The problem of minimizing a convex function over a convex set with the extra requirement that some variable must be integral, in spite of its intrinsic difficulty, enjoys enough structural properties to deserve the development of dedicated solvers. However, most known methods cannot offer any quality guarantee of the approximate solution they provide after a prescribed time.
In the first part of the talk, I will present two different methodologies to address this concern, both inspired from first-order methods for purely convex optimization problems. The results we obtain from these techniques hint at a general conjecture on the complexity of solving Mixed-Integer Convex problems approximately. In a second part of the talk, I will show a new approach for studying duality of mixed-integer convex problems, which also hint at our general conjecture.
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