A discrete Farkas lemma
نویسندگان
چکیده
منابع مشابه
A Discrete Farkas Lemma
Given A ∈ Zm×n and b ∈ Zm, we consider the issue of existence of a nonnegative integral solution x ∈ Nn to the system of linear equations Ax = b. We provide a discrete and explicit analogue of the celebrated Farkas lemma for linear systems in Rn and prove that checking existence of integral solutions reduces to solving an explicit linear programming problem of fixed dimension, known in advance.
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The Farkas Lemma is extended to simultaneous linear operator and polyhedral sublinear operator inequalities by Boolean valued analysis. Introduction The Farkas Lemma, also known as the Farkas–Minkowski Lemma, plays a key role in linear programming and the relevant areas of optimization (cp. [1]). Some rather simple proof of the lemma is given in [2]. Using the technique of Boolean valued analys...
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Farkas’ lemma is one of the key results in optimization. Yet, it is not a trivial conclusion, and its proof contains certain difficulties. In this note we propose a new proof which is based on elementary arguments.
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This note presents a partially new proof of Farkas’ lemma, based on no other tools than elementary linear algebra (matrixand vector calculus). No properties of the real numbers other than those shared by the rational numbers are used. The general approach is the same as in the paper by A. Dax from 1997 (SIAM Review, 39(3):503-507), but instead of using an active set method for proving the exist...
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The main purpose of this paper is to extend the conventional separation theorems concerning the convex subset of Rm to generalized separation theorems concerning the convex subset of Rm × Sn. This is accomplished by the introduction of the generalized inner product in Rm × Sn. Then we derive the famous Farkas' lemma in nonlinear semidefinite programming, which may be very important for the anal...
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ژورنال
عنوان ژورنال: Discrete Optimization
سال: 2004
ISSN: 1572-5286
DOI: 10.1016/j.disopt.2004.04.002