Graphs generated by inconsistent systems of linear inequalities

1983 ◽  
Vol 33 (2) ◽  
pp. 146-150
Author(s):  
D. N. Gainanov ◽  
V. Yu. Novokshenov ◽  
L. I. Tyagunov
Keyword(s):  
Linear Inequalities ◽  
Inconsistent Systems ◽  
Systems Of Linear Inequalities

Iterative solution of inconsistent systems of linear inequalities

PAMM ◽  
2013 ◽  
Vol 13 (1) ◽  
pp. 407-408 ◽  
Author(s):  
Doina Carp ◽  
Constantin Popa ◽  
Cristina Serban
Keyword(s):  
Iterative Solution ◽  
Linear Inequalities ◽  
Inconsistent Systems ◽  
Systems Of Linear Inequalities

scholarly journals Modified Han algorithm for inconsistent linear inequalities

2015 ◽  
Vol 31 (1) ◽  
pp. 45-52
Author(s):  
DOINA CARP ◽  
◽  
CONSTANTIN POPA ◽  
CRISTINA SERBAN ◽  
◽  
...  
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Numerical Experiments ◽  
Linear Inequalities ◽  
Convergence Properties ◽  
Minimal Norm ◽  
Direct Solver ◽  
Least Squares Solution ◽  
Inconsistent Systems ◽  
Systems Of Linear Inequalities

In this paper we present a modified version of S. P. Han iterative method for solving inconsistent systems of linear inequalities. Our method uses an iterative Kaczmarz-type solver to approximate the minimal norm least squares solution of the problems involved in each iteration of Han’s algorithm. We prove some convergence properties for the sequence of approximations generated in this way and present numerical experiments and comparisons with Han’s and other direct solver based methods for inconsistent linear inequalities.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

Sign in / Sign up

Export Citation Format

Share Document