{"type":"ScholarlyArticle","title":"Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables","authors":[{"name":"Frank Neumann","slug":"frank-neumann","given":"Frank","family":"Neumann","affiliation":"Optimisation and Logistics , University of Adelaide , Adelaide , Australia"},{"name":"Carsten Witt","slug":"carsten-witt","given":"Carsten","family":"Witt","affiliation":"DTU Compute , Technical University of Denmark , Kongens Lyngby , Denmark"}],"abstract":"Chance constrained optimization problems allow to model problems where constraints involving stochastic components should only be violated with a small probability. Evolutionary algorithms have been applied to this scenario and shown to achieve high quality results. With this paper, we contribute to the theoretical understanding of evolutionary algorithms for chance constrained optimization. We study the scenario of stochastic components that are independent and Normally distributed.\r\nConsidering the simple single-objective (1+1)~EA, we show that imposing an additional uniform constraint already leads to local optima for very restricted scenarios and an exponential optimization time. We therefore introduce a multi-objective formulation of the problem which trades off the expected cost and its variance. We show that multi-objective evolutionary algorithms are highly effective when using this formulation and obtain a set of solutions that contains an optimal solution for any possible confidence level imposed on the constraint. Furthermore, we prove that this approach can also be used to compute a set of optimal solutions for the chance constrained minimum spanning tree problem. Experimental investigations on instances of the NP-hard stochastic minimum weight dominating set problem  confirm the benefit of the multi-objective approach in practice.","doi":"10.24963/ijcai.2022/665","url":"https://ijcai.org/proceedings/2022/665","pdf":"https://ijcai.org/proceedings/2022/0665.pdf","markdown":"https://ijcai.org/proceedings/2022/665.md","bibtex":"https://ijcai.org/proceedings/2022/bibtex/665","volume":{"slug":"2022","title":"Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2022","url":"https://ijcai.org/proceedings/2022/","year":2022,"location":"Vienna","dates":"23-29 July 2022","editors":"Luc De Raedt","isbn":"978-1-956792-00-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2022"},"section":"Search","track":"Main Track","number":665,"pages":{"first":4800,"last":4806},"keywords":["Search: Evolutionary Computation"],"videos":[{"label":"Video #1","url":"https://slideslive.com/38986783","length_sec":87,"provider":"slideslive"},{"label":"Video #2","url":"https://slideslive.com/38984926","length_sec":359,"provider":"slideslive"}],"references":[{"position":1,"raw":"[Assimi et al., 2020] Hirad Assimi, Oscar Harper, Yue Xie, Aneta Neumann, and Frank Neumann. Evolutionary bi- objective optimization for the dynamic chance-constrained knapsack problem based on tail bound objectives. In ECAI, volume 325, pages 307-314. IOS Press, 2020.","title":"Evolutionary biobjective optimization for the dynamic chance-constrained knapsack problem based on tail bound objectives","authors":"Assimi","year":"2020","venue":"ECAI","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Ben-Tal et al., 2009] A. Ben-Tal, L. El Ghaoui, and A.S. Nemirovski. Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, Octo- ber 2009.","title":"Robust Optimization","authors":"A Ben-Tal, L El Ghaoui, A S Nemirovski","year":"2009","venue":"Princeton Series in Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Charnes and Cooper, 1959] Abraham Charnes and William W Cooper. Chance-constrained programming. Management science, 6(1):73-79, 1959.","title":"Abraham Charnes and William W Cooper","authors":"Cooper Charnes","year":"1959","venue":"Management science","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Doerr and Neumann, 2020] Benjamin Doerr and Frank Neumann, editors. Theory of Evolutionary Computation- Recent Developments in Discrete Optimization. Springer, 2020.","title":"Theory of Evolutionary Computation-Recent Developments in Discrete Optimization","authors":"Neumann Doerr","year":"2020","venue":null,"doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Doerr et al., 2012] Benjamin Doerr, Daniel Johannsen, and Carola Winzen. Multiplicative drift analysis. Algorith- mica, 64:673-697, 2012.","title":"Multiplicative drift analysis","authors":"Benjamin Doerr, Daniel Johannsen, Carola Winzen","year":"2012","venue":"Algorithmica","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Doerr et al., 2020] Benjamin Doerr, Carola Doerr, Aneta Neumann, Frank Neumann, and Andrew M. Sutton. Opti- mization of chance-constrained submodular functions. In AAAI, pages 1460-1467. AAAI Press, 2020.","title":"Optimization of chance-constrained submodular functions","authors":"Benjamin Doerr, Carola Doerr, Aneta Neumann, Frank Neumann, Andrew M Sutton","year":"2020","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Doerr, 2020] Benjamin Doerr. Probabilistic tools for the analysis of randomized optimization heuristics. In Ben- jamin Doerr and Frank Neumann, editors, Theory of Evo- lutionary Computation -Recent Developments in Discrete Optimization, pages 1-87. Springer, 2020.","title":"Probabilistic tools for the analysis of randomized optimization heuristics","authors":"Benjamin Doerr","year":"2020","venue":"Theory of Evolutionary Computation -Recent Developments in Discrete Optimization","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Friedrich and Neumann, 2015] Tobias Friedrich and Frank Neumann. Maximizing submodular functions under ma- troid constraints by evolutionary algorithms. Evolutionary Computation, 23(4):543-558, 2015.","title":"Maximizing submodular functions under matroid constraints by evolutionary algorithms","authors":"Neumann Friedrich, Tobias Friedrich, Frank Neumann","year":"2015","venue":"Evolutionary Computation","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Friedrich et al., 2010] Tobias Friedrich, Jun He, Nils Heb- binghaus, Frank Neumann, and Carsten Witt. Approxi- mating covering problems by randomized search heuristics using multi-objective models. Evolutionary Computation, 18(4):617-633, 2010.","title":"Approximating covering problems by randomized search heuristics using multi-objective models","authors":"Friedrich","year":"2010","venue":"Evolutionary Computation","doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Giel, 2003] O. Giel. Expected runtimes of a simple multi- objective evolutionary algorithm. In CEC, volume 3, pages 1918-1925, 2003.","title":"Expected runtimes of a simple multiobjective evolutionary algorithm","authors":"O Giel","year":"2003","venue":"CEC","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Ishii et al., 1981] Hiroaki Ishii, Shogo Shiode, Toshio Nishida, and Yoshikazu Namasuya. Stochastic spanning tree problem. Discret. Appl. Math., 3(4):263-273, 1981.","title":"Stochastic spanning tree problem","authors":"Hiroaki Ishii, Shogo Shiode, Toshio Nishida, Yoshikazu Namasuya","year":"1981","venue":"Discret. Appl. Math","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Jansen, 2013] Thomas Jansen. Analyzing Evolutionary Al- gorithms -The Computer Science Perspective. Springer, 2013.","title":"Analyzing Evolutionary Algorithms -The Computer Science Perspective","authors":"Thomas Jansen","year":"2013","venue":null,"doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Kratsch and Neumann, 2013] Stefan Kratsch and Frank Neumann. Fixed-parameter evolutionary algorithms and the vertex cover problem. Algorithmica, 65(4):754-771, 2013.","title":"Fixed-parameter evolutionary algorithms and the vertex cover problem","authors":"Neumann Kratsch, Stefan Kratsch, Frank Neumann","year":"2013","venue":"Algorithmica","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Laumanns et al., 2004] Marco Laumanns, Lothar Thiele, and Eckart Zitzler. Running time analysis of multiobjec- tive evolutionary algorithms on pseudo-boolean functions. IEEE Trans. Evol. Comput., 8(2):170-182, 2004.","title":"Running time analysis of multiobjective evolutionary algorithms on pseudo-boolean functions","authors":"Marco Laumanns, Lothar Thiele, Eckart Zitzler","year":"2004","venue":"IEEE Trans. Evol. Comput","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Liu et al., 2013] Bo Liu, Qingfu Zhang, Francisco V. Fernández, and Georges G. E. Gielen. An efficient evo- lutionary algorithm for chance-constrained bi-objective stochastic optimization. IEEE Trans. Evolutionary Com- putation, 17(6):786-796, 2013.","title":"An efficient evolutionary algorithm for chance-constrained bi-objective stochastic optimization","authors":"Bo Liu, Qingfu Zhang, Francisco V Fernández, Georges G E Gielen","year":"2013","venue":"IEEE Trans. Evolutionary Computation","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Neumann and Neumann, 2020] Aneta Neumann and Frank Neumann. Optimising monotone chance-constrained sub- modular functions using evolutionary multi-objective al- gorithms. In PPSN (1), pages 404-417. Springer, 2020.","title":"Optimising monotone chance-constrained submodular functions using evolutionary multi-objective algorithms","authors":"Aneta Neumann, Frank Neumann","year":"2020","venue":"PPSN (1)","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Neumann and Sutton, 2019] Frank Neumann and An- drew M. Sutton. Runtime analysis of the (1 + 1) evolutionary algorithm for the chance-constrained knapsack problem. In FOGA, pages 147-153. ACM, 2019.","title":"Frank Neumann and Andrew M. Sutton. Runtime analysis of the (1 + 1) evolutionary algorithm for the chance-constrained knapsack problem","authors":"Sutton Neumann","year":"2019","venue":"FOGA","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Neumann and Wegener, 2006] Frank Neumann and Ingo Wegener. Minimum spanning trees made easier via multi- objective optimization. Nat. Comput., 5(3):305-319, 2006.","title":"Minimum spanning trees made easier via multi-objective optimization","authors":"Wegener Neumann, Frank Neumann, Ingo Wegener","year":"2006","venue":"Nat. Comput","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Neumann and Witt, 2010] Frank Neumann and Carsten Witt. Bioinspired Computation in Combinatorial Opti- mization -Algorithms and Their Computational Complex- ity. Springer, 2010.","title":"Bioinspired Computation in Combinatorial Optimization -Algorithms and Their Computational Complexity","authors":"Witt Neumann, Frank Neumann, Carsten Witt","year":"2010","venue":null,"doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Neumann et al., 2022] Aneta Neumann, Yue Xie, and Frank Neumann. Evolutionary algorithms for limiting the effect of uncertainty for the knapsack problem with stochastic profits. CoRR, abs/2204.05597, 2022.","title":"Evolutionary algorithms for limiting the effect of uncertainty for the knapsack problem with stochastic profits","authors":"Aneta Neumann, Yue Xie, Frank Neumann","year":"2022","venue":null,"doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Neumann, 2007] Frank Neumann. Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem. Eur. J. Oper. Res., 181(3):1620-1629, 2007.","title":"Expected runtimes of a simple evolutionary algorithm for the multi-objective minimum spanning tree problem","authors":"Frank Neumann","year":"2007","venue":"Eur. J. Oper. Res","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Poojari and Varghese, 2008] Chandra A. Poojari and Boby Varghese. Genetic algorithm based technique for solv- ing chance constrained problems. Eur. J. Oper. Res., 185(3):1128-1154, 2008.","title":"Genetic algorithm based technique for solving chance constrained problems","authors":"Chandra A Poojari, Boby Varghese","year":"2008","venue":"Eur. J. Oper. Res","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Qian et al., 2015] Chao Qian, Yang Yu, and Zhi-Hua Zhou. Subset selection by Pareto optimization. In NIPS, pages 1774-1782, 2015.","title":"Subset selection by Pareto optimization","authors":"Yang Chao Qian, Zhi-Hua Yu, Zhou","year":"2015","venue":"NIPS","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Qian et al., 2017] Chao Qian, Jing-Cheng Shi, Yang Yu, and Ke Tang. On subset selection with general cost con- straints. In IJCAI, pages 2613-2619, 2017.","title":"On subset selection with general cost constraints","authors":"Jing-Cheng Chao Qian, Yang Shi, Ke Yu, Tang","year":"2017","venue":"IJCAI","doi":null,"url":null,"internal_url":"/proceedings/2017/364"},{"position":25,"raw":"[Roostapour et al., 2022] Vahid Roostapour, Aneta Neu- mann, Frank Neumann, and Tobias Friedrich. Pareto op- timization for subset selection with dynamic cost con- straints. Artif. Intell., 302:103597, 2022.","title":"Pareto optimization for subset selection with dynamic cost constraints","authors":"Aneta Vahid Roostapour, Frank Neumann, Tobias Neumann, Friedrich","year":"2022","venue":"Artif. Intell","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Rossi and Ahmed, 2015] Ryan A. Rossi and Nesreen K. Ahmed. The network data repository with interactive graph analytics and visualization. In AAAI, pages 4292- 4293. AAAI Press, 2015.","title":"The network data repository with interactive graph analytics and visualization","authors":"Ryan A Rossi, Nesreen K Ahmed","year":"2015","venue":"AAAI","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Xie et al., 2019] Yue Xie, Oscar Harper, Hirad Assimi, Aneta Neumann, and Frank Neumann. Evolutionary al- gorithms for the chance-constrained knapsack problem. In GECCO, pages 338-346. ACM, 2019.","title":"Evolutionary algorithms for the chance-constrained knapsack problem","authors":"Yue Xie, Oscar Harper, Hirad Assimi, Aneta Neumann, Frank Neumann","year":"2019","venue":"GECCO","doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Xie et al., 2020] Yue Xie, Aneta Neumann, and Frank Neu- mann. Specific single-and multi-objective evolutionary algorithms for the chance-constrained knapsack problem. In GECCO, pages 271-279. ACM, 2020.","title":"Specific single-and multi-objective evolutionary algorithms for the chance-constrained knapsack problem","authors":"Yue Xie, Aneta Neumann, Frank Neumann","year":"2020","venue":"GECCO","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Xie et al., 2021a] Yue Xie, Aneta Neumann, and Frank Neumann. Heuristic strategies for solving complex inter- acting stockpile blending problem with chance constraints. In GECCO, pages 1079-1087. ACM, 2021.","title":"Heuristic strategies for solving complex interacting stockpile blending problem with chance constraints","authors":"Xie","year":"2021","venue":"GECCO","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Xie et al., 2021b] Yue Xie, Aneta Neumann, Frank Neu- mann, and Andrew M. Sutton. Runtime analysis of RLS and the (1+1) EA for the chance-constrained knapsack problem with correlated uniform weights. In GECCO, pages 1187-1194. ACM, 2021.","title":"Runtime analysis of RLS and the (1+1) EA for the chance-constrained knapsack problem with correlated uniform weights","authors":"Xie","year":"2021","venue":"GECCO","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Zhou et al., 2019] Zhi-Hua Zhou, Yang Yu, and Chao Qian. Evolutionary learning: Advances in theories and algo- rithms. Springer, 2019.","title":"Evolutionary learning: Advances in theories and algorithms","authors":"Zhi-Hua Zhou, Yang Yu, Chao Qian","year":"2019","venue":null,"doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}