A geometric programming approach to the Van der Waerden conjecture on doubly stochastic matrices

Abstract

LetD _p be the set of all doubly stochastic square matrices of orderp i.e. the set of allp × p matrices with non-negative entries with row and column sums equal to unity. The permanent of ap × p matrixA = (a _ij ) is defined byP(A) = Σ _τ∈Sp II ^p _i=1 a _iτ(i) whereS _p is the symmetric group of orderp. Van der Waerden conjectured thatP(A) ≥ p !/p ^p for all A ∈A∈D _p with equality occurring if and only ifA = J _p , whereJ _p is the matrix all of whose entries are equal to 1/p.

The validity of this conjecture has been shown for a few values ofp and for generalp under certain assumptions. In this paper the problem of finding the minimum of the permanent of a doubly stochastic matrix has been formulated as a reversed geometric program with a single constraint and an equivalent dual program is given. A related problem of reversed homogeneous posynomial programming problem is also studied.