Maximum Entropy Convex Decompositions of Doubly Stochastic and Nonnegative Matrices

1987 ◽  
Vol 19 (3) ◽  
pp. 403-407 ◽  
Author(s):  
P B Slater
Keyword(s):  
Maximum Entropy ◽  
Extreme Points ◽  
Social Classes ◽  
Nonnegative Matrices ◽  
Iterative Proportional Fitting ◽  
Mathematical Structures ◽  
Stochastic Version ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Fitting Procedure

Two maximum entropy convex decompositions are computed with the use of the iterative proportional fitting procedure. First, a doubly stochastic version of a 5 × 5 British social mobility table is represented as the sum of 120 5 × 5 permutation matrices. The most heavily weighted permutations display a bandwidth form, indicative of relatively strong movements within social classes and between neighboring classes. Then the mobility table itself is expressed as the sum of 6 985 5 × 5 transportation matrices—possessing the same row and column sums as the mobility table. A particular block-diagonal structure is evident in the matrices assigned the greatest weight. The methodology can be applied as well to the representation of other nonnegative matrices in terms of their extreme points, and should be extendable to higher-order mathematical structures—for example, operators and functions.

On the Extreme Rays of the Metric Cone

1980 ◽  
Vol 32 (1) ◽  
pp. 126-144 ◽  
Author(s):  
David Avis
Keyword(s):  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Convex Polyhedra ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Image Position ◽  
Polyhedral Convex Set ◽  
Extreme Rays

A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and neatest example of this is the class of doubly stochastic matrices. This polyhedron can be defined by the system of equationsBirkhoff [2] and Von Neuman have shown that the extreme points of this bounded polyhedron are just the n × n permutation matrices. The importance of this result for mathematical programming is that it tells us that the maximum of any linear form over P will occur for a permutation matrix X.

Maximum-Entropy Representations in Convex Polytopes: Applications to Spatial Interaction

1989 ◽  
Vol 21 (11) ◽  
pp. 1541-1546 ◽  
Author(s):  
P B Slater
Keyword(s):  
Maximum Entropy ◽  
Convex Combination ◽  
Spatial Interaction ◽  
Stochastic Matrix ◽  
Convex Polytopes ◽  
Convex Polytope ◽  
Extreme Points ◽  
Doubly Stochastic ◽  
Interaction Modeling

Of all representations of a given point situated in a convex polytope, as a convex combination of extreme points, there exists one for which the probability or weighting distribution has maximum entropy. The determination of this multiplicative or exponential distribution can be accomplished by inverting a certain bijection—developed by Rothaus and by Bregman—of convex polytopes into themselves. An iterative algorithm is available for this procedure. The doubly stochastic matrix with a given set of transversals (generalized diagonal products) can be found by means of this method. Applications are discussed of the Rothaus -Bregman map to a proof of Birkhoff's theorem and to the calculation of trajectories of points leading to stationary or equilibrium values of the generalized permanent, in particular in spatial interaction modeling.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

An Analogue of Birkhoff's Problem III for Infinite Markov Matrices1

1969 ◽  
Vol 12 (5) ◽  
pp. 625-633
Author(s):  
Choo-Whan Kim
Keyword(s):  
Convex Hull ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Permutation Matrices ◽  
Doubly Stochastic Matrices

A celebrated theorem of Birkhoff ([1], [6]) states that the set of n × n doubly stochastic matrices is identical with the convex hull of the set of n × n permutation matrices. Birkhoff [2, p. 266] proposed the problem of extending his theorem to the set of infinite doubly stochastic matrices. This problem, which is often known as Birkhoffs Problem III, was solved by Isbell ([3], [4]), Rattray and Peck [7], Kendall [5] and Révész [8].

Insights into European interbank network contagion

2015 ◽  
Vol 41 (8) ◽  
pp. 754-772 ◽  
Author(s):  
Dionisis Philippas ◽  
Yiannis Koutelidakis ◽  
Alexandros Leontitsis
Keyword(s):  
Financial Stability ◽  
Preferential Attachment ◽  
Policy Implications ◽  
Contagion Effect ◽  
Iterative Proportional Fitting ◽  
Content Type ◽  
Scale Free ◽  
Banking Systems ◽  
Fitting Procedure ◽  
Capital Ratios

Purpose – The purpose of this paper is to analyse the importance of interbank connections and shocks on banks’ capital ratios to financial stability by looking at a network comprising a large number of European and UK banks. Design/methodology/approach – The authors model interbank contagion using insights from the Susceptible Infected Recovered model. The authors construct scale-free networks with preferential attachment and growth, applying simulated interbank data to capture the size and scale of connections in the network. The authors proceed to shock these networks per country and perform Monte Carlo simulations to calculate mean total losses and duration of infection. Finally, the authors examine the effects of contagion in terms of Core Tier 1 Capital Ratios for the affected banking systems. Findings – The authors find that shocks in smaller banking systems may cause smaller overall losses but tend to persist longer, leading to important policy implications for crisis containment. Originality/value – The authors infer the interbank domestic and cross-border exposures of banks employing an iterative proportional fitting procedure, called the RAS algorithm. The authors use an extend sample of 169 European banks, that also captures effects on the UK as well as the Eurozone interbank markets. Finally, the authors provide evidence of the contagion effect on each bank by allowing heterogeneity. The authors compare the bank’s relative financial strength with the contagion effect which is modelled by the number and the volume of bilateral connections.

Constrained Optimization of Multi-Degree-of-Freedom Mechanisms for Near-Time-Optimal Motions

1994 ◽  
Vol 116 (2) ◽  
pp. 412-418 ◽  
Author(s):  
S. Sundar ◽  
Z. Shiller
Keyword(s):  
Constrained Optimization ◽  
Optimization Problem ◽  
Extreme Points ◽  
Close Correlation ◽  
Projection Methods ◽  
Degree Of Freedom ◽  
Design Parameters ◽  
Maximum Acceleration ◽  
Fitting Procedure ◽  
Time Optimal

This paper presents a method to design multi-degree-of-freedom mechanisms for near-time optimal motions. The design objective is to select system parameters, such as link lengths and actuator sizes, that will minimize the optimal motion time of the mechanism along a given path. The exact time-optimization problem is approximated by a simpler procedure that maximizes the acceleration near the end points. Representing the directions of maximum acceleration with the acceleration lines, and the reachability constraints as explicit functions of the design parameters, we transform the constrained optimization to a simpler curve-fitting procedure. This problem is formulated analytically, permitting the use of efficient gradient-based optimizations instead of the zero order optimization that is otherwise required. It is shown that with the appropriate choice of variables, the reachability constraints for planar mechanisms are linear in the design parameters. Consequently, the reachability of the entire path can be guaranteed by satisfying the reachability of only two extreme points along the path. This greatly simplifies the optimization problem since it reduces the dimensionality of the constraints and it permits the use of efficient projection methods. Examples for optimizing the dimensions of a five-bar planar mechanism demonstrate close correlation between the approximate and the exact solutions and better computational efficiency of the constrained optimization over previous penalty-based methods.

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