Author
David Bartl
Other affiliations: University of Ostrava
Bio: David Bartl is an academic researcher from Silesian University. The author has contributed to research in topics: Farkas' lemma & Lemma (mathematics). The author has an hindex of 4, co-authored 11 publications receiving 70 citations. Previous affiliations of David Bartl include University of Ostrava.
Papers
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TL;DR: The purpose of this paper is to present a generalization of the Farkas lemma with a short algebraic proof and to mention the key theorem and two theorems of the alternative, namely Motzkin's theorem and Tucker's theorem.
Abstract: The purpose of this paper is to present a generalization of the Farkas lemma with a short algebraic proof. The generalization lies in the fact that we formulate the Farkas lemma in the setting of two vector spaces over a common linearly ordered field where one of the vector spaces is also linearly ordered. At the end of the paper, we mention the key theorem and two theorems of the alternative, namely Motzkin's theorem and Tucker's theorem.
29 citations
TL;DR: In this article, a generalized version of Farkas' lemma was proposed and proved in a purely linear-algebraic way, and the duality theorem was proved for linear programming with a finite number of linear constraints.
Abstract: 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.
13 citations
TL;DR: A very short algebraic proof of a generalisation of the Farkas Lemma: the non-positivity of a finite homogeneous system of linear inequalities implies thenon-positive of a linear mapping whose image space is another linearly ordered vector space.
Abstract: We present a very short algebraic proof of a generalisation of the Farkas Lemma: we set it in a vector space of finite or infinite dimension over a linearly ordered (possibly skew) field; the non-positivity of a finite homogeneous system of linear inequalities implies the non-positivity of a linear mapping whose image space is another linearly ordered vector space. In conclusion, we briefly discuss other algebraic proofs of the result, its special cases and related results.
10 citations
05 Mar 2021
TL;DR: In this paper, the authors compare three methods for deriving a priority vector in the theoretical framework of pairwise comparisons: the Geometric Mean Method (GMM), Eigenvalue Method (EVM) and Best-Worst Method (BWM) with respect to two features: sensitivity and order violation.
Abstract: In this paper, we compare three methods for deriving a priority vector in the theoretical framework of pairwise comparisons—the Geometric Mean Method (GMM), Eigenvalue Method (EVM) and Best–Worst Method (BWM)—with respect to two features: sensitivity and order violation. As the research method, we apply One-Factor-At-a-Time (OFAT) sensitivity analysis via Monte Carlo simulations; the number of compared objects ranges from 3 to 8, and the comparison scale coincides with Saaty’s fundamental scale from 1 to 9 with reciprocals. Our findings suggest that the BWM is, on average, significantly more sensitive statistically (and thus less robust) and more susceptible to order violation than the GMM and EVM for every examined matrix (vector) size, even after adjustment for the different numbers of pairwise comparisons required by each method. On the other hand, differences in sensitivity and order violation between the GMM and EMM were found to be mostly statistically insignificant.
9 citations
TL;DR: In this article, a short algebraic proof of the Farkas Lemma is presented, which is a better exposition of the proof than the one presented in this paper, and the author would therefore like to publish the new form of proof in this note.
Abstract: The author published ‘A Short Algebraic Proof of the Farkas Lemma’ [SIAM J. Optim. 19 (2008), pp. 234–239]. The author then found, in his opinion, a better exposition of the proof. He would therefore like to publish the new form of the proof in this note.
7 citations
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TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.
Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.
1,479 citations
TL;DR: In this article, the authors present linear programming in infinite-dimensional spaces and show that linear programming can be used to solve problems in the real world as well as in the virtual world.
Abstract: (1989). Linear Programming in Infinite-Dimensional Spaces. Journal of the Operational Research Society: Vol. 40, No. 1, pp. 109-110.
170 citations
TL;DR: The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces and most of them are also new in finite dimensions.
Abstract: This paper concerns second-order analysis for a remarkable class of variational systems in finite-dimensional and infinite-dimensional spaces, which is particularly important for the study of optimization and equilibrium problems with equilibrium constraints. Systems of this type are described via variational inequalities over polyhedral convex sets and allow us to provide a comprehensive local analysis by using appropriate generalized differentiation of the normal cone mappings for such sets. In this paper we efficiently compute the required coderivatives of the normal cone mappings exclusively via the initial data of polyhedral sets in reflexive Banach spaces. This provides the main tools of second-order variational analysis allowing us, in particular, to derive necessary and sufficient conditions for robust Lipschitzian stability of solution maps to parameterized variational inequalities with evaluating the exact bound of the corresponding Lipschitzian moduli. The efficient coderivative calculations and characterizations of robust stability obtained in this paper are the first results in the literature for the problems under consideration in infinite-dimensional spaces. Most of them are also new in finite dimensions.
88 citations
TL;DR: In this article, the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces has been studied in the context of normal cone mappings, and necessary and sufficient conditions for the stability of these inequalities have been derived.
Abstract: In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11] , but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11] , [12] .
32 citations
TL;DR: The purpose of this paper is to present a generalization of the Farkas lemma with a short algebraic proof and to mention the key theorem and two theorems of the alternative, namely Motzkin's theorem and Tucker's theorem.
Abstract: The purpose of this paper is to present a generalization of the Farkas lemma with a short algebraic proof. The generalization lies in the fact that we formulate the Farkas lemma in the setting of two vector spaces over a common linearly ordered field where one of the vector spaces is also linearly ordered. At the end of the paper, we mention the key theorem and two theorems of the alternative, namely Motzkin's theorem and Tucker's theorem.
29 citations