ABSTRACT
The area of runtime analysis has made important contributions to the theoretical understanding of evolutionary algoirthms for stochastic problems in recent years. Important real-world applications involve chance constraints where the goal is to optimize a function under the condition that constraints are only violated with a small probability. We rigorously analyze the runtime of the (1+1) EA for the chance-constrained knapsack problem. In this setting, the weights are stochastic, and the objective is to maximize a linear profit function while minimizing the probability of a constraint violation in the total weight. We investigate a number of special cases for this problem, paying attention to how the structure of the chance constraint influences the runtime behavior of the (1+1) EA. Our results reveal that small changes to the profit value can result in hard-to-escape local optima.
- Anne Auger and Benjamin Doerr (Eds.). 2011. Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific, Singapore. Google ScholarDigital Library
- Xiaodi Bai, Xiaojin Zheng, and Xiaoling Sun. 2012. A survey on probabilistically constrained optimization problems. Numerical Algebra, Control & Optimization 2, 4 (2012), 767--778.Google ScholarCross Ref
- Abraham Charnes and William W Cooper. 1959. Chance-constrained programming. Management Science 6, 1 (1959), 73--79. Google ScholarDigital Library
- Tobias Friedrich, Timo Kötzing, Martin S. Krejca, and Andrew M. Sutton. 2016. Robustness of Ant Colony Optimization to Noise. Evolutionary Computation 24, 2 (2016), 237--254. Google ScholarDigital Library
- Tobias Friedrich, Timo Kötzing, J.A. Gregor Lagodzinski, Frank Neumann, and Martin Schirneck. 2018. Analysis of the (1+1) EA on subclasses of linear functions under uniform and linear constraints. Theoretical Computer Science (2018). Google ScholarDigital Library
- Grani Adiwena Hanasusanto, Vladimir Roitch, Daniel Kuhn, and Wolfram Wiesemann. 2015. A distributionally robust perspective on uncertainty quantification and chance constrained programming. Mathematical Programming 151, 1 (2015), 35--62. Google ScholarDigital Library
- Grani Adiwena Hanasusanto, Vladimir Roitch, Daniel Kuhn, and Wolfram Wiesemann. 2017. Ambiguous Joint Chance Constraints Under Mean and Dispersion Information. Operations Research 65, 3 (2017), 751--767. Google ScholarDigital Library
- Jun He, Boris Mitavskiy, and Yuren Zhou. 2014. A theoretical assessment of solution quality in evolutionary algorithms for the knapsack problem. In Proceedings of the IEEE Congress on Evolutionary Computation (CEC). IEEE, 141--148.Google ScholarCross Ref
- Thomas Jansen. 2013. Analyzing Evolutionary Algorithms - The Computer Science Perspective. Springer, Berlin. Google ScholarDigital Library
- Olivier Klopfenstein and Dritan Nace. 2008. A robust approach to the chance-constrained knapsack problem. Operations Research Letters 36, 5 (2008), 628 -- 632. Google ScholarDigital Library
- Rajeev Kumar and Nilanjan Banerjee. 2005. Running Time Analysis of a Multiobjective Evolutionary Algorithm on Simple and Hard Problems. In Proceedings of the Eighth Conference on Foundations of Genetic Algorithms (FOGA) (Lecture Notes in Computer Science), Vol. 3469. Springer, Berlin, 112--131. Google ScholarDigital Library
- Pu Li, Harvey Arellano-Garcia, and Günter Wozny. 2008. Chance constrained programming approach to process optimization under uncertainty. Computers & Chemical Engineering 32, 1 (2008), 25 -- 45.Google ScholarCross Ref
- Pu Li, Moritz Wendt, and Günter Wozny. 2000. Robust model predictive control under chance constraints. Computers & Chemical Engineering 24, 2 (2000), 829 -- 834.Google ScholarCross Ref
- Andrei Lissovoi and Carsten Witt. 2015. Runtime analysis of ant colony optimization on dynamic shortest path problems. Theoretical Computer Science 561 (2015), 73 -- 85. Google ScholarDigital Library
- Andrei Lissovoi and Carsten Witt. 2016. MMAS Versus Population-Based EA on a Family of Dynamic Fitness Functions. Algorithmica 75, 3 (01 Jul 2016), 554--576. Google ScholarDigital Library
- Andrei Lissovoi and Carsten Witt. 2017. A Runtime Analysis of Parallel Evolutionary Algorithms in Dynamic Optimization. Algorithmica 78, 2 (01 Jun 2017), 641--659. Google ScholarDigital Library
- Bo Liu, Qingfu Zhang, Francisco V. Fernández, and Georges G. E. Gielen. 2013. An Efficient Evolutionary Algorithm for Chance-Constrained Bi-Objective Stochastic Optimization. IEEE Trans. Evolutionary Computation 17, 6 (2013), 786--796. Google ScholarDigital Library
- Frank Neumann and Andrew M. Sutton. 2018. Runtime Analysis of Evolutionary Algorithms for the Knapsack Problem with Favorably Correlated Weights. In Proceedings of the Fifteenth International Conference on Parallel Problem Solving from Nature (Lecture Notes in Computer Science), Anne Auger, Carlos M. Fonseca, Nuno Lourenço, Penousal Machado, Luís Paquete, and Darrell Whitley (Eds.), Vol. 11102. Springer, 141--152.Google Scholar
- Frank Neumann and Carsten Witt. 2010. Bioinspired Computation in Combinatorial Optimization: Algorithms and Their Computational Complexity (1st ed.). Springer, Berlin. Google ScholarDigital Library
- Prabhakar Raghavan and Rajeev Motwani. 1995. Randomized algorithms. Cambridge University Press, Cambridge. Google ScholarDigital Library
- Vahid Roostapour, Aneta Neumann, and Frank Neumann. 2018. On the Performance of Baseline Evolutionary Algorithms on the Dynamic Knapsack Problem. In Proceedings of the Fifteenth International Conference on Parallel Problem Solving from Nature (Lecture Notes in Computer Science), Anne Auger, Carlos M. Fonseca, Nuno Lourenço, Penousal Machado, Luís Paquete, and Darrell Whitley (Eds.), Vol. 11101. Springer, Berlin, 158--169.Google Scholar
- Vahid Roostapour, Aneta Neumann, Frank Neumann, and Tobias Friedrich. 2018. Pareto Optimization for Subset Selection with Dynamic Cost Constraints. CoRR abs/1811.07806 (2018). arXiv:1811.07806 http://arxiv.org/abs/1811.07806 Conference version appears at AAAI 2019.Google Scholar
- Vahid Roostapour, Mojgan Pourhassan, and Frank Neumann. 2018. Analysis of Evolutionary Algorithms in Dynamic and Stochastic Environments. CoRR abs/1806.08547 (2018). arXiv:1806.08547 http://arxiv.org/abs/1806.08547Google Scholar
- Abraham Wald. 1944. On Cumulative Sums of Random Variables. Ann. Math. Statist. 15, 3 (09 1944), 283--296.Google Scholar
- Abraham Wald. 1945. Some Generalizations of the Theory of Cumulative Sums of Random Variables. Ann. Math. Statist. 16, 3 (09 1945), 287--293.Google Scholar
- Yue Xie, Oscar Harper, Hirad Assimi, Aneta Neumann, and Frank Neumann. 2019. Evolutionary Algorithms for the Chance-Constrained Knapsack Problem. CoRR abs/1902.04767 (2019). arXiv:1902.04767 http://arxiv.org/abs/1902.04767 Conference version appears at GECCO 2019.Google Scholar
- Yuren Zhou and Jun He. 2007. A Runtime Analysis of Evolutionary Algorithms for Constrained Optimization Problems. IEEE Transactions on Evolutionary Computation 11, 5 (2007), 608--619. Google ScholarDigital Library
Index Terms
- Runtime analysis of the (1 + 1) evolutionary algorithm for the chance-constrained knapsack problem
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