組合優化

組合優化（英語：Combinatorial optimization）是數學優化的一個子領域，在應用數學和理論計算機科學的領域中，組合優化是在一個有限的對象集中找出最優對象的一類問題。^[1]在很多組合優化的問題中，窮舉搜索／枚舉法是不可行的。組合優化的問題的特徵是可行解的集是離散或者可以簡化到離散的，並且目標是找到最優解。常見的例子有旅行推銷員問題和最小生成樹。

組合優化涉及運籌學、算法理論和計算複雜性理論，在人工智能、機器學習、拍賣理論、軟件工程、超大規模集成電路、應用數學和理論計算機科學等多個領域有重要的應用。

組合優化的難處主要是加入拓撲分析的情況，不同的拓撲形態下，不同部分的約束關係便不同，算法也就要調整。如果給定一個拓撲形態，組合優化往往退化成一個整數優化的問題。

應用

物流^[2]
供應鏈最佳化（英語：Supply chain optimization）^[3]

特定問題

參考文獻

^ Schrijver 2003，第1頁.
^ Sbihi, Abdelkader; Eglese, Richard W. Combinatorial optimization and Green Logistics (PDF). 4OR. 2007, 5 (2): 99–116 [2019-12-26]. S2CID 207070217. doi:10.1007/s10288-007-0047-3. （原始內容存檔 (PDF)於2019-12-26）.
^ Eskandarpour, Majid; Dejax, Pierre; Miemczyk, Joe; Péton, Olivier. Sustainable supply chain network design: An optimization-oriented review (PDF). Omega. 2015, 54: 11–32 [2019-12-26]. doi:10.1016/j.omega.2015.01.006. （原始內容存檔 (PDF)於2019-12-26）.

引注

Beasley, J. E. Integer programming (lecture notes). [2022-10-16]. （原始內容存檔於2022-10-16）.
Cook, William J.; Cunningham, William H.; Pulleyblank, William R.; Schrijver, Alexander. Combinatorial Optimization. Wiley. 1997. ISBN 0-471-55894-X.
Cook, William. Optimal TSP Tours. University of Waterloo. 2016 [2022-10-16]. （原始內容存檔於2012-07-22）. (Information on the largest TSP instances solved to date.)
Crescenzi, Pierluigi; Kann, Viggo; Halldórsson, Magnús; Karpinski, Marek; Woeginger, Gerhard (編). A Compendium of NP Optimization Problems. [2022-10-16]. （原始內容存檔於2007-04-05）. (This is a continuously updated catalog of approximability results for NP optimization problems.)
Das, Arnab; Chakrabarti, Bikas K (編). Quantum Annealing and Related Optimization Methods. Lecture Notes in Physics 679. Springer. 2005. Bibcode:2005qnro.book.....D.
Das, Arnab; Chakrabarti, Bikas K. Colloquium: Quantum annealing and analog quantum computation. Rev. Mod. Phys. 2008, 80 (3): 1061. Bibcode:2008RvMP...80.1061D. CiteSeerX 10.1.1.563.9990 . S2CID 14255125. arXiv:0801.2193 . doi:10.1103/RevModPhys.80.1061.
Lawler, Eugene. Combinatorial Optimization: Networks and Matroids. Dover. 2001. ISBN 0-486-41453-1.
Lee, Jon. A First Course in Combinatorial Optimization. Cambridge University Press. 2004. ISBN 0-521-01012-8.
Papadimitriou, Christos H.; Steiglitz, Kenneth. Combinatorial Optimization : Algorithms and Complexity. Dover. July 1998. ISBN 0-486-40258-4.
Schrijver, Alexander. Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics 24. Springer. 2003. ISBN 9783540443896.
Schrijver, Alexander. On the history of combinatorial optimization (till 1960) (PDF). Aardal, K.; Nemhauser, G.L.; Weismantel, R. (編). Handbook of Discrete Optimization. Elsevier. 2005: 1–68 [2022-10-16]. （原始內容存檔 (PDF)於2020-11-24）.
Schrijver, Alexander. A Course in Combinatorial Optimization (PDF). February 1, 2006 [2022-10-16]. （原始內容存檔 (PDF)於2022-12-02）.
Sierksma, Gerard; Ghosh, Diptesh. Networks in Action; Text and Computer Exercises in Network Optimization. Springer. 2010. ISBN 978-1-4419-5512-8.
Gerard Sierksma; Yori Zwols. Linear and Integer Optimization: Theory and Practice. CRC Press. 2015. ISBN 978-1-498-71016-9.
Pintea, C-M. Advances in Bio-inspired Computing for Combinatorial Optimization Problem. Intelligent Systems Reference Library. Springer. 2014 [2022-10-16]. ISBN 978-3-642-40178-7. （原始內容存檔於2021-04-27）.

[1] Schrijver 2003，第1頁.

[2] Sbihi, Abdelkader; Eglese, Richard W. Combinatorial optimization and Green Logistics (PDF). 4OR. 2007, 5 (2): 99–116 [2019-12-26]. S2CID 207070217. doi:10.1007/s10288-007-0047-3. （原始內容存檔 (PDF)於2019-12-26）.

[3] Eskandarpour, Majid; Dejax, Pierre; Miemczyk, Joe; Péton, Olivier. Sustainable supply chain network design: An optimization-oriented review (PDF). Omega. 2015, 54: 11–32 [2019-12-26]. doi:10.1016/j.omega.2015.01.006. （原始內容存檔 (PDF)於2019-12-26）.

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