Projection method for solving systems of linear inequalities

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 $\boldsymbol{\pi}(N)$ .

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On systems of linear inequalities in infinitely many variables and generalized Hausdorff means

Mathematische Zeitschrift ◽

10.1007/bf01112818 ◽

1963 ◽

Vol 82 (1) ◽

pp. 1-7 ◽

Cited By ~ 7

Author(s):

Kurt Endl

Keyword(s):

Linear Inequalities ◽

Systems Of Linear Inequalities ◽

Hausdorff Means

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On finite-rowed systems of linear inequalities in infinitely many variables. II

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1933-1501695-4 ◽

1933 ◽

Vol 35 (2) ◽

pp. 452-452 ◽

Cited By ~ 2

Author(s):

I. J. Schoenberg

Keyword(s):

Linear Inequalities ◽

Systems Of Linear Inequalities

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Lower Bounds for the Football Pool Problem for 7 and 8 Matches

The Electronic Journal of Combinatorics ◽

10.37236/945 ◽

2007 ◽

Vol 14 (1) ◽

Author(s):

Wolfgang Haas

Keyword(s):

Lower Bounds ◽

Covering Radius ◽

Linear Inequalities ◽

Minimal Cardinality ◽

Covering Code ◽

Ternary Code ◽

Systems Of Linear Inequalities

Let $k_3(n)$ denote the minimal cardinality of a ternary code of length $n$ and covering radius one. In this paper we show $k_3(7)\ge 156$ and $k_3(8)\ge 402$ improving on the best previously known bounds $k_3(7)\ge 153$ and $k_3(8)\ge 398$. The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the number of elements of a covering code, that lie in $k$-dimensional subspaces of F${}_3^n$.

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