  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.

Optimization and Control