29 research outputs found
The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem
The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most
prominent algorithms to solve multi-objective optimization problems. Recently,
the first mathematical runtime guarantees have been obtained for this
algorithm, however only for synthetic benchmark problems.
In this work, we give the first proven performance guarantees for a classic
optimization problem, the NP-complete bi-objective minimum spanning tree
problem. More specifically, we show that the NSGA-II with population size computes all extremal points of the Pareto front in
an expected number of iterations, where
is the number of vertices, the number of edges, and is the
maximum edge weight in the problem instance. This result confirms, via
mathematical means, the good performance of the NSGA-II observed empirically.
It also shows that mathematical analyses of this algorithm are not only
possible for synthetic benchmark problems, but also for more complex
combinatorial optimization problems.
As a side result, we also obtain a new analysis of the performance of the
global SEMO algorithm on the bi-objective minimum spanning tree problem, which
improves the previous best result by a factor of , the number of extremal
points of the Pareto front, a set that can be as large as . The
main reason for this improvement is our observation that both multi-objective
evolutionary algorithms find the different extremal points in parallel rather
than sequentially, as assumed in the previous proofs.Comment: Author-generated version of a paper appearing in the proceedings of
IJCAI 202
More effective randomized search heuristics for graph coloring through dynamic optimization
Dynamic optimization problems have gained significant attention in evolutionary computation as evolutionary algorithms (EAs) can easily adapt to changing environments. We show that EAs can solve the graph coloring problem for bipartite graphs more efficiently by using dynamic optimization. In our approach the graph instance is given incrementally such that the EA can reoptimize its coloring when a new edge introduces a conflict. We show that, when edges are inserted in a way that preserves graph connectivity, Randomized Local Search (RLS) efficiently finds a proper 2-coloring for all bipartite graphs. This includes graphs for which RLS and other EAs need exponential expected time in a static optimization scenario. We investigate different ways of building up the graph by popular graph traversals such as breadth-first-search and depth-first-search and analyse the resulting runtime behavior. We further show that offspring populations (e. g. a (1 + λ) RLS) lead to an exponential speedup in λ. Finally, an island model using 3 islands succeeds in an optimal time of Θ(m) on every m-edge bipartite graph, outperforming offspring populations. This is the first example where an island model guarantees a speedup that is not bounded in the number of islands
Runtime Analysis for the NSGA-II: Proving, Quantifying, and Explaining the Inefficiency For Many Objectives
The NSGA-II is one of the most prominent algorithms to solve multi-objective
optimization problems. Despite numerous successful applications, several
studies have shown that the NSGA-II is less effective for larger numbers of
objectives. In this work, we use mathematical runtime analyses to rigorously
demonstrate and quantify this phenomenon. We show that even on the simple
-objective generalization of the discrete OneMinMax benchmark, where every
solution is Pareto optimal, the NSGA-II also with large population sizes cannot
compute the full Pareto front (objective vectors of all Pareto optima) in
sub-exponential time when the number of objectives is at least three. The
reason for this unexpected behavior lies in the fact that in the computation of
the crowding distance, the different objectives are regarded independently.
This is not a problem for two objectives, where any sorting of a pair-wise
incomparable set of solutions according to one objective is also such a sorting
according to the other objective (in the inverse order)
Runtime Analysis for Self-adaptive Mutation Rates
We propose and analyze a self-adaptive version of the
evolutionary algorithm in which the current mutation rate is part of the
individual and thus also subject to mutation. A rigorous runtime analysis on
the OneMax benchmark function reveals that a simple local mutation scheme for
the rate leads to an expected optimization time (number of fitness evaluations)
of when is at least for
some constant . For all values of , this
performance is asymptotically best possible among all -parallel
mutation-based unbiased black-box algorithms.
Our result shows that self-adaptation in evolutionary computation can find
complex optimal parameter settings on the fly. At the same time, it proves that
a relatively complicated self-adjusting scheme for the mutation rate proposed
by Doerr, Gie{\ss}en, Witt, and Yang~(GECCO~2017) can be replaced by our simple
endogenous scheme.
On the technical side, the paper contributes new tools for the analysis of
two-dimensional drift processes arising in the analysis of dynamic parameter
choices in EAs, including bounds on occupation probabilities in processes with
non-constant drift
Complexity Theory for Discrete Black-Box Optimization Heuristics
A predominant topic in the theory of evolutionary algorithms and, more
generally, theory of randomized black-box optimization techniques is running
time analysis. Running time analysis aims at understanding the performance of a
given heuristic on a given problem by bounding the number of function
evaluations that are needed by the heuristic to identify a solution of a
desired quality. As in general algorithms theory, this running time perspective
is most useful when it is complemented by a meaningful complexity theory that
studies the limits of algorithmic solutions.
In the context of discrete black-box optimization, several black-box
complexity models have been developed to analyze the best possible performance
that a black-box optimization algorithm can achieve on a given problem. The
models differ in the classes of algorithms to which these lower bounds apply.
This way, black-box complexity contributes to a better understanding of how
certain algorithmic choices (such as the amount of memory used by a heuristic,
its selective pressure, or properties of the strategies that it uses to create
new solution candidates) influences performance.
In this chapter we review the different black-box complexity models that have
been proposed in the literature, survey the bounds that have been obtained for
these models, and discuss how the interplay of running time analysis and
black-box complexity can inspire new algorithmic solutions to well-researched
problems in evolutionary computation. We also discuss in this chapter several
interesting open questions for future work.Comment: This survey article is to appear (in a slightly modified form) in the
book "Theory of Randomized Search Heuristics in Discrete Search Spaces",
which will be published by Springer in 2018. The book is edited by Benjamin
Doerr and Frank Neumann. Missing numbers of pointers to other chapters of
this book will be added as soon as possibl
Fourier Analysis Meets Runtime Analysis: Precise Runtimes on Plateaus
We propose a new method based on discrete Fourier analysis to analyze the
time evolutionary algorithms spend on plateaus. This immediately gives a
concise proof of the classic estimate of the expected runtime of the
evolutionary algorithm on the Needle problem due to Garnier, Kallel, and
Schoenauer (1999).
We also use this method to analyze the runtime of the evolutionary
algorithm on a new benchmark consisting of plateaus of effective size
which have to be optimized sequentially in a LeadingOnes fashion.
Using our new method, we determine the precise expected runtime both for
static and fitness-dependent mutation rates. We also determine the
asymptotically optimal static and fitness-dependent mutation rates. For , the optimal static mutation rate is approximately . The optimal
fitness dependent mutation rate, when the first fitness-relevant bits have
been found, is asymptotically . These results, so far only proven for
the single-instance problem LeadingOnes, are thus true in a much broader
respect. We expect similar extensions to be true for other important results on
LeadingOnes. We are also optimistic that our Fourier analysis approach can be
applied to other plateau problems as well.Comment: 40 page