can often only be solved approximately by heuristic approaches. Evolutionary algorithms have been widely used to tackle multi-objective problems. These algorithms use different measures to ensure diversity in the objective space but are not guided by a formal notion of approximation. We present a new framework of an evolutionary algorithm for multi-objective optimization that allows to work with a formal notion of approximation. Our experimental results show that our approach outperforms state-of-theart evolutionary algorithms in terms of the quality of the approximation that is obtained in particular for problems with many objectives.

In order to allow a comparison of (otherwise incomparable) sets, many evolutionary multiobjective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms. The most widely used indicator is the hypervolume indicator. It measures the volume of the dominated portion of the objective space bounded from below by a reference point. Though the hypervolume indicator is very popular, it has not been shown that maximizing the hypervolume indicator of sets of bounded size is indeed equivalent to the overall objective of finding a good approximation of the Pareto front. To address this question, we compare the optimal approximation ratio with the approximation ratio achieved by two-dimensional sets maximizing the hypervolume indicator. We bound the optimal multiplicative approximation ratio of n points by 1+Θ(1/n) for arbitrary Pareto fronts. Furthermore, we prove that the same asymptotic approximation ratio is achieved by sets of n points that maximize the hypervolume indicator. However, there is a provable gap between the two approximation ratios which is even exponential in the ratio between the largest and the smallest value of the front. We also examine the additive approximation ratio of the hypervolume indicator in two dimensions and prove that it achieves the optimal additive approximation ratio apart from a small ratio � n/(n−2), where n is the size of the population. Hence the hypervolume indicator can be used to achieve a good additive but not a good multiplicative approximation of a Pareto front. This motivates the introduction of a “logarithmic hypervolume indicator ” which provably achieves a good multiplicative approximation ratio.

It was recently proven that sets of points maximizing the hypervolume indicator do not give a good multiplicative approximation of the Pareto front. We introduce a new“logarithmic hypervolume indicator”and prove that it achieves a close-to-optimal multiplicative approximation ratio. This is experimentally verified on several benchmark functions by comparing the approximation quality of the multi-objective covariance matrix evolution strategy (MO-CMA-ES) with the classic hypervolume indicator and the MO-CMA-ES with the logarithmic hypervolume indicator.

We revisit the classic problem of sampling from a discrete distribution: Given n non-negative w-bit integers x1,..., xn, the task is to build a data structure that allows sampling i with probability proportional to xi. The classic solution is Walker’s alias method that takes, when implemented on a Word RAM, O(n) preprocessing time, O(1) expected query time for one sample, and n(w+2 lg n+o(1)) bits of space. Using the terminology of succinct data structures, this solution has redundancy 2n lg n + o(n) bits, i.e., it uses 2n lg n + o(n) bits in addition to the information theoretic minimum required for storing the input. In this paper, we study whether this space usage can be improved. In the systematic case, in which the input is read-only, we present a novel data structure using r + O(w) redundant

Multi-objective optimization deals with the task of computing a set of solutions that represents possible trade-offs with respect to a given set of objective functions. Set-based approaches such as evolutionary algorithms are very popular for solving multi-objective optimization problems. Convergence of set-based approaches for multi-objective optimization is essential for their success. We take an order-theoretic view on the convergence of set-based multi-objective optimization and examine how the use of indicator functions can help to direct the search towards Pareto optimal sets. In doing so, we point out that set-based multi-objective optimization working on the dominance relation of search points has to deal with a cyclic behavior that may lead to worsening with respect to the Pareto-dominance relation defined on sets. Later on, we show in which situations well-known binary and unary indicators can help to avoid this cyclic behavior and therefore guarantee convergence of the algorithm. We also study the impact of deteriorative cycles on the runtime behavior and give an example in which they provably slow down the optimization process.

3.1. Modeling and landscape analysis 4

Abstract not found