969 research outputs found
A Faster Algorithm for Calculating Hypervolume
We present an algorithm for calculating hypervolume exactly, the Hypervolume by Slicing Objectives (HSO) algorithm, that is faster than any that has previously been published. HSO processes objectives instead of points, an idea that has been considered before but that has never been properly evaluated in the literature. We show that both previously studied exact hypervolume algorithms are exponential in at least the number of objectives and that although HSO is also exponential in the number of objectives in the worst case, it runs in significantly less time, i.e., two to three orders of magnitude less for randomly generated and benchmark data in three to eight objectives. Thus, HSO increases the utility of hypervolume, both as a metric for general optimization algorithms and as a diversity mechanism for evolutionary algorithm
Approximating the least hypervolume contributor: NP-hard in general, but fast in practice
The hypervolume indicator is an increasingly popular set measure to compare
the quality of two Pareto sets. The basic ingredient of most hypervolume
indicator based optimization algorithms is the calculation of the hypervolume
contribution of single solutions regarding a Pareto set. We show that exact
calculation of the hypervolume contribution is #P-hard while its approximation
is NP-hard. The same holds for the calculation of the minimal contribution. We
also prove that it is NP-hard to decide whether a solution has the least
hypervolume contribution. Even deciding whether the contribution of a solution
is at most (1+\eps) times the minimal contribution is NP-hard. This implies
that it is neither possible to efficiently find the least contributing solution
(unless ) nor to approximate it (unless ).
Nevertheless, in the second part of the paper we present a fast approximation
algorithm for this problem. We prove that for arbitrarily given \eps,\delta>0
it calculates a solution with contribution at most (1+\eps) times the minimal
contribution with probability at least . Though it cannot run in
polynomial time for all instances, it performs extremely fast on various
benchmark datasets. The algorithm solves very large problem instances which are
intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions)
within a few seconds.Comment: 22 pages, to appear in Theoretical Computer Scienc
Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization
The use of surrogate based optimization (SBO) is widely spread in engineering design to reduce the number of computational expensive simulations. However, "real-world" problems often consist of multiple, conflicting objectives leading to a set of competitive solutions (the Pareto front). The objectives are often aggregated into a single cost function to reduce the computational cost, though a better approach is to use multiobjective optimization methods to directly identify a set of Pareto-optimal solutions, which can be used by the designer to make more efficient design decisions (instead of weighting and aggregating the costs upfront). Most of the work in multiobjective optimization is focused on multiobjective evolutionary algorithms (MOEAs). While MOEAs are well-suited to handle large, intractable design spaces, they typically require thousands of expensive simulations, which is prohibitively expensive for the problems under study. Therefore, the use of surrogate models in multiobjective optimization, denoted as multiobjective surrogate-based optimization, may prove to be even more worthwhile than SBO methods to expedite the optimization of computational expensive systems. In this paper, the authors propose the efficient multiobjective optimization (EMO) algorithm which uses Kriging models and multiobjective versions of the probability of improvement and expected improvement criteria to identify the Pareto front with a minimal number of expensive simulations. The EMO algorithm is applied on multiple standard benchmark problems and compared against the well-known NSGA-II, SPEA2 and SMS-EMOA multiobjective optimization methods
A Convergence indicator for Multi-Objective Optimisation Algorithms
The algorithms of multi-objective optimisation had a relative growth in the
last years. Thereby, it's requires some way of comparing the results of these.
In this sense, performance measures play a key role. In general, it's
considered some properties of these algorithms such as capacity, convergence,
diversity or convergence-diversity. There are some known measures such as
generational distance (GD), inverted generational distance (IGD), hypervolume
(HV), Spread(), Averaged Hausdorff distance (), R2-indicator,
among others. In this paper, we focuses on proposing a new indicator to measure
convergence based on the traditional formula for Shannon entropy. The main
features about this measure are: 1) It does not require tho know the true
Pareto set and 2) Medium computational cost when compared with Hypervolume.Comment: Submitted to TEM
Hypervolume-based Multi-objective Bayesian Optimization with Student-t Processes
Student- processes have recently been proposed as an appealing alternative
non-parameteric function prior. They feature enhanced flexibility and
predictive variance. In this work the use of Student- processes are explored
for multi-objective Bayesian optimization. In particular, an analytical
expression for the hypervolume-based probability of improvement is developed
for independent Student- process priors of the objectives. Its effectiveness
is shown on a multi-objective optimization problem which is known to be
difficult with traditional Gaussian processes.Comment: 5 pages, 3 figure
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