Abstract
Magic square is an old and interesting mathematical problem, which has the same value of all the sums of the elements in each row, column and diagonal. An unfinished magic square denotes that it gives us some clues and we need to fill the empty cells. In order to solve the unfinished magic squares more efficiently, we propose a solution based on sparse optimization. Using the properties of magic square, we establish a model of constraint programming. Then we transform the constraints into sparse linear constraints, meanwhile use l 0 norm minimization as the objective function to ensure the sparsity of the solution. Moreover, we use l 1 norm to approximate l 0 norm on the basis of RIP and KGG condition. This paper uses the primal-dual interior point method of linear programming, the branch and bound algorithm of binary programming and dual simplex method of integer linear programming to solve the magic square problems. The experimental results show that dual simplex method of integer linear programming can reach almost 100 % success rate. In addition, we propose a kind of special magic square problem and we apply this idea to construct and solve this problem, and obtain the good results.
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Acknowledgments
This research was supported in part by the Chinese National Natural Science Foundation under Grant nos. 61402395, 61472343 and 61379066, Natural Science Foundation of Jiangsu Province under contracts BK20151314,BK20140492 and BK20130452, Natural Science Foundation of Education Department of Jiangsu Province under contract 13KJB520026, and the New Century Talent Project of Yangzhou University.
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Liang, Y., Xu, X., Liao, Z., He, P. (2016). An Efficient Sparse Optimization Method for Unfinished Magic Squares. In: Yin, H., et al. Intelligent Data Engineering and Automated Learning – IDEAL 2016. IDEAL 2016. Lecture Notes in Computer Science(), vol 9937. Springer, Cham. https://doi.org/10.1007/978-3-319-46257-8_4
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DOI: https://doi.org/10.1007/978-3-319-46257-8_4
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