Gabriele Eichfelder An important aspect of optimization algorithms, for instance evolutionary algorithms, are termination criteria that measure the proximity of the found solution to the optimal solution set. A frequently used approach is the numerical verification of necessary optimality conditions such as the Karush-Kuhn-Tucker (KKT) conditions. In this paper, we present a proximity measure which characterizes the … Read more
This paper presents a novel trust-region method for the optimization of multiple expensive functions. We apply this method to a biobjective optimization problem in fluid mechanics, the optimal mixing of particles in a flow in a closed container. The three-dimensional time-dependent flows are driven by Lorentz forces that are generated by an oscillating permanent magnet … Read more
Optimization problems with multiple objectives which are expensive, i.e. where function evaluations are time consuming, are difficult to solve. Finding at least one locally optimal solution is already a difficult task. In case only one of the objective functions is expensive while the others are cheap, for instance analytically given, this can be used in … Read more
This paper is on multiobjective bilevel optimization, i.e. on bilevel optimization problems with multiple objectives on the lower or on the upper level, or even on both levels. We give an overview on the major optimality notions used in multiobjective optimization. We provide characterization results for the set of optimal solutions of multiobjective optimization problems … Read more
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. We present a branch-and-bound method based on the use of properly defined lower bounds. We do not simply rely on convex relaxations, … Read more
In real life applications optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one … Read more
A set of 78 test examples is presented for the trust region method MHT described in J. Thomann, G. Eichfelder, A trust region algorithm for heterogeneous multi-objective optimization, 2018 (available as preprint: http://optimization-online.org/DB_HTML/2018/03/6495.html) . It is designed for multi-objective heterogeneous optimization problems where one of the objective functions is an expensive black-box function, for example … Read more
This paper presents a new trust region method for multiobjective heterogeneous optimization problems. One of the objective functions is an expensive black-box function, for example given by a time-consuming simulation. For this function derivative information cannot be used and the computation of function values involves high computational effort. The other objective functions are given analytically … Read more
A new branch-and-bound based algorithm for smooth nonconvex multiobjective optimization problems with convex constraints is presented. The algorithm computes an $(\varepsilon,\delta)$-approximation of all globally optimal solutions. We introduce the algorithm which uses selection rules, discarding and termination tests. The discarding tests are the most important aspect, as they examine in different ways whether a box … Read more
For most optimisation methods an essential assumption is the vector space structure of the feasible set. This condition is not fulfilled if we consider optimisation problems over the sphere. We present an algorithm for solving a special global problem over the sphere, namely the determination of Frechet means, which are points minimising the mean distance … Read more