6,723 research outputs found
A discrete Farkas lemma
Given and , we consider the issue of
existence of a nonnegative integral solution to the system of
linear equations . We provide a discrete and explicit analogue of the
celebrated Farkas lemma for linear systems in and prove that checking
existence of integral solutions reduces to solving an explicit linear
programming problem of fixed dimension, known in advance.Comment: 9 pages; ICCSA 2003 conference, Montreal, May 200
On Farkas Lemma and Dimensional Rigidity of Bar Frameworks
We present a new semidefinite Farkas lemma involving a side constraint on the
rank. This lemma is then used to present a new proof of a recent
characterization, by Connelly and Gortler, of dimensional rigidity of bar
frameworks.Comment: First Draf
An easy way to obtain strong duality results in linear, linear semidefinite and linear semi-infinite programming
In linear programming it is known that an appropriate non-homogeneous Farkas Lemma leads to a short proof of the strong duality results for a pair of primal and dual programs. By using a corresponding generalized Farkas lemma we give a similar proof of the strong duality results for semidefinite programs under constraint qualifications. The proof includes optimality conditions. The same approach leads to corresponding results for linear semi-infinite programs. For completeness, the proofs for linear programs and the proofs of all auxiliary lemmata for the semidefinite case are included
A short simple proof of closedness of convex cones and Farkas' lemma
Proving that a finitely generated convex cone is closed is often considered the most difficult part of geometric proofs of Farkas' lemma. We provide a short simple proof of this fact and (for completeness) derive Farkas' lemma from it using well-known arguments
A short simple proof of closedness of convex cones and Farkas' lemma
Proving that a finitely generated convex cone is closed is often considered
the most difficult part of geometric proofs of Farkas' lemma. We provide a
short simple proof of this fact and (for completeness) derive Farkas' lemma
from it using well-known arguments.Comment: 2 pages; v2: note largely rewritten, provided more context, improved
presentation, added 5 reference
From Farkasâ lemma to linear programming: An exercise in diagrammatic algebra
Farkasâ lemma is a celebrated result on the solutions of systems of linear inequalities, which finds application pervasively in mathematics and computer science. In this work we show how to formulate and prove Farkasâ lemma in diagrammatic polyhedral algebra, a sound and complete graphical calculus for polyhedra. Furthermore, we show how linear programs can be modeled within the calculus and how some famous duality results can be proved
Around a Farkas type Lemma
The first two authors of this paper asserted in Lemma 4 of "New Farkas-type
constraint qualifications in convex infinite programming" (DOI:
10.1051/cocv:2007027) that a given reverse convex inequality is consequence of
a given convex system satisfying the Farkas-Minkowski constraint qualification
if and only if certain set depending on the data contains a particular point of
the vertical axis. This paper identifies a hidden assumption in this reverse
Farkas lemma which always holds in its applications to nontrivial optimization
problems. Moreover, it shows that the statement remains valid when the
Farkas-Minkowski constraint qualification fails by replacing the mentioned set
by its closure. This hidden assumption is also characterized in terms of the
data. Finally, the paper provides some applications to convex infinite systems
and to convex infinite optimization problems.Comment: 0 figure
Dominant Strategy Mechanisms with Multidimensional Types
This paper provides a characterization of dominant strategy mechanisms with quasi-linear utilities and multi-dimensional types for a variety of preference domains. These characterizations are in terms of a monotonicity property on the underlying allocation rule.Dominant Strategy, Farkas Lemma, Combinatorial Auctions.
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