Permanence Analysis for Continuous and Discrete-time Generalized Lotka-Volterra Models with Delay and Switching of Parameters

The problem of finding a feasible solution to a linear inequality system arises in numerous contexts. We consider solving linear semi-infinite inequality systems via an extension of the relaxation method for finite linear inequality systems. The difficulties are discussed and a convergence result is derived under fairly general assumptions on a large class of linear semi-infinite inequality systems.

Bimonotone linear inequalities and sublattices of [formula omitted

We introduce a concept of dual consistency of systems of linear inequalities with full generality. We show that a cardinality constrained polytope is represented by a certain system of linear inequalities if and only if the systems of linear inequalities associated with the cardinalities are dual consistent. Typical dual consistent systems of inequalities are those which describe polymatroids, generalized polymatroids, and dual greedy polyhedra with certain choice functions. We show that the systems of inequalities for cardinality-constrained ordinary bipartite matching polytopes are not dual consistent in general, and give additional inequalities to make them dual consistent. Moreover, we show that ordinary systems of inequalities for the cardinality-constrained (poly)matroid intersection are not dual consistent, which disproves a conjecture of Maurras, Spiegelberg, and Stephan about a linear representation of the cardinality-constrained polymatroid intersection.

We introduce a concept of dual consistency of systems of linear inequalities with full generality. We show that a cardinality constrained polytope is represented by a certain system of linear inequalities if and only if the systems of linear inequalities associated with the cardinalities are dual consistent. Typical dual consistent systems of inequalities are those which describe polymatroids, generalized polymatroids, and dual greedy polyhedra with certain choice functions. We show that the systems of inequalities for cardinality-constrained ordinary bipartite matching polytopes are not dual consistent in general, and give additional inequalities to make them dual consistent. Moreover, we show that ordinary systems of inequalities for the cardinality-constrained (poly)matroid intersection are not dual consistent, which disproves a conjecture of Maurras, Spiegelberg, and Stephan about a linear representation of the cardinality-constrained polymatroid intersection.

A typical recurrent neural network called zeroing neural network (ZNN) was developed for time-varying problem-solving in a previous study. Many applications result in time-varying linear equation and inequality systems that should be solved in real time. This paper provides a ZNN model for determining the solution of time-varying linear equation and inequality systems. By introducing a nonnegative slack variable, the time-varying linear equation and inequality systems are transformed into a mixed nonlinear system. The ZNN model is established via the definition of an indefinite error function and the usage of an exponential decay formula. Theoretical results indicate the convergence property of the proposed ZNN model. Comparative simulation results prove the ZNN effectiveness and superiority for time-varying linear equation and inequality systems. Furthermore, the proposed ZNN model is employed to robot manipulators, thus showing the ZNN applicability.

A representation of an arbitrary system of strict linear inequalities in R n as a system of points is proposed. The representation is obtained by using a so-called polarity. Based on this representation an algorithm for constructing a committee of a plane system of linear inequalities is given. A solution of two problems on minimal committee of a plane system is proposed. The obtained solutions to these problems can be found by means of the proposed algorithm.

A method for solving the system of linear equations and linear inequalities

We consider the methods for matrix correction or correction of all parameters of systems of linear equations and inequalities. We show that the problem of matrix correction of an inconsistent system of linear inequalities with the nonnegativity condition is reduced to a linear programming problem. Some stability measure is defined for a given solution to a system of linear inequalities as the minimal possible variation of parameters under which this solution does not satisfy the system. The problem of finding the most stable solution to the system is considered. The results are applied to constructing an optimal separating hyperplane in the feature space that is the most stable to the changes of features of the objects.

We present a novel method for prioritizing both linear equality and inequality systems and provide one algorithm for its resolution. This algorithm can be summarized as a sequence of optimal resolutions for each linear system following their priority order. We propose an optimality criterion that is adapted to linear inequality systems and characterize the resulting optimal sets at every priority level. We have successfully applied our method to plan local motions for the humanoid robot HPR-2. We will demonstrate the validity of the method using an original scenario where linear inequality constraints are solved at lower priority than equality constraints.

As a basic mathematical structure, the system of inequalities over symmetric cones and its solution can provide an effective method for solving the startup problem of interior point method which is used to solve many optimization problems. In this paper, a non-interior continuation algorithm is proposed for solving the system of inequalities under the order induced by a symmetric cone. It is shown that the proposed algorithm is globally convergent and well-defined. Moreover, it can start from any point and only needs to solve one system of linear equations at most at each iteration. Under suitable assumptions, global linear and local quadratic convergence is established with Euclidean Jordan algebras. Numerical results indicate that the algorithm is efficient. The systems of random linear inequalities were tested over the second-order cones with sizes of 10,100, ...,1 000 respectively and the problems of each size were generated randomly for 10 times. The average iterative numbers show that the proposed algorithm can generate a solution at one step for solving the given linear class of problems with random initializations. It seems possible that the continuation algorithm can solve larger scale systems of linear inequalities over the second-order cones quickly. Moreover, a system of nonlinear inequalities was also tested over Cartesian product of two simple second-order cones, and numerical results indicate that the proposed algorithm can deal with the nonlinear cases.

The strong Slater condition plays a significant role in the stability analysis of linear semi-infinite inequality systems. This piece of work studies the set of strong Slater points, whose non-emptiness guarantees the fullfilment of the strong Slater condition. Given a linear inequality system, we firstly establish some basic properties of the set of strong Slater points. Then, we derive dual characterizations for this set in terms of the data of the system, following similar characterizations provided also for the set of Slater points and the solution set of the given system, which are based on the polarity operators for evenly convex and closed convex sets. Finally, we present two geometric interpretations and apply our results to analyze the strict inequality systems defined by lower semicontinuous convex functions.

A Linear Pricing Rule is established for the No Strong Arbitrage Principle (NSAP) in a finite state, single period asset pricing model. The (NSAP) condition is a statement about the inconsistency of a particular system of linear inequalities. The novelty here lies in the use of the Kuhn-Motzkin-Fourier elimination technique that derives the corresponding dual linear inequality system using elementary methods only. The advantage is that a familiar computational scheme yields the relationship between the (NSAP) inequalities and their dual system. Indeed, the method uncovers why the dual inequality system is, in fact, a dual system in the first place. Students and researchers unfamiliar with systems of dual linear inequalities and Theorems of the Alternative may find the approach taken here as a way to better understand the motivation and use of these techniques.

It should be apparent by now that systems of linear inequalities emerge naturally in revealed preference theory. They constitute the essence of Afriat's Theorem, for example; and we formulated revealed preference problems using systems of linear inequalities in Chapters 3, 6, 7, and 8. In this chapter, we describe how revealed preference problems can generally be understood as a system of inequalities. From a purely computational perspective, one can very often solve a revealed preference problem by algorithmically solving the corresponding system of inequalities.

The purpose of this chapter is to present some fundamental theorems for systems of linear integer inequalities. A discrete polyhedral set SI = S ∩ Zn corresponds to each integer polyhedral set S (i.e. a polyhedral set generated by a system of linear inequalities with integer coefficients). SI will be called the discrete polyhedral set attached to S. The set S is always classically convex. Chapter 2 proved that SI is not classically convex but it is convex with respect to Zn. Conversely, if X ⊆ Zn is an integer convex set, then the set conv (X) is an integer polyhedral set.

The reduction procedures presented use the Sup-Inf-Method developed by Bledsoe, (see [BLE] or [SHO],) a complete decision procedure for systems of linear inequalities over the rational numbers. A system of linear inequalities over the integers given it can find out its unsatisfiability only, if it is unsatisfiable if considered as system over the rationals. But there are other reduction procedures outlined in chapter 3, which make possible to increase the class of systems over the integers recognized to be satisfiable by the Sup-InfMethod. These procedures and some results taken from the theory of polyhedral sets allow to construct a procedure deciding most systems of linear inequalities over the integers without use of quantifier elimination. This algorithm is presented in chapter 5 together with its foundations. Additionally the class of systems of inequalities decided by it is characterized.