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Solving linear optimization over arithmetic constraint formula

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Abstract

Since Balas extended the classical linear programming problem to the disjunctive programming (DP) problem where the constraints are combinations of both logic AND and OR, many researchers explored this optimization problem under various theoretical or application scenarios such as generalized disjunctive programming (GDP), optimization modulo theories (OMT), robot path planning, real-time systems, etc. However, the possibility of combining these differently-described but form-equivalent problems into a single expression remains overlooked. The contribution of this paper is two folded. First, we convert the linear DP/GDP model, linear-arithmetic OMT problem and related application problems into an equivalent form, referred to as the linear optimization over arithmetic constraint formula (LOACF). Second, a tree-search-based algorithm named RS-LPT is proposed to solve LOACF. RS-LPT exploits the techniques of interval analysis and nonparametric estimation for reducing the search tree and lowering the number of visited nodes. Also, RS-LPT alleviates bad construction of search tree by backtracking and pruning dynamically. We evaluate RS-LPT against two most common DP/GDP methods, three state-of-the-art OMT solvers and the disjunctive transformation based method on optimization benchmarks with different types and scales. Our results favor RS-LPT as compared to existing competing methods, especially for large scale cases.

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Acknowledgements

The authors are very grateful to the anonymous referees for their helpful comments and suggestions for improving the paper. This work is jointly supported by the Natural Science Foundation of China under Grants61303057, 61379048 and 61672508, and the CAS/SAFEA International Partnership Program for Creative Research Teams.

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Authors and Affiliations

  1. State Key Laboratory of Computer Sciences, Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Li Chen, Yinrun Lyu, Chong Wang, Jingzheng Wu & Yongji Wang

  2. Laboratory of Parallel Software and Computational Science, Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Changyou Zhang

  3. University of Chinese Academy of Sciences, Beijing, 100049, China

    Li Chen, Yinrun Lyu, Chong Wang, Jingzheng Wu, Changyou Zhang & Yongji Wang

  4. CCSIT, University of Dammam, Dammam, Saudi Arabia

    Nasro Min-Allah & Jamal Alhiyafi

  5. National Engineering Research Center for Fundamental Software, Institute of Software, Chinese Academy of Sciences, Beijing, 100190, China

    Li Chen, Yinrun Lyu, Chong Wang, Jingzheng Wu & Yongji Wang

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  1. Li Chen
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Chen, L., Lyu, Y., Wang, C. et al. Solving linear optimization over arithmetic constraint formula. J Glob Optim 69, 69–102 (2017). https://doi.org/10.1007/s10898-017-0499-8

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