Abstract:
In modern terminology, the conditions of the classical Hardy, Littlewood and Pólya theorem on the permutation of three systems guarantee the strict solvability of the optimisation problem for a bilinear form with a symmetric Toeplitz matrix of a special type. The bilinear form with the specified matrix takes extreme values on substitutions of two types, depending on whether the components of two vectors have the same or opposite orderings. Here the vectors deter mine the values of the variables of the bilinear form. The previous part of the article describes the conditions for achie ving the minimum of the functional of the quadratic choice problem on the first of these substitutions. These conditions generalise all previously obtained results of a similar plan for the quadratic form minimisation problem and the quadratic assignment problem. This section of the paper considers conditions, imposing of which on the elements of a fourindex matrix, guarantee the achievement of the minimum of the quadratic choice problem functional on the second substi tution given in the threesystem permutation theorem. The results presented in the two sections of the article describe by far the widest classes of fourindex matrices for which the quadratic choice problem functional takes extreme values on fixed substitutions.
Fulltext:
PDF file (1065 kB)