Copositive matrices with nondecreasing off-diagonal entries admit a PSD plus nonnegative decomposition, which implies exactness of a natural relaxation for separable quadratic optimization over the simplex.
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Evaluating rational functions without subtraction over the tropical semiring yields combinatorial interpretations that can serve as proofs.
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Copositive Matrices with Ordered Off-Diagonal Entries
Copositive matrices with nondecreasing off-diagonal entries admit a PSD plus nonnegative decomposition, which implies exactness of a natural relaxation for separable quadratic optimization over the simplex.
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Detropicalization as a proof technique
Evaluating rational functions without subtraction over the tropical semiring yields combinatorial interpretations that can serve as proofs.