Recent publications of the group

Further information can be found on the webpages of the individual authors.

Software

L. Warnow

ASMO - A Solver for Multiobjective Optimization

ASMO is a solver for solving multiobjective optimization problems (MOP). It is based on ideas and algorithms from [1] and [2] using scalarization approaches.

This implementation is realized entirely as MATLAB code. It is licenced under the GNU Lesser General Public Licence and free to use.

You can read more about its features and download the files on GitHub.

  1. Gabriele Eichfelder, An Adaptive Scalarization Method in Multi-Objective Optimization, SIAM Journal on Optimization, Volume 19, Issue 4, 1694-1718, 2009.
  2. Gabriele Eichfelder, Adaptive Scalarization Methods in Multiobjective Optimization. Springer, 242 p., ISBN: 978-3-540-79157-7, 2008.
M. De Santis

BB-MOQIP- A Branch-and-Bound method for Multiobjective Convex Quadratic Integer Problems

BB-MOQIP is a solver for mutliobjective convex quadratic integer problems presented in the paper De Santis, M., Eichfelder, G., A Decision Space Algorithm for Multiobjective Convex Quadratic Integer Optimization, OptimizationOnline, 2020.

This implementation is realized as MATLAB code. It is published on GitHub and licenced under the GNU Lesser General Public Licence and free to use.

C. Kästner; E. Quintana

MOBO - Multiobjective Bilevel Optimization Solver

MOBO is a solver for continuous multiobjective bilevel optimization problems based on the algorithm published in G. Eichfelder, Multiobjective bilevel optimization, Mathematical Programming 123(2), 419-449, 2010. It is licenced under the GNU Lesser General Public Licence, can be found on GitHub and is free to use.

J. Niebling

MOMIX - A Solver for Multiobjective Mixed Integer Convex Optimization

MOMIX is a solver for Multiobjective Mixed Integer Convex Optimization. It is a branch-and-bound method based on the use of properly defined lower bounds, constructed by convex relaxations and by linear outer approximations of the image set in an adaptive way. The algorithm guarantee correctness in terms of detecting both the efficient and the nondominated set of multiobjective mixed integer convex problems according to a prescribed precision. This implementation is realized as MATLAB code. It is published on GitHub and licenced under the GNU Lesser General Public Licence and free to use.

The implementation is based on the paper M. De Santis, G. Eichfelder, J. Niebling, S. Rocktäschel, Solving Multiobjective Mixed Integer Convex Optimization Problems, SIAM Journal on Optimization, 30(4), 3122-3145, 2020. See also OptimizationOnline

 

Spherical Branch And Bound Algorithms

This Github project includes an R package containing a branch and bound algorithm for computing Fréchet-p-means on the circle and the 2-sphere. Moreover, it provides a wrapper to easily extend these algorithms also to spheres of higher dimension.  The repository is supplementary to the paper Eichfelder, G., Hotz, T., Wieditz, J., An algorithm for computing Fréchet means on the sphere, 2019, and was implemented by Johannes Wieditz.

     

Publications of the group (from the database of the library)

Information on the publications of the individual group members can be found on the individual pages. For the team see here.
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Eichfelder, Gabriele;
Twenty years of continuous multiobjective optimization in the twenty-first century. - In: EURO journal on computational optimization, ISSN 2192-4414, Bd. 9 (2021), 100014, S. 1-15

The survey highlights some of the research topics which have attracted attention in the last two decades within the area of mathematical optimization of multiple objective functions. We give insights into topics where a huge progress can be seen within the last years. We give short introductions to the specific sub-fields as well as some selected references for further reading. Primarily, the survey covers the progress in the development of algorithms. In particular, we discuss publicly available solvers and approaches for new problem classes such as non-convex and mixed integer problems. Moreover, bilevel optimization problems and the handling of uncertainties by robust approaches and their relation to set optimization are presented. In addition, we discuss why numerical approaches which do not use scalarization techniques are of interest.



https://doi.org/10.1016/j.ejco.2021.100014
Eichfelder, Gabriele; Groetzner, Patrick;
A note on completely positive relaxations of quadratic problems in a multiobjective framework. - In: Journal of global optimization, ISSN 1573-2916, (2021), insges. 12 S.

In a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.



https://doi.org/10.1007/s10898-021-01091-2
Eichfelder, Gabriele; Warnow, Leo;
An approximation algorithm for multi-objective optimization problems using a box-coverage. - In: Journal of global optimization, ISSN 1573-2916, (2021), insges. 29 S.

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin.



https://doi.org/10.1007/s10898-021-01109-9
Eichfelder, Gabriele; Rocktäschel, Stefan;
Solving set-valued optimization problems using a multiobjective approach. - In: Optimization, ISSN 1029-4945, Bd. 0 (2021), 0, S. 1-32

Set-valued optimization using the set approach is a research topic of high interest due to its practical relevance and numerous interdependencies to other fields of optimization. However, it is a very difficult task to solve these optimization problems even for specific cases. In this paper, we study set-valued optimization problems and develop a multiobjective optimization problem that is strongly related to it. We prove that the set of weakly minimal solutions of this subproblem is closely related to the set of weakly minimal elements of the set-valued optimization problem and that these sets can get arbitrarily close in a certain sense. Subsequently, we introduce a concept of approximations of the solution set of the set-valued optimization problem. We define a quality measure in the image space that can be used to compare finite approximations of this kind and outline a procedure to enhance a given approximation. We conclude the paper with some numerical examples.



https://doi.org/10.1080/02331934.2021.1988596
Gerlach, Tobias; Rocktäschel, Stefan;
On convexity and quasiconvexity of extremal value functions in set optimization. - In: Applied set-valued analysis and optimization, ISSN 2562-7783, Bd. 3 (2021), 3, S. 293-308

We study different classes of convex and quasiconvex set-valued maps defined by means of the l-less relation and the u-less relation. The aim of this paper is to formulate necessary and especially sufficient conditions for the convexity/quasiconvexity of extremal value functions.



https://doi.org/10.23952/asvao.3.2021.3.04
Hoff, Daniel; Wendland, Holger;
A meshfree method for a PDE-constrained optimization problem. - In: SIAM journal on numerical analysis, ISSN 1095-7170, Bd. 59 (2021), 4, S. 1896-1917

We describe a new approximation method for solving a PDE-constrained optimization problem numerically. Our method is based on the adjoint formulation of the optimization problem, leading to a system of weakly coupled, elliptic PDEs. These equations are then solved using kernel-based collocation. We derive an error analysis and give numerical examples.



https://doi.org/10.1137/20M1363510
Bouza, Gemayqzel; Quintana, Ernest; Tammer, Christiane;
A steepest descent method for set optimization problems with set-valued mappings of finite cardinality. - In: Journal of optimization theory and applications, ISSN 1573-2878, Bd. 190 (2021), 3, S. 711-743

In this paper, we study a first-order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems and introduce two concepts of critical points. Furthermore, we propose a descent method and provide a convergence result to points satisfying the optimality conditions previously derived. Some numerical examples illustrating the performance of the method are also discussed. This paper is a modified and polished version of Chapter 5 in the dissertation by Quintana (On set optimization with set relations: a scalarization approach to optimality conditions and algorithms, Martin-Luther-Universität Halle-Wittenberg, 2020).



https://doi.org/10.1007/s10957-021-01887-y
De Santis, Marianna; Eichfelder, Gabriele;
A decision space algorithm for multiobjective convex quadratic integer optimization. - In: Computers & operations research, ISSN 0305-0548, Bd. 134 (2021), 105396

We present a branch-and-bound algorithm for minimizing multiple convex quadratic objective functions over integer variables. Our method looks for efficient points by fixing subsets of variables to integer values and by using lower bounds in the form of hyperplanes in the image space derived from the continuous relaxations of the restricted objective functions. We show that the algorithm stops after finitely many fixings of variables with detecting both the full efficient and the nondominated set of multiobjective strictly convex quadratic integer problems. A major advantage of the approach is that the expensive calculations are done in a preprocessing phase so that the nodes in the branch-and-bound tree can be enumerated fast. We show numerical experiments on biobjective instances and on instances with three and four objectives.



https://doi.org/10.1016/j.cor.2021.105396
Eichfelder, Gabriele; Kirst, Peter; Meng, Laura; Stein, Oliver;
A general branch-and-bound framework for continuous global multiobjective optimization. - In: Journal of global optimization, ISSN 1573-2916, Bd. 80 (2021), 1, S. 195-227

Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.



https://doi.org/10.1007/s10898-020-00984-y
Eichfelder, Gabriele; Jahn, Johannes;
Optimality conditions in discrete-continuous nonlinear optimization. - In: Minimax theory and its applications, ISSN 2199-1413, Bd. 6 (2021), 1, S. 127-144