The value function of a mixed integer program: I

Blair, Charles E.; Jeroslow, Robert G.

doi:10.1016/0012-365x(77)90028-0

Cited by 101 publications

(56 citation statements)

References 8 publications

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“…To this end, we need quantitative continuity and growth properties of optimal value functions and solution sets of parametric mixed-integer linear programs. Such properties are known for parametric right-hand sides [4,5,20] and parametric costs separately [1,2,6]. Since to our knowledge simultaneous perturbation results with respect to right-hand sides and costs are less familiar, we discuss such properties of optimal value functions in Proposition 1.…”

Section: Introductionmentioning

confidence: 94%

Quantitative stability of fully random mixed-integer two-stage stochastic programs

Römisch

Vigerske

2007

Optimization Letters

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Section: Introductionmentioning

confidence: 94%

Quantitative stability of fully random mixed-integer two-stage stochastic programs

Römisch

Vigerske

2007

Optimization Letters

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“…Proof Under our standard assumptions, finiteness, upper semicontinuity and statement about the discontinuity points of f follow directly from the classical results in [7] and [13]. By the same results, there exist constants α, β > 0 such that…”

Section: Assumptionmentioning

confidence: 75%

Weak Continuity of Risk Functionals with Applications to Stochastic Programming

Claus¹,

Krätschmer²,

Schultz³

2017

SIAM J. Optim.

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Measuring and managing risk has become crucial in modern decision making under stochastic uncertainty. In two-stage stochastic programming, mean risk models are essentially defined by a parametric recourse problem and a quantification of risk. From the perspective of qualitative robustness theory, we discuss sufficient conditions for continuity of the resulting objective functions with respect to perturbation of the underlying probability measure. Our approach covers a fairly comprehensive class of both stochasticprogramming related risk measures and relevant recourse models. Not only this unifies previous approaches but also extends known stability results for two-stage stochastic programs to models with mixed-integer quadratic recourse and mixed-integer convex recourse, respectively.

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“…Finally, we define three families of functions that provide an alternative vocabulary for describing "integrality"; namely, Chvátal functions, Gomory functions and Jeroslow functions. These families derive significance here from their ability to articulate value functions of integer and mixed-integer programs (as seen in Blair and Jeroslow [6,7], Blair [8], Blair and Jeroslow [9]).…”

Section: Key Definitionsmentioning

confidence: 99%

Mixed-integer bilevel representability

2019

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We study the representability of sets that admit extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints, and polyhedral reverse convex constraints are all finite unions of polyhedra. Conversely, any finite union of polyhedra can be represented using any one of these three paradigms. We then prove that the feasible region of bilevel problems with integer constraints exclusively in the upper level is a finite union of sets representable by mixed-integer programs and vice versa. Further, we prove that, up to topological closures, we do not get additional modeling power by allowing integer variables in the lower level as well. To establish the last statement, we prove that the family of sets that are finite unions of mixed-integer representable sets forms an algebra of sets (up to topological closures). *

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The value function of a mixed integer program: I

Cited by 101 publications

References 8 publications

Quantitative stability of fully random mixed-integer two-stage stochastic programs

Quantitative stability of fully random mixed-integer two-stage stochastic programs

Weak Continuity of Risk Functionals with Applications to Stochastic Programming

Mixed-integer bilevel representability

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