Stiemke's Alternative Theorem
Stiemke's alternative theorem is a fundamental result in the duality theory of linear programming. It states that for a matrix A, exactly one of the following holds: there exists a nonzero nonnegative vector x such that Ax=0; or there exists a vector y such that A^T y>0 (strictly positive). Along with Gordan's theorem and Farkas' lemma, it belongs to the family of alternative theorems and is often used to determine the existence of solutions to linear systems.
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Stiemke's alternative theorem states that for a matrix A, exactly one of the following holds: there exists x≥0, x≠0 such that Ax=0; or there exists y such that A^T y>0 (strictly positive).
The paper's first core result: a reward-increasing update can avoid harming all individual tasks if and only if there is no strictly positive task weighting that equalizes the policy's fitness across tasks.
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Stiemke's alternative theorem states that for a matrix A, exactly one of the following holds: there exists x≥0, x≠0 such that Ax=0; or there exists y such that A^T y>0 (strictly positive).
50%UnverifiedThe paper's first core result: a reward-increasing update can avoid harming all individual tasks if and only if there is no strictly positive task weighting that equalizes the policy's fitness across tasks.
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