Integer Hulls, Z-Polyhedra and Presburger Arithmetic in Action

A polyhedron is called integer if its every vertex has integer coordinates. We consider integer polyhedra PI = conv(P ∩ ℤd) defined implicitly; that is, no system of linear inequalities is known for PI but some is known for P. Some estimates are given for the number of vertices of PI.

Fourier-Motzkin elimination is a fundamental operation in polyhedral geometry. It can be performed by several equivalent procedures, which can be regarded as an adaptation of Gaussian elimination to systems of linear inequalities. These procedures tend to generate large numbers of redundant inequalities. Efficiently detecting these redundancies is essential to obtain software implementation of practical interest. In this paper, we propose a detection technique. We demonstrate its benefits over alternative approaches. A detailed experimentation is reported.

An Elementary Exposition of the No Strong Arbitrage Principle for Financial Markets

Scheduling of a LQ Control Algorithm for Efficient FPGA Implementation

Vector-valued Markov decision processes and the systems of linear inequalities

We show in detail that the category of general Roth systems or the category of semi-stable systems of linear inequalities of slope zero is a neutral Tannakian category. On the way, we present a new proof of the semi-stability of the tensor product of semi-stable systems. The proof is based on a numerical criterion for a system of linear inequalities to be semi-stable.

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

The consistency of multi-index transport-type systems of linear inequalities is investigated using an approach based on the principle of cycle-based reducibility of systems of linear inequalities to flow algorithms. Sufficient conditions of reducibility, determining a subclass of multi-index systems that can be solved using this approach are found.

The paper considers a multicriteria multimodal transport problem, which contains three types of transport and two objective functions. The article investigates the possibility of transforming a multi-criteria multimodal transport problem into a classical transport problem by constructing a system of linear matrix equations and inequalities. It is shown that measurements for the corresponding system of linear matrix equations and inequalities are directly dependent on the number of objective functions of the considered problem. The paper proposes the algorithm for finding the problem reference plans, which is based on finding the reference plans for different types of vehicles. The multicriteria problem is solved using the method of weighted coefficients and the method of sequential concessions according to the relevant criteria. Tools of the computer mathematics system Mathcad were used for the numerical experiment on solving demonstration problems by the methods proposed in the work and for comparing the obtained results. The application of the proposed methods in this paper for solving multicriteria multimodal transport problems allows to reduce the number of iterations in the numerical solution of these problems and to overcome the different measurement problem of the objective functions in their superposition.

Existence of optimal strategies in bimatrix game and applications

Problems that reduce to consistency or inconsistency of systems of linear equations or inequalities arise in many divisions of theoretical informatics. The examples are problems in linear programming, machine learning, multicriteria optimization, etc. There exist different stability measures for the property of consistency or inconsistency, and different information constituents are possible (all the input parameters, the coefficient matrix, the vector of constraints). In this paper, variations of all parameters are examined in combination with an additional constraint important in applications, namely, the nonnegativity of feasible points.

Consider a rational convex polygon given by a system of linear inequalities \(A \vec {x} \le \vec {b}\), where A is a matrix over \({{\mathbb Z}}\), with m rows and 2 columns, and \(\vec {b}\) is an integer vector. The coordinates \(b_1, \ldots , b_m\) of \(\vec {b}\) are treated as parameters while the coefficients of A have fixed values. We observe that for every \(1 \le i \le m\), there exists a positive integer \(T_i\) so that, when each \(b_1, \ldots , b_m\) is large enough, the vertex sets V and \(V'\) of the respective integer hulls of \(P := P(b_1,\ldots ,b_{i-1},b_i,b_{i+1}, \ldots , b_m)\) and \(P' := P(b_1,\ldots ,b_{i-1},b_i \,+\,T_i,b_{i+1}, \ldots , b_m)\), respectively, are in a “simple” one-to-one correspondence. We state and prove explicit formulas for the pseudo-period \(T_i\) and that correspondence between V and \(V'\). This result and the ingredients of its proof lead us to propose a new algorithm for computing the integer hull of a rational convex polygon.

This is the third of five chapters on optimal programming (the typical mathematics of economics) and related issues as related to choice making. It discusses linear programming, which might appear to be a special case of convex programming, but is more substantial, and is really an embodiment of the theory of systems of linear inequalities (as reflected here). This chapter initiates the subject with reference to systems of linear inequalities and natural questions about them, and all LP (linear programming) theorems are encountered simply in pursuing those. Theorems about linear inequalities that have uses directly on their own are also derived (and are illustrated in many places in this book). The eight sections of the chapter are: linear inequalities; separation theorems; theorems of alternatives; polyhedra and polytopes; LP Duality Theorem; the pivot operation; the Simplex Algorithm; and BASIC program.

Entropic perturbation method for solving a system of linear inequalities

The row pivoting method for linear programming