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›› 2016, Vol. 31 ›› Issue (5): 987-1011.

Special Issue: Theory and Algorithms

• Theory and Algorithms •

### An Efficient Approach for Solving Optimization over Linear Arithmetic Constraints

Li Chen, Jing-Zheng Wu, Yin-Run Lv, and Yong-Ji Wang*, Senior Member, CCF

1. State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing 100190, China University of Chinese Academy of Sciences, Beijing 100190, China;
National Engineering Research Center of Fundamental Software, Institute of Software Chinese Academy of Sciences Beijing 100190, China
• Received:2015-04-02 Revised:2015-12-08 Online:2016-09-05 Published:2016-09-05
• Contact: Yong-Ji Wang E-mail:ywang@itechs.iscas.ac.cn
• About author:Li Chen is currently a Ph.D. student at the Institute of Software, Chinese Academy of Sciences, Beijing. He received his B.S. degree in mathematics from the University of Science and Technology of China, Hefei. His primary research interests include satisfiability modulo theories (SMT), constrained optimization, real-time systems, and robotics.
• Supported by:

This work is supported by the National Natural Science Foundation of China under Grant No. 61170072, and the Chinese Academy of Sciences/State Administration of Foreign Experts Affairs (CAS/SAFEA) International Partnership Program for Creative Research Teams.

Satisfiability Modulo Theories (SMT) have been widely investigated over the last decade. Recently researchers have extended SMT to the optimization problem over linear arithmetic constraints. To the best of our knowledge, Symba and OPT-MathSAT are two most efficient solvers available for this problem. The key algorithms used by Symba and OPT-MathSAT consist of the loop of two procedures: 1) critical finding for detecting a critical point, which is very likely to be globally optimal, and 2) global checking for confirming the critical point is really globally optimal. In this paper, we propose a new approach based on the Simplex method widely used in operation research. Our fundamental idea is to find several critical points by constructing and solving a series of linear problems with the Simplex method. Our approach replaces the algorithms of critical finding in Symba and OPT-MathSAT, and reduces the runtime of critical finding and decreases the number of executions of global checking. The correctness of our approach is proved. The experiment evaluates our implementation against Symba and OPT-MathSAT on a critical class of problems in real-time systems. Our approach outperforms Symba on 99.6% of benchmarks and is superior to OPT-MathSAT in large-scale cases where the number of tasks is more than 24. The experimental results demonstrate that our approach has great potential and competitiveness for the optimization problem.

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