Explore the vast range of titles available.

Pattern Recognition on Oriented Matroids covers a range of innovative problems in combinatorics, poset and graph theories, optimization, and number theory that constitute a far-reaching extension of the arsenal of committee methods in pattern recognition.

We study properties of systems of linear constraints that are minimally infeasible with respect to some subset S of constraints (i.e., systems that are infeasible but that become feasible on removal of any constraint in S ). We then apply these results and a theorem of Conforti, Cornuéjols, Kapoor, and Vu kovi to a class of 0, 1 matrices, for which the linear relaxation of the set-partitioning polytope LSP ( A )= { x | Ax = 1 , x 0} is integral. In this way, we obtain combinatorial properties of those matrices in the class that are minimal (w.r.t. taking row submatrices) with the property that the set-partitioning polytope associated with them is infeasible.

This paper presents a branch-and-cut algorithm for the NP-hard maximum feasible subsystem problem: For a given infeasible linear inequality system, determine a feasible subsystem containing as many inequalities as possible. The complementary problem, where one has to remove as few inequalities as possible in order to make the system feasible, can be formulated as a set covering problem. The rows of this formulation correspond to irreducible infeasible subsystems, which can be exponentially many. It turns out that the main issue of a branch-and-cut algorithm for the maximum feasible subsystem problem (Max FS) is to efficiently find such infeasible subsystems. We present three heuristics for the corresponding NP-hard separation problem and discuss cutting planes from the literature, such as set covering cuts of Balas and Ng, Gomory cuts, and $\\{0,\\frac{1}{2}\\}$-cuts. Furthermore, we compare a heuristic of Chinneck and a simple greedy algorithm. The main contribution of this paper is an extensive computational study on a variety of instances arising in a number of applications.

A classical problem in the study of an infeasible system of linear inequalities is to determine irreducible infeasible subsystems of inequalities (IISs), i.e., infeasible subsets of inequalities whose proper subsets are feasible. In this article, we examine a particular situation where only a given subsystem is of interest for the analysis of infeasibility. For this, we define relatively irreducible infeasible subsystems (RIISs) as infeasible subsystems of inequalities that are irreducible with respect to a given subsystem. It is a generalization of the definition of an IIS, since an IIS is irreducible with respect to the full system. We provide a practical characterization of RIISs, making the link with the alternative polyhedron commonly used in the detection of IISs. We then turn to the study of the RIISs that can be obtained from the Phase I of the simplex algorithm. We answer an open question regarding the covering of the clusters of IISs and show that this result cannot be generalized to RIISs. We thus develop a practical algorithm to find a covering of the clusters of RIISs. Our findings are numerically illustrated on the Netlib infeasible linear programs.

We study the optimum correction of infeasible systems of linear inequalities through making minimal changes in the coefficient matrix and the right-hand side vector by using the Frobenius norm. It leads to a special structured unconstrained nonlinear and nonconvex problem, which can be reformulated as a one-dimensional parametric minimization problem such that each objective function corresponds to a trust region subproblem. We show that, under some assumptions, the parametric function is differentiable and strictly unimodal. We present optimally conditions, propose lower and upper bounds on the optimal value and discuss attainability of the optimal value. To solve the original problem, we propose a binary search method accompanied by a type of Newton–Lagrange method for solving the subproblem. The numerical results illustrate the effectiveness of the suggested method.

A classical theorem by Block and Levin (Block, H. D., S. A. Levin. 1970. On the boundedness of an iterative procedure for solving a system of linear inequalities. Proc. Amer. Math. Soc. 26 229–235) states that certain variants of the relaxation method for solving systems of linear inequalities generate bounded sequences of intermediate solutions, even when applied to infeasible systems. Using a new approach, we prove a more general version of this result and answer an old open problem of quantifying the bound as a function of the input data.