TY - JOUR
T1 - Finding Symmetry Groups of Some Quadratic Programming Problems
AU - Eremeev , Anton V.
AU - Yurkov , Alexander S.
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 370
EP - 392
PY - 2023
DA - 2023/04
SN - 16
DO - http://doi.org/10.4208/nmtma.OA-2022-0092
UR - https://global-sci.org/intro/article_detail/nmtma/21581.html
KW - Non-convex programming, orthogonal transformation, symmetry group, Lie group.
AB - <p style="text-align: justify;">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 decrease the problem dimension, reduce the size of the search space by means of linear cuts. 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. We formulate conditions, which allow us to transform the
original problem to a new system of coordinates, such that the symmetries may be
sought only among orthogonal transformations. In particular, these conditions are
satisfied if the sum of all matrices of quadratic forms, involved in the constraints,
is a positive definite matrix. We describe the structure and some useful properties
of the group of symmetries of the problem. Besides that, the methods of detection
of such symmetries are outlined for different special cases as well as for the general
case.</p>