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The latter variant is mentioned for completeness; it is not actually a "Farkas lemma" since it contains only equalities. Its proof is an exercise in linear algebra. There are also Farkas-like lemmas for integer programs. [9]: 12--14 For systems of equations, the lemma is simple: Assume that A and b have rational coefficients.
Let (,) be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation, (,) =and the inhomogeneous equation (,) = ().The Fredholm alternative is the statement that, for every non-zero fixed complex number, either the first equation has a non-trivial solution, or the second equation has a solution for all ().
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space.There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap.
The proof establishes that, once the simplex algorithm finishes with a solution to the primal LP, it is possible to read from the final tableau, a solution to the dual LP. So, by running the simplex algorithm, we obtain solutions to both the primal and the dual simultaneously. [1]: 87–89 Another proof uses the Farkas lemma. [1]: 94
These results are used by the theory of convex minimization along with geometric notions from functional analysis (in Hilbert spaces) such as the Hilbert projection theorem, the separating hyperplane theorem, and Farkas' lemma. [citation needed]
This fact is known as Farkas' lemma. A subtle point in the representation by vectors is that the number of vectors may be exponential in the dimension, so the proof that a vector is in the cone might be exponentially long.
Burnside's lemma also known as the Cauchy–Frobenius lemma; Frattini's lemma (finite groups) Goursat's lemma; Mautner's lemma (representation theory) Ping-pong lemma (geometric group theory) Schreier's subgroup lemma; Schur's lemma (representation theory) Zassenhaus lemma
Farkas' lemma – can be proved using FM elimination. Real closed field – the cylindrical algebraic decomposition algorithm performs quantifier elimination over polynomial inequalities, not just linear.