When is the Permanent Function Convex on the Set of Doubly Stochastic Matrices?

1979 ◽  
Vol 86 (2) ◽  
pp. 119 ◽  
Author(s):  
Edward T. H. Wang
Keyword(s):  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Permanent Function ◽  
Function Convex

Inequalities for permanents and permanental minors

1965 ◽  
Vol 61 (3) ◽  
pp. 741-746 ◽  
Author(s):  
R. A. Brualdi ◽  
M. Newman
Keyword(s):  
Convex Function ◽  
Convex Set ◽  
Identity Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Counter Example ◽  
Doubly Stochastic Matrices ◽  
Permanent Function ◽  
Image Position

Let Ωndenote the convex set of alln×ndoubly stochastic matrices: chat is, the set of alln×nmatrices with non-negative entries and row and column sums 1. IfA= (aij) is an arbitraryn×nmatrix, then thepermanentofAis the scalar valued function ofAdefined bywhere the subscriptsi1,i2, …,inrun over all permutations of 1, 2, …,n. The permanent function has been studied extensively of late (see, for example, (1), (2), (3), (4), (6)) and it is known that ifA∈ Ωnthen 0 <cn≤ per (A) ≤ 1, where the constantcndepends only onn. It is natural to inquire if per (A) is a convex function ofAforA∈ Ωn. That this is not the case was shown by a counter-example given by Marcus and quoted by Perfect in her paper ((5)). In this paper, however, she shows that per (½I+ ½A) ≤ ½ + ½ per (A) for allA∈ Ωn. HereI=Inis the identity matrix of ordern.

scholarly journals Positional Voting and Doubly Stochastic Matrices

2021 ◽  
Vol 128 (4) ◽  
pp. 337-351
Author(s):  
Jacqueline Anderson ◽  
Brian Camara ◽  
John Pike
Keyword(s):  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Positional Voting
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