Abstract
Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut the search space by linear cuts or obtain new local optima from the ones previously found. While the previous studies of symmetries in the mathematical programming usually dealt with permutations of coordinates of the solutions space, the present paper considers a larger group of invertible linear transformations. We study a special case of the quadratic programming problem, where the objective function and constraints are given by quadratic forms, and the sum of all matrices of quadratic forms, involved in the constraints, is a positive definite matrix. In this setting, it is sufficient to consider only orthogonal transformations of the solution space. In this group of orthogonal transformations, we describe the structure of the subgroup which gives the symmetries of the problem. Besides that, a method for finding such symmetries is outlined, and illustrated in two simple examples.
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Notes
- 1.
For abstract groups, such a unique connection exists only in the case of simply connected groups; otherwise, an abstract exponent cannot be uniquely determined. But in our case of a matrix group, the matrix exponent is uniquely determined.
- 2.
This is because \( Q_{33} \) is \(-1\), rather than 1 as in the continuous subgroup.
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Acknowledgments
The authors thank V.M. Gichev for helpful comments on the preliminary version of the manuscript. The work on Sects. 2 and 3 was funded in accordance with the state task of the Omsk Scientific Center SB RAS (project AAAA-A19-119052890058-2). The rest of the work was funded by the program of fundamental scientific research of the SB RAS, I.5.1., project 0314-2019-0019.
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Eremeev, A.V., Yurkov, A.S. (2020). On Symmetry Groups of Some Quadratic Programming Problems. In: Kononov, A., Khachay, M., Kalyagin, V., Pardalos, P. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2020. Lecture Notes in Computer Science(), vol 12095. Springer, Cham. https://doi.org/10.1007/978-3-030-49988-4_3
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