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**11 - 14**of**14**### Lexicographic Pivoting Rules, LexPr

, 1998

"... this process is repeated endlessly. Because the simplex method produces a sequence with monotonically improving objective values, the objective stays constant in a cycle, thus each pivot in the cycle must be degenerate. The possibility of cycling was recognized shortly after the invention of the sim ..."

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this process is repeated endlessly. Because the simplex method produces a sequence with monotonically improving objective values, the objective stays constant in a cycle, thus each pivot in the cycle must be degenerate. The possibility of cycling was recognized shortly after the invention of the simplex algorithm. Cycling examples were given by E.M.L. Beale [2] and by A.J. Hoffman [10]. Recently a scheme to construct cycling LO examples is presented in [9]. These examples made evident that extra techniques are needed to ensure finite termination of simplex methods. The first and widely used such tool is the lexicographic simplex rule. Other techniques, like the leastindex anti-cycling rules and more general recursive schemes were developed more recently. Lexicographic simplex methods. First we need to define an ordering, the so-called lexicographic ordering of vectors. Lexicograph

### On Circuit Valuation of Matroids

, 2000

"... The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R[f01g)-valued vectors defined on the circuits of the un ..."

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The concept of valuated matroids was introduced by Dress and Wenzel as a quantitative extension of the base exchange axiom for matroids. This paper gives several sets of cryptomorphically equivalent axioms of valuated matroids in terms of (R[f01g)-valued vectors defined on the circuits of the underlying matroid, where R is a totally ordered additive group. The dual of a valuated matroid is characterized by an orthogonality of (R [ f01g)- valued vectors on circuits. Minty's characterization for matroids by the painting property is generalized for valuated matroids.