Certifying feasibility and objective value of linear programs

Althaus, Ernst; Dumitriu, Daniel

doi:10.1016/j.orl.2012.03.004

Cited by 2 publications

(2 citation statements)

References 13 publications

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“…For chemical reaction networks with mass-action kinetics, the concentration dynamics are governed by dynamical systems (2) with polynomial maps f κ (x) = A κ x B , as defined in (1). We introduce some terms that are standard in the chemical engineering literature.…”

Section: Motivation From Chemical Reaction Networkmentioning

confidence: 99%

“…Therefore, we can verify the infeasibility of (11) by checking the infeasibility of the system of linear inequalities obtained by replacing the inequalities > 0 and < 0 by ≥ ǫ and ≤ −ǫ, respectively, for an arbitrary ǫ ∈ R + . In this setting, one can apply methods for exact linear programming, which makes use of Farkas' lemma to guarantee the infeasibility of linear programs by way of rational certificates; see for example [1,3,36] and the exact linear programming solver QSopt ex [4]. An alternative is to develop and adapt exact linear programming methods for strict inequalities using Theorems of the Alternative (Transposition theorems); see for example [51,64].…”

Section: Algorithmic Verification Of Sign Conditionsmentioning

confidence: 99%

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Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

et al. 2015

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We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes' rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.

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Section: Motivation From Chemical Reaction Networkmentioning

confidence: 99%

Section: Algorithmic Verification Of Sign Conditionsmentioning

confidence: 99%