Finite System Of Linear Inequalities Research Articles

Nonparametric tests for Optimal Predictive Ability

A nonparametric method for comparing multiple forecast models is developed and implemented. The hypothesis of Optimal Predictive Ability generalizes the Superior Predictive Ability hypothesis from a single given loss function to an entire class of loss functions. Distinction is drawn between General Loss functions, Convex Loss functions, and Symmetric Convex Loss functions. The research hypothesis is formulated in terms of moment inequality conditions. The empirical moment conditions are reduced to an exact and finite system of linear inequalities based on piecewise-linear loss functions. The hypothesis can be tested in a statistically consistent way using a blockwise Empirical Likelihood Ratio test statistic. A computationally feasible test procedure computes the test statistic using Convex Optimization methods, and estimates conservative, data-dependent critical values using a majorizing chi-square limit distribution and a moment selection method. An empirical application to inflation forecasting reveals that a very large majority of thousands of forecast models are redundant, leaving predominantly Phillips Curve-type models, when convexity and symmetry are assumed.

Nonparametric Tests for Superior Predictive Ability

Abstract A nonparametric method for comparing multiple forecast models is developed and implemented. The hypothesis of Optimal Predictive Ability generalizes the Superior Predictive Ability hypothesis from a single given loss function to an entire class of loss functions. Distinction is drawn between General Loss functions, Convex Loss functions and Symmetric Convex Loss functions. The research hypothesis is formulated in terms of moment inequality conditions. The empirical moment conditions are reduced to an exact and finite system of linear inequalities based on piecewise-linear loss functions. The hypothesis can be tested in a statistically consistent way using a blockwise Empirical Likelihood Ratio test statistic. A computationally feasible test procedure computes the test statistic using Convex Optimization methods, and estimates conservative, data-dependent critical values using a majorizing chi-square limit distribution and a moment selection method. An empirical application to inflation forecasting reveals that a very large majority of thousands of forecast models are redundant, leaving predominantly Phillips Curve type models, when convexity and symmetry are assumed.

Farkas' Lemma, other theorems of the alternative, and linear programming in infinite-dimensional spaces: a purely linear-algebraic approach

We briefly consider several formulations of Farkas' Lemma first. Then we assume the setting of two vector spaces, one of them being linearly ordered, over a linearly ordered field till the end of this article. In this setting, we state a generalized version of Farkas' Lemma and prove it in a purely linear-algebraic way. Afterwards, we present Theorems of Motzkin, Tucker, Carver, Dax, and some other theorems of the alternative that characterize consistency of a finite system of linear inequalities. We also mention the Key Theorem, which is a related result. Finally, we use Farkas' Lemma to prove the Duality Theorem for linear programming (with a finite number of linear constraints). The Duality Theorem that is proved here covers, among others, linear programming in a real vector space of finite or infinite dimension and lexicographic linear programming.

On the Solution of a Locally Finite System of Linear Inequalities with Graph Structure

We propose a method for the solution of a locally finite system of linear inequalities that arises in the course of solution of problems of control over resource in networks with generalized Kirchhoff law. We present a criterion for a system of inequalities to have the graph structure.

Solution of a Locally Finite System of Linear Inequalities with Directed Multigraph Structure

A method is proposed for approximation of solutions of a locally finite system of linear inequalities with a multigraph structure. The results obtained are used for search for dynamic flows in networks that satisfy the generalized principle of conservation.

Algorithm for maximizing a linear function on the set of integral points of a convex polyhedron

In this paper we present a new algorithm for solving the following problem of integral linear programming: find the maximum of the linear function fo (x) = clx: qc2x~ -5 "-5 c,~x,~ ( i ) on the finite set of integral points of the convex polyhedron V, defined by the finite system of linear inequalities f~ (x) = ai:xl + ai~x2 + . . . + a~xn <~ hi, (2) ~= 1,m. It is assumed here that aij, b i, cj (i = i, m, j = I, n) belong to the ring of integers Z. In Sec. i the algorithm of [i] for finding the set of integral points of the polyhedron V is generalized; in Sec. 2 an algorithm for solving the problem of integral linear programming for the case when the set of its admissible solutions is bounded is described.

The General Solution of a Finite System of Linear Inequalities

A simple procedure for the computation of a basis of the general solution of a finite system of linear inequalities is presented. It generalizes the well-known Burger's algorithm for homogeneous systems.

Carathéodory’s Theorem with Linear Constraints

Carathéodory has shown that if x1, x2,…, xm, (m finite) are points of Rn and if somethen ∃μ∈Ω with at most n+1 nonzero components and for which (See [5]). The authors of [2] have extended this result to include the case where m= +∞. In theorems 1 and 2 below we establish somewhat similar results for the case in which Ω is further restricted by a finite system of linear inequalities (or equalities).

Establishing the matching polytope

This paper gives an elementary, inductive proof-“graphical” in spirit-of a theorem of Edmonds' which specifies the convex hull of the matchings of an arbitrary, finite, undirected graph in terms of a finite system of linear inequalities.

On the enumeration of majority games

(3) In this paper, we shall identify two games if there is a one-to-one correspondence between their players identifying their families of winning sets; to name a more definite object one must say, not game but game with ordered players Pi, * *, P,, or a similar phrase. Beyond this, note that n-player game (e.g. the 435-player game 'of the House of Representatives) may be converted into (n + k)-player game by adjoining k voteless players or dummies. The phrase an n-player game G does not here imply that all n players are non-dummies. (In the first part of the argument we need games with dummies; afterward we shall exclude them.) A majority game is a game for which one can assign numerical weights wl, , wn to the players so that the winning sets are precisely those sets which have more than half the total weight. Some of the wi may be negative or zero; it is easy to see that the corresponding players must be dummies. (Since every superset of a winning set wins, a player can have negative weight wi only if I wi I is so small that it makes no difference.) Given a game G, the question whether is a majority game is effectively decidable, since it turns on a finite system of linear inequalities. No better method than the obvious ones is known. It is clear that every majority game has non-negative integral weights-even positive integral weights, for any weights may be assigned to the dummies provided the weights of the other players are made large enough. The method described below for enumerating all majority games depends on the determination of the game with ordered players with weights (wo, w1, * , w ) by the systems (woV + wl, w2, ***, wn) and (wo -w1, w2, ***, wa). The choice