Gabriele Eichfelder: Multiobjective Mixed Integer Convex Optimization

Abstract
Multiobjective mixed integer convex optimization refers to mathematical programming problems where more than one convex objective function needs to be optimized simultaneously and some of the variables are constrained to take integer values. In this talk, we give a short introduction to the basic concepts of multiobjective optimization. We give insights why the famous approach of scalarization might not be an appropriate method to solve these problems. Instead, we present two methods to solve the problems directly. The first is a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, but we built linear outer approximations of the image set in an adaptive way. The second method is purely based on the criterion space. It uses ingredients from the well-known outer approximation algorithm from single-objective mixed-integer optimization and combines them with strategies to generate enclosures of nondominated sets by iteratively improving approximations. For both algorithms, we are able to guarantee correctness in terms of detecting the nondominated set of multiobjective mixed integer convex problems according to a prescribed precision.