A Geometric Programming Approach to Solve Van Der Warden Conjecture on Doubly Stochastic Matrices

Raghavacahari M

IIMA Working Papers from Indian Institute of Management Ahmedabad, Research and Publication Department

Abstract: Let Dn be the set of all doubly stochastic square matrices of order n i.e. the set of all n x n matrices with non-negative entries with row and column sums equal to unity. The permanent of an n x n matrix A = (aij) is defined where sn is the symmetric group of order n. van der Waerdon conjectured that P(A) > n !/nn for all ^EDn with equality occurring if and only if A = Jn, where Jn is the matrix all of whose entries are equal to 1/n. The validity of this conjecture has been shown for a few values of n and for general n 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 formulation is given. A related problem of reversed homogeneous posynomial programming problem is also studied.

Date: 1976-11-01
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