# Duality for mixed-integer convex minimization

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## Abstract

We extend in two ways the standard Karush–Kuhn–Tucker optimality conditions to problems with a convex objective, convex functional constraints, and the extra requirement that some of the variables must be integral. While the standard Karush–Kuhn–Tucker conditions involve separating hyperplanes, our extension is based on mixed-integer-free polyhedra. Our optimality conditions allow us to define an exact dual of our original mixed-integer convex problem.

## Mathematics Subject Classification

90C11 90C46## Notes

### Acknowledgments

We would like to thank the reviewers, whose careful comments improved significantly the presentation of our results.

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