Conditions for the effective solvability of the quadratic choice problem. Part 1

A class of four-index real matrices is described for which the effective solvability of the quadratic choice problem is guaranteed. This means achieving the extreme values of its functional on one of the permutations of a special kind, which are given in the classical theorem of Hardy, Littlewood and Polya on the permutation of three systems. The introduced conditions generalise all the previously proposed conditions imposed on the kind of matrices that guarantee strict solvability of the problems of minimising the bilinear form on the Cartesian product of the symmetric group (conditions of the theorem on the permutation of three systems), the quadratic form of the symmetric group, and also generalise the similar results obtained for the quadratic assignment problem.