Continuous curve-based approximations in convex multiobjective optimization

• Applicant: Professorin Dr. Gabriele Eichfelder
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 576381369

In this project, we want to examine the possibilities of approximating the set of nondominated points of a convex nonlinear multiobjective optimization problem by nonlinear curves. More precisely, we want to concentrate on biobjective convex nonlinear optimization problems, i.e. problems with two objective functions. We plan to approximate their nondominated set by nonlinear plane curves. There are three ways to describe plane curves: (i) explicitly, as the graph of a univariate real-valued function, (ii) implicitly, as the level set of a bivariate real-valued function, and (iii) parametrically, as the image set of a univariate vector-valued function. We want to investigate the first two descriptions of plane curves. In the case of (i) we want to use results for the approximation of convex functions. We plan to use the well-established sandwich methods for convex functions as a starting point and to smooth the polygonal functions which are constructed there, since smooth functions are more practical for more complex tasks. In case of (ii) we want to find parametric functions of which the level sets are, or include, convex plane curves. A promising class of functions are the weighted p-norms since their unit circles are closed convex curves. In both cases, we plan to develop an algorithm where in each step an approximation is constructed and the error between them and the actual set of nondominated points is reduced in every iteration. In addition, we want to use these approximations to solve more complex optimization problems. As a first application of nonlinear approximations of the set of nondominated points we want to approximate the set of nondominated points of convex mixed-integer biobjective optimization problems. To do so, the mixed-integer problem is partitioned into several continuous subproblems, so-called patch problems, by fixing integer assignments. Then, approximations of the nondominated set of some of the patch problems are constructed and these approximation sets are united. This union gives the approximation of the set of nondominated points of the convex mixed-integer biobjective optimization problem. A second application of the proposed approximations is the optimization over the set of efficient points, i.e. optimization problems for which the feasible set is the set of efficient points of a multiobjective, in our case biobjective, optimization problem. We plan to develop an approximation based Branch-and-Bound algorithm where the approximations are used to formulate relaxations of the original problem.

• DFG Programme: Research Grants