2019 [ to top ]

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  Friedrich, Tobias; Krejca, Martin S.; Rothenberger, Ralf; Arndt, Tobias; Hafner, Danijar; Kellermeier, Thomas; Krogmann, Simon; Razmjou, Armin Routing for On-Street Parking Search using Probabilistic Data. AI Communications 2019
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  A significant percentage of urban traffic is caused by the search for parking spots. One possible approach to improve this situation is to guide drivers along routes which are likely to have free parking spots. The task of finding such a route can be modeled as a probabilistic graph problem which is NP-complete. Thus, we propose heuristic approaches for solving this problem and evaluate them experimentally. For this, we use probabilities of finding a parking spot, which are based on publicly available empirical data from TomTom International B.V. Additionally, we propose a heuristic that relies exclusively on conventional road attributes. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. Last, we complement our experiments with results from a field study, comparing the success rates of our algorithms against real human drivers.
  @article{noauthororeditor, abstract = {A significant percentage of urban traffic is caused by the search for parking spots. One possible approach to improve this situation is to guide drivers along routes which are likely to have free parking spots. The task of finding such a route can be modeled as a probabilistic graph problem which is NP-complete. Thus, we propose heuristic approaches for solving this problem and evaluate them experimentally. For this, we use probabilities of finding a parking spot, which are based on publicly available empirical data from TomTom International B.V. Additionally, we propose a heuristic that relies exclusively on conventional road attributes. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. Last, we complement our experiments with results from a field study, comparing the success rates of our algorithms against real human drivers.}, author = {Friedrich, Tobias and Krejca, Martin S. and Rothenberger, Ralf and Arndt, Tobias and Hafner, Danijar and Kellermeier, Thomas and Krogmann, Simon and Razmjou, Armin}, journal = {AI Communications}, keywords = {arminrazmjou danijarhafner martinskrejca ralfrothenberger simonkrogmann thomaskellermeier tobiasarndt tobiasfriedrich year2019}, title = {Routing for On-Street Parking Search using Probabilistic Data}, year = 2019 }
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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S. Unbiasedness of Estimation-of-Distribution Algorithms. Theoretical Computer Science 2019
  [ BibTeX ] [ DOI ] [ Download ]
   
  @article{noauthororeditor, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S.}, journal = {Theoretical Computer Science}, keywords = {martinskrejca tcs timokoetzing tobiasfriedrich year2019}, title = {Unbiasedness of Estimation-of-Distribution Algorithms}, year = 2019 }
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  Peters, Jannik; Stephan, Daniel; Amon, Isabel; Gawendowicz, Hans; Lischeid, Julius; Salabarria, Julius; Umland, Jonas; Werner, Felix; Krejca, Martin S.; Rothenberger, Ralf; Kötzing, Timo; Friedrich, Tobias Mixed Integer Programming versus Evolutionary Computation for Optimizing a Hard Real-World Staff Assignment Problem. International Conference on Automated Planning and Scheduling (ICAPS) 2019
  [ Abstract ] [ BibTeX ] [ Download ]
   
  Assigning staff to engagements according to hard constraints while optimizing several objectives is a task encountered by many companies on a regular basis. Simplified versions of such assignment problems are NP-hard. Despite this, a typical approach to solving them consists of formulating them as mixed integer programming (MIP) problems and using a state-of-the-art solver to get solutions that closely approximate the optimum. In this paper, we consider a complex real-world staff assignment problem encountered by the professional service company KPMG, with the goal of finding an algorithm that solves it faster and with a better solution than a commercial MIP solver. We follow the evolutionary algorithm (EA) metaheuristic and design a search heuristic which iteratively improves a solution using domain-specific mutation operators. Furthermore, we use a flow algorithm to optimally solve a subproblem, which tremendously reduces the search space for the EA. For our real-world instance of the assignment problem, given the same total time budget of \($100$\)~hours, a parallel EA approach finds a solution that is only \($1.7$\),% away from an upper bound for the (unknown) optimum within under five hours, while the MIP solver Gurobi still has a gap of \($10.5$\),%.
  @inproceedings{peters2019mixed, abstract = {Assigning staff to engagements according to hard constraints while optimizing several objectives is a task encountered by many companies on a regular basis. Simplified versions of such assignment problems are NP-hard. Despite this, a typical approach to solving them consists of formulating them as mixed integer programming (MIP) problems and using a state-of-the-art solver to get solutions that closely approximate the optimum. In this paper, we consider a complex real-world staff assignment problem encountered by the professional service company KPMG, with the goal of finding an algorithm that solves it faster and with a better solution than a commercial MIP solver. We follow the evolutionary algorithm (EA) metaheuristic and design a search heuristic which iteratively improves a solution using domain-specific mutation operators. Furthermore, we use a flow algorithm to optimally solve a subproblem, which tremendously reduces the search space for the EA. For our real-world instance of the assignment problem, given the same total time budget of $100$~hours, a parallel EA approach finds a solution that is only $1.7$\,\% away from an upper bound for the (unknown) optimum within under five hours, while the MIP solver Gurobi still has a gap of $10.5$\,\%.}, author = {Peters, Jannik and Stephan, Daniel and Amon, Isabel and Gawendowicz, Hans and Lischeid, Julius and Salabarria, Julius and Umland, Jonas and Werner, Felix and Krejca, Martin S. and Rothenberger, Ralf and Kötzing, Timo and Friedrich, Tobias}, booktitle = {International Conference on Automated Planning and Scheduling (ICAPS)}, keywords = {danielstephan felixwerner hansgawendowicz icaps isabelamon jannikpeters jonasumland juliuslischeid lennartsalabarria martinskrejca ralfrothenberger timokoetzing tobiasfriedrich year2019}, title = {Mixed Integer Programming versus Evolutionary Computation for Optimizing a Hard Real-World Staff Assignment Problem}, year = 2019 }

2018 [ to top ]

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  Dang, Duc-Cuong; Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Lehre, Per Kristian; Oliveto, Pietro S.; Sudholt, Dirk; Sutton, Andrew M. Escaping Local Optima Using Crossover with Emergent Diversity. IEEE Transactions on Evolutionary Computation 2018: 484-497
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the \((\mu+1)\) GA using uniform crossover on the fitness function \(Jump_k\). All previous results in this context only hold for unrealistically low crossover probability \(p_c=O(k/n)\), while we give analyses for the setting of constant \(p_c < 1\) in all but one case. Our bounds show a dependence on the problem size \(n\), the jump length \(k\), the population size \(\mu\), and the crossover probability \(p_c\). For the typical case of constant \(k > 2\) and constant \(p_c\), we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of \(\mu\): \(O(n^{k-1})\) for duplicate elimination/minimisation, \(O(n^2 \log n)\) for maximising the convex hull, \(O(n \log n)\) for det. crowding (assuming \(p_c = k/n\)), \(O(n \log n)\) for maximising the Hamming distance, \(O(n \log n)\) for fitness sharing, \(O(n \log n)\) for the single-receiver island model. This proves a sizeable advantage of all variants of the \((\mu+1)\) GA compared to the (1+1) EA, which requires \(\Theta(n^k)\). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally.
  @article{dang2017escaping, abstract = {Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the \((\mu+1)\) GA using uniform crossover on the fitness function \(Jump_k\). All previous results in this context only hold for unrealistically low crossover probability \(p_c=O(k/n)\), while we give analyses for the setting of constant \(p_c < 1\) in all but one case. Our bounds show a dependence on the problem size \(n\), the jump length \(k\), the population size \(\mu\), and the crossover probability \(p_c\). For the typical case of constant \(k > 2\) and constant \(p_c\), we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of \(\mu\): \(O(n^{k-1})\) for duplicate elimination/minimisation, \(O(n^2 \log n)\) for maximising the convex hull, \(O(n \log n)\) for det. crowding (assuming \(p_c = k/n\)), \(O(n \log n)\) for maximising the Hamming distance, \(O(n \log n)\) for fitness sharing, \(O(n \log n)\) for the single-receiver island model. This proves a sizeable advantage of all variants of the \((\mu+1)\) GA compared to the (1+1) EA, which requires \(\Theta(n^k)\). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally.}, author = {Dang, Duc-Cuong and Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Lehre, Per Kristian and Oliveto, Pietro S. and Sudholt, Dirk and Sutton, Andrew M.}, editor = {Tan, Kay Chen}, journal = {IEEE Transactions on Evolutionary Computation}, keywords = {andrewmsutton martinskrejca timokoetzing tobiasfriedrich year2018}, month = {jun}, number = 3, pages = {484-497}, publisher = {IEEE}, title = {Escaping Local Optima Using Crossover with Emergent Diversity}, volume = 22, year = 2018 }
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  Krejca, Martin; Witt, Carsten Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax. Theoretical Computer Science 2018
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  The Univariate Marginal Distribution Algorithm (UMDA) -- a popular estimation-of-distribution algorithm -- is studied from a run time perspective. On the classical OneMax benchmark function on bit strings of length \(n\), a lower bound of \(\Omega(\lambda + \mu \sqrt{n} + n\log n)\), where \(\mu\) and \(\lambda\) are algorithm-specific parameters, on its expected run time is proved. This is the first direct lower bound on the run time of UMDA. It is stronger than the bounds that follow from general black-box complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation-of-distribution algorithms in general.
  @article{krejca2018lower, abstract = {The Univariate Marginal Distribution Algorithm (UMDA) -- a popular estimation-of-distribution algorithm -- is studied from a run time perspective. On the classical OneMax benchmark function on bit strings of length \(n\), a lower bound of \(\Omega(\lambda + \mu \sqrt{n} + n\log n)\), where \(\mu\) and \(\lambda\) are algorithm-specific parameters, on its expected run time is proved. This is the first direct lower bound on the run time of UMDA. It is stronger than the bounds that follow from general black-box complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation-of-distribution algorithms in general.}, author = {Krejca, Martin and Witt, Carsten}, editor = {Oliveto, Pietro S. and Sutton, Andrew M.}, journal = {Theoretical Computer Science}, keywords = {martinskrejca tcs year2018}, title = {Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax}, year = 2018 }
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  Doerr, Benjamin; Krejca, Martin S. Significance-based Estimation-of-Distribution Algorithms. Genetic and Evolutionary Computation Conference (GECCO) 2018: 1483-1490
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Estimation-of-distribution algorithms (EDAs) are randomized search heuristics that maintain a stochastic model of the solution space. This model is updated from iteration to iteration based on the quality of the solutions sampled according to the model. As previous works show, this short-term perspective can lead to erratic updates of the model, in particular, to bit-frequencies approaching a random boundary value. This can lead to significant performance losses. In order to overcome this problem, we propose a new EDA that takes into account a longer history of samples and updates its model only with respect to information which it classifies as statistically significant. We prove that this significance-based compact genetic algorithm (sig-cGA) optimizes the common benchmark functions OneMax and LeadingOnes both in \(O(n\log n)\) time, a result shown for no other EDA or evolutionary algorithm so far. For the recently proposed scGA – an EDA that tries to prevent erratic model updates by imposing a bias to the uniformly distributed model – we prove that it optimizes OneMax only in a time exponential in the hypothetical population size \(1/\rho\).
  @inproceedings{doerr2018significancebased, abstract = {Estimation-of-distribution algorithms (EDAs) are randomized search heuristics that maintain a stochastic model of the solution space. This model is updated from iteration to iteration based on the quality of the solutions sampled according to the model. As previous works show, this short-term perspective can lead to erratic updates of the model, in particular, to bit-frequencies approaching a random boundary value. This can lead to significant performance losses. In order to overcome this problem, we propose a new EDA that takes into account a longer history of samples and updates its model only with respect to information which it classifies as statistically significant. We prove that this significance-based compact genetic algorithm (sig-cGA) optimizes the common benchmark functions OneMax and LeadingOnes both in \(O(n\log n)\) time, a result shown for no other EDA or evolutionary algorithm so far. For the recently proposed scGA – an EDA that tries to prevent erratic model updates by imposing a bias to the uniformly distributed model – we prove that it optimizes OneMax only in a time exponential in the hypothetical population size \(1/\rho\).}, author = {Doerr, Benjamin and Krejca, Martin S.}, booktitle = {Genetic and Evolutionary Computation Conference (GECCO)}, keywords = {GECCO benjamindoerr martinskrejca year2018}, pages = {1483-1490}, title = {Significance-based Estimation-of-Distribution Algorithms}, year = 2018 }
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  Kötzing, Timo; Krejca, Martin S. First-Hitting Times for Finite State Spaces. Parallel Problem Solving From Nature (PPSN) 2018: 79-91
  [ BibTeX ] [ DOI ] [ Download ]
   
  @inproceedings{kotzing2018firsthitting, author = {Kötzing, Timo and Krejca, Martin S.}, booktitle = {Parallel Problem Solving From Nature (PPSN)}, keywords = {martinskrejca ppsn timokoetzing year2018}, pages = {79-91}, title = {First-Hitting Times for Finite State Spaces}, year = 2018 }
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  Kötzing, Timo; Krejca, Martin S. First-Hitting Times Under Additive Drift. Parallel Problem Solving From Nature (PPSN) 2018: 92-104
  [ BibTeX ] [ DOI ] [ Download ]
   
  @inproceedings{kotzingyear2018firsthitting, author = {Kötzing, Timo and Krejca, Martin S.}, booktitle = {Parallel Problem Solving From Nature (PPSN)}, keywords = {martinskrejca ppsn timokoetzing year2018}, pages = {92-104}, title = {First-Hitting Times Under Additive Drift}, year = 2018 }
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  Bläsius, Thomas; Eube, Jan; Feldtkeller, Thomas; Friedrich, Tobias; Krejca, Martin S.; Lagodzinski, J. A. Gregor; Rothenberger, Ralf; Severin, Julius; Sommer, Fabian; Trautmann, Justin Memory-restricted Routing With Tiled Map Data. IEEE International Conference on Systems, Man, and Cybernetics (SMC) 2018: 3347-3354
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Modern routing algorithms reduce query time by depending heavily on preprocessed data. The recently developed Navigation Data Standard (NDS) enforces a separation between algorithms and map data, rendering preprocessing inapplicable. Furthermore, map data is partitioned into tiles with respect to their geographic coordinates. With the limited memory found in portable devices, the number of tiles loaded becomes the major factor for run time. We study routing under these restrictions and present new algorithms as well as empirical evaluations. Our results show that, on average, the most efficient algorithm presented uses more than 20 times fewer tile loads than a normal A*.
  @inproceedings{blasius2018memory, abstract = {Modern routing algorithms reduce query time by depending heavily on preprocessed data. The recently developed Navigation Data Standard (NDS) enforces a separation between algorithms and map data, rendering preprocessing inapplicable. Furthermore, map data is partitioned into tiles with respect to their geographic coordinates. With the limited memory found in portable devices, the number of tiles loaded becomes the major factor for run time. We study routing under these restrictions and present new algorithms as well as empirical evaluations. Our results show that, on average, the most efficient algorithm presented uses more than 20 times fewer tile loads than a normal A*.}, author = {Bläsius, Thomas and Eube, Jan and Feldtkeller, Thomas and Friedrich, Tobias and Krejca, Martin S. and Lagodzinski, J. A. Gregor and Rothenberger, Ralf and Severin, Julius and Sommer, Fabian and Trautmann, Justin}, booktitle = {IEEE International Conference on Systems, Man, and Cybernetics (SMC)}, keywords = {fabiansommer gregorlagodzinski janeube juliusseverin justintrautmann martinskrejca ralfrothenberger smc thomasblaesius thomasfeldtkeller tobiasfriedrich year2018}, month = 10, pages = {3347-3354}, title = {Memory-restricted Routing With Tiled Map Data}, year = 2018 }

2017 [ to top ]

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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. The Compact Genetic Algorithm is Efficient under Extreme Gaussian Noise. IEEE Transactions on Evolutionary Computation 2017: 477-490
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noise-handling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noise-handling. Furthermore, we prove that using a population does not help. Finally we show that a simple EDA called the compact Genetic Algorithm can overcome the shortsightedness of mutation-only heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property.
  @article{friedrich2017compact, abstract = {Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noise-handling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noise-handling. Furthermore, we prove that using a population does not help. Finally we show that a simple EDA called the compact Genetic Algorithm can overcome the shortsightedness of mutation-only heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property.}, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Sutton, Andrew M.}, journal = {IEEE Transactions on Evolutionary Computation}, keywords = {andrewmsutton martinskrejca tec timokoetzing tobiasfriedrich year2017}, number = 3, pages = {477-490}, title = {The Compact Genetic Algorithm is Efficient under Extreme Gaussian Noise}, volume = 21, year = 2017 }
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  Krejca, Martin S.; Witt, Carsten Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax. Foundations of Genetic Algorithms (FOGA) 2017: 65-79
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  The Univariate Marginal Distribution Algorithm (UMDA), a popular estimation of distribution algorithm, is studied from a run time perspective. On the classical OneMax benchmark function, a lower bound of \(Omega (\mu \sqrt n + n \log n)\), where \(\mu\) is the population size, on its expected run time is proved. This is the first direct lower bound on the run time of the UMDA. It is stronger than the bounds that follow from general black-box complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation of distribution algorithms in general.
  @inproceedings{DBLP:conf/foga/KrejcaW17, abstract = {The Univariate Marginal Distribution Algorithm (UMDA), a popular estimation of distribution algorithm, is studied from a run time perspective. On the classical OneMax benchmark function, a lower bound of \(\Omega (\mu \sqrt n + n \log n)\), where \(\mu\) is the population size, on its expected run time is proved. This is the first direct lower bound on the run time of the UMDA. It is stronger than the bounds that follow from general black-box complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation of distribution algorithms in general.}, author = {Krejca, Martin S. and Witt, Carsten}, booktitle = {Foundations of Genetic Algorithms (FOGA)}, keywords = {foga martinskrejca year2017}, pages = {65-79}, publisher = {ACM Press}, title = {Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax}, year = 2017 }

2016 [ to top ]

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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. Robustness of Ant Colony Optimization to Noise. Evolutionary Computation 2016: 237-254
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Recently Ant Colony Optimization (ACO) algorithms have been proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses focus on combinatorial problems, such as path finding. We analyze an ACO algorithm in a setting where we try to optimize the simple OneMax test function, but with additive posterior noise sampled from a Gaussian distribution. Without noise the classical \((\mu+1)\)-EA outperforms any ACO algorithm, with smaller \(\mu\) being better; however, with large noise, the \((\mu+1)\)-EA fails, even for high values of \(\mu\) (which are known to help against small noise). In this paper we show that ACO is able to deal with arbitrarily large noise in a graceful manner, that is, as long as the evaporation factor \(p\) is small enough dependent on the parameter \(\delta^2\) of the noise and the dimension \(n\) of the search space \((p = o(1/(n(n + \delta \log n)^2 \log n)))\), optimization will be successful.
  @article{DBLP:journals/ec/FriedrichKKS16, abstract = {Recently Ant Colony Optimization (ACO) algorithms have been proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses focus on combinatorial problems, such as path finding. We analyze an ACO algorithm in a setting where we try to optimize the simple OneMax test function, but with additive posterior noise sampled from a Gaussian distribution. Without noise the classical \((\mu+1)\)-EA outperforms any ACO algorithm, with smaller \(\mu\) being better; however, with large noise, the \((\mu+1)\)-EA fails, even for high values of \(\mu\) (which are known to help against small noise). In this paper we show that ACO is able to deal with arbitrarily large noise in a graceful manner, that is, as long as the evaporation factor \(p\) is small enough dependent on the parameter \(\delta^2\) of the noise and the dimension \(n\) of the search space \((p = o(1/(n(n + \delta \log n)^2 \log n)))\), optimization will be successful.}, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Sutton, Andrew M.}, journal = {Evolutionary Computation}, keywords = {EC andrewmsutton martinskrejca timokoetzing tobiasfriedrich year2016}, number = 2, pages = {237-254}, title = {Robustness of Ant Colony Optimization to Noise}, volume = 24, year = 2016 }
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  Arndt, Tobias; Hafner, Danijar; Kellermeier, Thomas; Krogmann, Simon; Razmjou, Armin; Krejca, Martin S.; Rothenberger, Ralf; Friedrich, Tobias Probabilistic Routing for On-Street Parking Search. European Symposium on Algorithms (ESA) 2016: 6:1-6:13
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  An estimated \(30\%\) of urban traffic is caused by search for parking spots. Traffic could be reduced by suggesting effective routes leading along potential parking spots. In this paper, we formalize parking search as a probabilistic problem on a road graph and show that it is NP-complete. We explore heuristics that optimize for the driving duration and the walking distance to the destination. Routes are constrained to reach a certain probability threshold of finding a spot. Empirically estimated probabilities of successful parking attempts are provided by TomTom on a per-street basis. We release these probabilities as a dataset of about 80,000 roads covering the Berlin area. This allows to evaluate parking search algorithms on a real road network with realistic probabilities for the first time. However, for many other areas, parking probabilities are not openly available. Because they are effortful to collect, we propose an algorithm that relies on conventional road attributes only. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. This leads to the conclusion that conventional road attributes may be sufficient to compute reasonably good parking search routes.
  @inproceedings{DBLP:conf/esa/ArndtHKKRKRF16, abstract = {An estimated \(30\%\) of urban traffic is caused by search for parking spots. Traffic could be reduced by suggesting effective routes leading along potential parking spots. In this paper, we formalize parking search as a probabilistic problem on a road graph and show that it is NP-complete. We explore heuristics that optimize for the driving duration and the walking distance to the destination. Routes are constrained to reach a certain probability threshold of finding a spot. Empirically estimated probabilities of successful parking attempts are provided by TomTom on a per-street basis. We release these probabilities as a dataset of about 80,000 roads covering the Berlin area. This allows to evaluate parking search algorithms on a real road network with realistic probabilities for the first time. However, for many other areas, parking probabilities are not openly available. Because they are effortful to collect, we propose an algorithm that relies on conventional road attributes only. Our experiments show that this algorithm comes close to the baseline by a factor of 1.3 in our cost measure. This leads to the conclusion that conventional road attributes may be sufficient to compute reasonably good parking search routes.}, author = {Arndt, Tobias and Hafner, Danijar and Kellermeier, Thomas and Krogmann, Simon and Razmjou, Armin and Krejca, Martin S. and Rothenberger, Ralf and Friedrich, Tobias}, booktitle = {European Symposium on Algorithms (ESA)}, keywords = {ESA arminrazmjou danijarhafner martinskrejca ralfrothenberger simonkrogmann thomaskellermeier tobiasarndt tobiasfriedrich year2016}, pages = {6:1-6:13}, title = {Probabilistic Routing for On-Street Parking Search}, year = 2016 }
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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. The Benefit of Recombination in Noisy Evolutionary Search. Genetic and Evolutionary Computation Conference (GECCO) 2016: 161-162
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noise-handling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noise-handling. Furthermore, we prove that using a population does not help. Finally we show that a simple EDA called the Compact Genetic Algorithm can overcome the shortsightedness of mutation-only heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property.
  @inproceedings{DBLP:conf/gecco/FriedrichKKS16, abstract = {Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings, randomized search heuristics such as evolutionary algorithms are a popular choice because they are often assumed to exhibit some kind of resistance to noise. Empirical evidence suggests that some algorithms, such as estimation of distribution algorithms (EDAs) are robust against a scaling of the noise intensity, even without resorting to explicit noise-handling techniques such as resampling. In this paper, we want to support such claims with mathematical rigor. We introduce the concept of graceful scaling in which the run time of an algorithm scales polynomially with noise intensity. We study a monotone fitness function over binary strings with additive noise taken from a Gaussian distribution. We show that myopic heuristics cannot efficiently optimize the function under arbitrarily intense noise without any explicit noise-handling. Furthermore, we prove that using a population does not help. Finally we show that a simple EDA called the Compact Genetic Algorithm can overcome the shortsightedness of mutation-only heuristics to scale gracefully with noise. We conjecture that recombinative genetic algorithms also have this property.}, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Sutton, Andrew M.}, booktitle = {Genetic and Evolutionary Computation Conference (GECCO)}, keywords = {andrewmsutton gecco martinskrejca timokoetzing tobiasfriedrich year2016}, pages = {161-162}, title = {The Benefit of Recombination in Noisy Evolutionary Search}, year = 2016 }
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  Dang, Duc-Cuong; Friedrich, Tobias; Krejca, Martin S.; Kötzing, Timo; Lehre, Per Kristian; Oliveto, Pietro S.; Sudholt, Dirk; Sutton, Andrew Michael Escaping Local Optima with Diversity Mechanisms and Crossover. Genetic and Evolutionary Computation Conference (GECCO) 2016: 645-652
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the \((\mu+1)\) GA using uniform crossover on the fitness function \(Jump_k\). All previous results in this context only hold for unrealistically low crossover probability \(p_c=O(k/n)\), while we give analyses for the setting of constant \(p_c < 1\) in all but one case. Our bounds show a dependence on the problem size \(n\), the jump length \(k\), the population size \(\mu\), and the crossover probability \(p_c\). For the typical case of constant \(k > 2\) and constant \(p_c\), we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of \(\mu\): \(O(n^{k-1})\) for duplicate elimination/minimisation, \(O(n^2 \log n)\) for maximising the convex hull, \(O(n \log n)\) for det. crowding (assuming \(p_c = k/n\)), \(O(n \log n)\) for maximising the Hamming distance, \(O(n \log n)\) for fitness sharing, \(O(n \log n)\) for the single-receiver island model. This proves a sizeable advantage of all variants of the \((\mu+1)\) GA compared to the (1+1) EA, which requires \(\Theta(n^k)\). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally.
  @inproceedings{DBLP:conf/gecco/DangFKKLOSS16, abstract = {Population diversity is essential for the effective use of any crossover operator. We compare seven commonly used diversity mechanisms and prove rigorous run time bounds for the \((\mu+1)\) GA using uniform crossover on the fitness function \(Jump_k\). All previous results in this context only hold for unrealistically low crossover probability \(p_c=O(k/n)\), while we give analyses for the setting of constant \(p_c < 1\) in all but one case. Our bounds show a dependence on the problem size \(n\), the jump length \(k\), the population size \(\mu\), and the crossover probability \(p_c\). For the typical case of constant \(k > 2\) and constant \(p_c\), we can compare the resulting expected optimisation times for different diversity mechanisms assuming an optimal choice of \(\mu\): \(O(n^{k-1})\) for duplicate elimination/minimisation, \(O(n^2 \log n)\) for maximising the convex hull, \(O(n \log n)\) for det. crowding (assuming \(p_c = k/n\)), \(O(n \log n)\) for maximising the Hamming distance, \(O(n \log n)\) for fitness sharing, \(O(n \log n)\) for the single-receiver island model. This proves a sizeable advantage of all variants of the \((\mu+1)\) GA compared to the (1+1) EA, which requires \(\Theta(n^k)\). In a short empirical study we confirm that the asymptotic differences can also be observed experimentally.}, author = {Dang, Duc-Cuong and Friedrich, Tobias and Krejca, Martin S. and Kötzing, Timo and Lehre, Per Kristian and Oliveto, Pietro S. and Sudholt, Dirk and Sutton, Andrew Michael}, booktitle = {Genetic and Evolutionary Computation Conference (GECCO)}, keywords = {andrewmsutton gecco martinskrejca timokoetzing tobiasfriedrich year2016}, pages = {645-652}, publisher = {ACM Press}, title = {Escaping Local Optima with Diversity Mechanisms and Crossover}, year = 2016 }
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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Nallaperuma, Samadhi; Neumann, Frank; Schirneck, Martin Fast Building Block Assembly by Majority Vote Crossover. Genetic and Evolutionary Computation Conference (GECCO) 2016: 661-668
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Different works have shown how crossover can help with building block assembly. Typically, crossover might get lucky to select good building blocks from each parent, but these lucky choices are usually rare. In this work we consider a crossover operator which works on three parent individuals. In each component, the offspring inherits the value present in the majority of the parents; thus, we call this crossover operator majority vote. We show that, if good components are sufficiently prevalent in the individuals, majority vote creates an optimal individual with high probability. Furthermore, we show that this process can be amplified: as long as components are good independently and with probability at least \(1/2+\delta\), we require only \(O(\log 1/\delta + \log \log n)\) successive stages of majority vote to create an optimal individual with high probability! We show how this applies in two scenarios. The first scenario is the Jump test function. With sufficient diversity, we get an optimization time of \(O(n \log n)\) even for jump sizes as large as \(O(n^{(1/2-\epsilon)})\). Our second scenario is a family of vertex cover instances. Majority vote optimizes this family efficiently, while local searches fail and only highly specialized two-parent crossovers are successful.
  @inproceedings{DBLP:conf/gecco/FriedrichKKNNS16, abstract = {Different works have shown how crossover can help with building block assembly. Typically, crossover might get lucky to select good building blocks from each parent, but these lucky choices are usually rare. In this work we consider a crossover operator which works on three parent individuals. In each component, the offspring inherits the value present in the majority of the parents; thus, we call this crossover operator majority vote. We show that, if good components are sufficiently prevalent in the individuals, majority vote creates an optimal individual with high probability. Furthermore, we show that this process can be amplified: as long as components are good independently and with probability at least \(1/2+\delta\), we require only \(O(\log 1/\delta + \log \log n)\) successive stages of majority vote to create an optimal individual with high probability! We show how this applies in two scenarios. The first scenario is the Jump test function. With sufficient diversity, we get an optimization time of \(O(n \log n)\) even for jump sizes as large as \(O(n^{(1/2-\epsilon)})\). Our second scenario is a family of vertex cover instances. Majority vote optimizes this family efficiently, while local searches fail and only highly specialized two-parent crossovers are successful.}, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Nallaperuma, Samadhi and Neumann, Frank and Schirneck, Martin}, booktitle = {Genetic and Evolutionary Computation Conference (GECCO)}, keywords = {GECCO frankneumann martinschirneck martinskrejca samadhinallaperuma timokoetzing tobiasfriedrich year2016}, pages = {661-668}, publisher = {ACM Press}, title = {Fast Building Block Assembly by Majority Vote Crossover}, year = 2016 }
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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S. EDAs cannot be Balanced and Stable. Genetic and Evolutionary Computation Conference (GECCO) 2016: 1139-1146
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Estimation of Distribution Algorithms (EDAs) work by iteratively updating a distribution over the search space with the help of samples from each iteration. Up to now, theoretical analyses of EDAs are scarce and present run time results for specific EDAs. We propose a new framework for EDAs that captures the idea of several known optimizers, including PBIL, UMDA, \(\lambda\)-MMASIB, cGA, and \((1,\lambda)\)-EA. Our focus is on analyzing two core features of EDAs: a balanced EDA is sensitive to signals in the fitness; a stable EDA remains uncommitted under a biasless fitness function. We prove that no EDA can be both balanced and stable. The LeadingOnes function is a prime example where, at the beginning of the optimization, the fitness function shows no bias for many bits. Since many well-known EDAs are balanced and thus not stable, they are not well-suited to optimize LeadingOnes. We give a stable EDA which optimizes LeadingOnes within a time of \(O(n\,\log n)\).
  @inproceedings{DBLP:conf/gecco/FriedrichKK16, abstract = {Estimation of Distribution Algorithms (EDAs) work by iteratively updating a distribution over the search space with the help of samples from each iteration. Up to now, theoretical analyses of EDAs are scarce and present run time results for specific EDAs. We propose a new framework for EDAs that captures the idea of several known optimizers, including PBIL, UMDA, \(\lambda\)-MMASIB, cGA, and \((1,\lambda)\)-EA. Our focus is on analyzing two core features of EDAs: a balanced EDA is sensitive to signals in the fitness; a stable EDA remains uncommitted under a biasless fitness function. We prove that no EDA can be both balanced and stable. The LeadingOnes function is a prime example where, at the beginning of the optimization, the fitness function shows no bias for many bits. Since many well-known EDAs are balanced and thus not stable, they are not well-suited to optimize LeadingOnes. We give a stable EDA which optimizes LeadingOnes within a time of \(O(n\,\log n)\).}, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S.}, booktitle = {Genetic and Evolutionary Computation Conference (GECCO)}, keywords = {GECCO martinskrejca timokoetzing tobiasfriedrich year2016}, pages = {1139-1146}, publisher = {ACM Press}, title = {EDAs cannot be Balanced and Stable}, type = {Publication}, year = 2016 }
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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. Graceful Scaling on Uniform versus Steep-Tailed Noise. Parallel Problem Solving From Nature (PPSN) 2016: 761-770
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Recently, different evolutionary algorithms (EAs) have been analyzed in noisy environments. The most frequently used noise model for this was additive posterior noise (noise added after the fitness evaluation) taken from a Gaussian distribution. In particular, for this setting it was shown that the \((\mu + 1)\)-EA on OneMax does not scale gracefully (higher noise cannot efficiently be compensated by higher \(\mu\)). In this paper we want to understand whether there is anything special about the Gaussian distribution which makes the \((\mu + 1)\)-EA not scale gracefully. We keep the setting of posterior noise, but we look at other distributions. We see that for exponential tails the \((\mu + 1)\)-EA on OneMax does also not scale gracefully, for similar reasons as in the case of Gaussian noise. On the other hand, for uniform distributions (as well as other, similar distributions) we see that the \((\mu + 1)\)-EA on OneMax does scale gracefully, indicating the importance of the noise model.
  @inproceedings{DBLP:conf/ppsn/FriedrichKKS16, abstract = {Recently, different evolutionary algorithms (EAs) have been analyzed in noisy environments. The most frequently used noise model for this was additive posterior noise (noise added after the fitness evaluation) taken from a Gaussian distribution. In particular, for this setting it was shown that the \((\mu + 1)\)-EA on OneMax does not scale gracefully (higher noise cannot efficiently be compensated by higher \(\mu\)). In this paper we want to understand whether there is anything special about the Gaussian distribution which makes the \((\mu + 1)\)-EA not scale gracefully. We keep the setting of posterior noise, but we look at other distributions. We see that for exponential tails the \((\mu + 1)\)-EA on OneMax does also not scale gracefully, for similar reasons as in the case of Gaussian noise. On the other hand, for uniform distributions (as well as other, similar distributions) we see that the \((\mu + 1)\)-EA on OneMax does scale gracefully, indicating the importance of the noise model.}, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Sutton, Andrew M.}, booktitle = {Parallel Problem Solving From Nature (PPSN)}, keywords = {PPSN andrewmsutton martinskrejca timokoetzing tobiasfriedrich year2016}, pages = {761-770}, title = {Graceful Scaling on Uniform versus Steep-Tailed Noise}, year = 2016 }
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  Dang, Duc-Cuong; Lehre, Per Kristian; Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Oliveto, Pietro S.; Sudholt, Dirk; Sutton, Andrew M. Emergence of Diversity and its Benefits for Crossover in Genetic Algorithms. Parallel Problem Solving From Nature (PPSN) 2016: 890-900
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for a standard \((\mu+1)\) GA and the \(Jump_k\) test function. By studying the stochastic process underlying the size of the largest collection of identical genotypes we show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order \(\Omega(n/ \log n)\) compared to mutation-only algorithms like the \((1+1)\) EA.
  @inproceedings{DBLP:conf/ppsn/DangFKKLOSS16, abstract = {Population diversity is essential for avoiding premature convergence in Genetic Algorithms (GAs) and for the effective use of crossover. Yet the dynamics of how diversity emerges in populations are not well understood. We use rigorous runtime analysis to gain insight into population dynamics and GA performance for a standard \((\mu+1)\) GA and the \(Jump_k\) test function. By studying the stochastic process underlying the size of the largest collection of identical genotypes we show that the interplay of crossover followed by mutation may serve as a catalyst leading to a sudden burst of diversity. This leads to improvements of the expected optimisation time of order \(\Omega(n/ \log n)\) compared to mutation-only algorithms like the \((1+1)\) EA.}, author = {Dang, Duc-Cuong and Lehre, Per Kristian and Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Oliveto, Pietro S. and Sudholt, Dirk and Sutton, Andrew M.}, booktitle = {Parallel Problem Solving From Nature (PPSN)}, keywords = {PPSN andrewmsutton martinskrejca timokoetzing tobiasfriedrich}, pages = {890-900}, title = {Emergence of Diversity and its Benefits for Crossover in Genetic Algorithms}, year = 2016 }

2015 [ to top ]

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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. Robustness of Ant Colony Optimization to Noise. Genetic and Evolutionary Computation Conference (GECCO) 2015: 17-24

  Best Paper Award (ACO/SI Track)
  [ Abstract ] [ BibTeX ] [ DOI ] [ Download ]
   
  Recently Ant Colony Optimization (ACO) algorithms have been proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses focus on combinatorial problems, such as path finding. We analyze an ACO algorithm in a setting where we try to optimize the simple OneMax test function, but with additive posterior noise sampled from a Gaussian distribution. Without noise the classical \((\mu+1)\)-EA outperforms any ACO algorithm, with smaller \(\mu\) being better; however, with large noise, the \((\mu+1)\)-EA fails, even for high values of \(\mu\) (which are known to help against small noise). In this paper we show that ACO is able to deal with arbitrarily large noise in a graceful manner, that is, as long as the evaporation factor \(\mu\) is small enough dependent on the parameter \(\delta^2\) of the noise and the dimension \(n\) of the search space \((p = o(1/(n(n + \delta \log n)^2 \log n)))\), optimization will be successful.
  @inproceedings{DBLP:conf/gecco/FriedrichKKS15, abstract = {Recently Ant Colony Optimization (ACO) algorithms have been proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses focus on combinatorial problems, such as path finding. We analyze an ACO algorithm in a setting where we try to optimize the simple OneMax test function, but with additive posterior noise sampled from a Gaussian distribution. Without noise the classical \((\mu+1)\)-EA outperforms any ACO algorithm, with smaller \(\mu\) being better; however, with large noise, the \((\mu+1)\)-EA fails, even for high values of \(\mu\) (which are known to help against small noise). In this paper we show that ACO is able to deal with arbitrarily large noise in a graceful manner, that is, as long as the evaporation factor \(\mu\) is small enough dependent on the parameter \(\delta^2\) of the noise and the dimension \(n\) of the search space \((p = o(1/(n(n + \delta \log n)^2 \log n)))\), optimization will be successful.}, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Sutton, Andrew M.}, booktitle = {Genetic and Evolutionary Computation Conference (GECCO)}, keywords = {andrewmsutton bestpaper first gecco imported martinskrejca timokoetzing tobiasfriedrich year2015}, note = {Best Paper Award (ACO/SI Track)}, pages = {17-24}, publisher = {{ACM}}, title = {Robustness of Ant Colony Optimization to Noise}, year = 2015 }
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  Friedrich, Tobias; Kötzing, Timo; Krejca, Martin S.; Sutton, Andrew M. The Benefit of Recombination in Noisy Evolutionary Search. International Symposium of Algorithms and Computation (ISAAC) 2015: 140-150
  [ Abstract ] [ BibTeX ] [ URL ] [ DOI ] [ Download ]
   
  Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings meta-heuristics are a popular choice for deriving good optimization algorithms, most notably evolutionary algorithms which mimic evolution in nature. Empirical evidence suggests that genetic recombination is useful in uncertain environments because it can stabilize a noisy fitness signal. With this paper we want to support this claim with mathematical rigor. The setting we consider is that of noisy optimization. We study a simple noisy fitness function that is derived by adding Gaussian noise to a monotone function. First, we show that a classical evolutionary algorithm that does not employ sexual recombination (the \((\mu+1)\)-EA) cannot handle the noise efficiently, regardless of the population size. Then we show that an evolutionary algorithm which does employ sexual recombination (the Compact Genetic Algorithm, short: cGA) can handle the noise using a graceful scaling of the population.
  @inproceedings{DBLP:conf/isaac/FriedrichKKS15, abstract = {Practical optimization problems frequently include uncertainty about the quality measure, for example due to noisy evaluations. Thus, they do not allow for a straightforward application of traditional optimization techniques. In these settings meta-heuristics are a popular choice for deriving good optimization algorithms, most notably evolutionary algorithms which mimic evolution in nature. Empirical evidence suggests that genetic recombination is useful in uncertain environments because it can stabilize a noisy fitness signal. With this paper we want to support this claim with mathematical rigor. The setting we consider is that of noisy optimization. We study a simple noisy fitness function that is derived by adding Gaussian noise to a monotone function. First, we show that a classical evolutionary algorithm that does not employ sexual recombination (the \((\mu+1)\)-EA) cannot handle the noise efficiently, regardless of the population size. Then we show that an evolutionary algorithm which does employ sexual recombination (the Compact Genetic Algorithm, short: cGA) can handle the noise using a graceful scaling of the population.}, address = {Berlin, Heidelberg}, author = {Friedrich, Tobias and Kötzing, Timo and Krejca, Martin S. and Sutton, Andrew M.}, booktitle = {International Symposium of Algorithms and Computation (ISAAC)}, editor = {Elbassioni, Khaled and Makino, Kazuhisa}, keywords = {andrewmsutton isaac martinskrejca timokoetzing tobiasfriedrich year2015}, pages = {140-150}, publisher = {Springer Berlin Heidelberg}, title = {The Benefit of Recombination in Noisy Evolutionary Search}, year = 2015 }