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Univ.-Prof. Dr. rer. nat. habil. Gabriele Eichfelder


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Variable Ordering Structures in Vector Optimization. Springer, May 2014, 190 p., in the series Vector Optimization

This book provides an introduction to vector optimization with variable ordering structures, i.e., to optimization problems with a vector-valued objective function where the elements in the objective space are compared based on a variable ordering structure: instead of a partial ordering defined by a convex cone, we see a whole family of convex cones, one attached to each element of the objective space.The book starts by presenting several applications that have recently sparked new interest in these optimization problems, and goes on to discuss fundamentals and important results on a wide range of topics. The theory developed includes various optimality notions, linear and nonlinear scalarization functionals, optimality conditions of Fermat and Lagrange type, existence and duality results. The book closes with a collection of numerical approaches for solving these problems in practice.



Adaptive Scalarization Methods in Multiobjective Optimization. Springer, 2008, 242 p., ISBN: 978-3-540-79157-7 in the series Vector Optimization

This book presents new adaptive solution methods for multiobjective optimization problems based on parameter dependent scalarizations. With the help of sensitivity results an adaptive parameter control is developed so that high-quality approximations of the efficient set are generated. These examinations are based on a general scalarization approach for arbitrary partial orderings defined by a closed pointed convex cone in the objective space. The application of the results to many other well-known scalarization
methods is also presented. Background material of multiobjective optimization and scalarization approaches is concisely summarized at the beginning. The effectiveness of these new methods is demonstrated by test problems and a recent problem in intensity-modulated radiotherapy. The book concludes with a further application: a procedure for solving multiobjective bilevel optimization problems.





Publikationsliste (Stand: 04.01.2021) pdf

for Software see here

Numerical algorithms for multiobjective optimization problems

  • A general branch-and-bound framework for continuous global multiobjective optimization (with Peter Kirst, Laura Meng, Oliver Stein), Journal of Global Optimization, DOI: 10.1007/s10898-020-00984-y, 2021.
  • Proximity measures based on KKT points for constrained multi-objective optimization problems,
    (with Leo Warnow), Journal of Global Optimization, 2020, DOI 10.1007/s10898-020-00971-3.
  • Solving Multiobjective Mixed Integer Convex Optimization Problems (with Marianna De Santis, Julia Niebling and Stefan Rocktäschel),
    SIAM Journal on Optimization, 30(4), 3122-3145, see also OptimizationOnline , 2020.
  • Methods for multiobjective bilevel optimization, Chapter 15 in: "Bilevel optimization: advances and next challenges", S. Dempe and A. Zemkoho (Eds.), p. 423-449, Springer, 2020, see also  OptimizationOnline.
  • An algorithmic approach to multiobjective optimization with decision uncertainty (with Julia Niebling and Stefan Rocktäschel),
    Journal of Global Optimization,  77,  325, 2020.  here, and as full-text view-only version here. See also OptimizationOnline
  • Representation of the Pareto front for heterogeneous multi-objective optimization (with Jana Thomann), 
    Journal of Applied and Numerical Optimization, 1(3), 293-323, 2019.
  • Numerical Results for the Multi-Objective Trust Region Algorithm MHT (with Jana Thomann),
    Data in Briefs, 25 , 18 pages, 2019.
  • A Trust-Region Algorithm for Heterogeneous Multiobjective Optimization (with Jana Thomann),
    SIAM Journal on Optimization, 29(2), 1017 - 1047, 2019 
  • A Branch-and-Bound based Algorithm for Nonconvex Multiobjective Optimization (with Julia Niebling),
    SIAM Journal on Optimization, 29(1), 794 - 821, 2019. 
  • Using a B&B algorithm from multiobjective optimization to solve constrained optimization problems (with Kathrin Klamroth and Julia Niebling),
    AIP Conference Proceedings 2070, 020028 (2019) 2019
  • A Branch-and-Bound Algorithm for Biobjective Problems (with Julia Niebling),
    Proceedings of the XIII Global Optimization Workshop GOW'16, 57-60, 2016.
  • Numerical procedures in multiobjective optimization with variable ordering structures,
    Journal of Optimization Theory and Applications, Volume 162(2), 489-514, 2014. [pdf]
  • Multiobjective bilevel optimization,   
    Mathematical Programming Ser. A, Volume 123, Issue 2, 419-449, 2010.
  • An Adaptive Scalarization Method in Multi-Objective Optimization,   
    SIAM Journal on Optimization,Volume 19, Issue 4, 1694-1718, 2009. [pdf] (Copyright SIAM)
  • Scalarizations For Adaptively Solving Multi-Objective Optimization Problems,
    Computational Optimization and Applications, Vol. 44, No. 2, 249-273, 2009.  [pdf]
  • A Constraint Method in Nonlinear Multi-Objective Optimization, in:   
    Multiobjective Programming and Goal Programming, Theoretical Results and Practical Applications, Lecture Notes in Economics and Mathematical Systems Vol. 618, Barichard, V. et al. (Eds.), 3-12, Springer, Heidelberg, 2009 [Abstract].
  • ε-Constraint Method with Adaptive Parameter Control and an Application to Intensity-Modulated Radiotherapy,
    In: Multicriteria Decision Making and Fuzzy Systems, Theory, Methods and Applications, eds.: K.-H. Küfer, H. Rommelfanger, C. Tammer and K. Winkler, 25 - 42, Shaker, Aachen, 2006.

Vector optimization

  • Computationally Expensive Functions and Large Scale Test Instances (with Dimo Brockhoff ,Carlos M. Fonseca, Susan R. Hunter, Enrico Rigoni, Michael Stiglmayr) (Seminarbericht) In: Scalability in Multiobjective Optimization, Dagstuhl Seminar 20031, 79-88,2020.
  • Personalization of multicriteria decision support systems (with Matthias Ehrgott, Karl-Heinz Küfer, Christoph Lofi, Kaisa Miettinen,
    Luís Paquete, Stefan Ruzika, Serpil Sayın, Ralph E. Steuer, Theodor J. Stewart, Michael Stiglmayr, and Daniel Vanderpooten) (Seminarbericht) In: Personalized Multiobjective Optimization: An Analytics Perspect, Dagstuhl Seminar 18031, 55-70, 2018.
  • Ordering Structures and Their Applications (with M. Pilecka), in: Applications of Nonlinear Analysis, Rassias, T.M. (Edt.), Springer,  2018, 256-304.
  • Decision uncertainty in multiobjective optimization (with C. Krüger, A. Schöbel),
    Journal of Global Optimization, 69(2), 485-510, 2017.  Read online
  • Ekeland's variational principle for vector optimization with variable ordering structure (with T.Q. Bao, B. Soleimani and Chr. Tammer), Journal of Convex Analysis, Vol. 24(2), 393-415, 2017.
  • Characterization of properly optimal elements with variable ordering structures (with Tobias Gerlach), 
    Optimization, Vol. 65(3), 571-588, 2016. See also
  • On the effects of combining objectives in multi-objective optimization (with Stephan Dempe and Jörg Fliege),
    Mathematical Methods of Operations Research, Volume 82, Issue 1 (2015), 1-18
  • Heterogeneous Functions (with X. Gandibleux, M.J. Geiger, J. Jahn, A. Jaszkiewicz, J. Knowles, P.K. Shukla, H. Trautmann und S. Wessing) (Seminarbericht) In: Understanding Complexity in Multiobjective Optimization, Dagstuhl Seminar 15031, 121-129, 2015.
  • Properly optimal elements in vector optimization with variable ordering structures (with Refail Kasimbeyli),
    Journal of Global Optimization, Volume 60(4), 689-712, 2014. [pdf]
  • Optimality conditions for vector optimization problems with variable ordering structures   
    (with Truong Xuan Duc Ha),
    Optimization, Vol. 62(5), 597-627, 2013. [pdf]
  • Variable ordering structures in vector optimization, Chapter 4 in:   
    Recent Developments in Vector Optimization, Ansari, Q.H.,Yao, J.-C. (Eds.), p. 95-126, Springer, 2012.
  • Vector optimization problems and their solution concepts (with Johannes Jahn), Chapter 1 in:   
    Recent Developments in Vector Optimization, Ansari, Q.H.,Yao, J.-C. (Eds.), p. 1-27, Springer, 2012.
  • Cone-valued maps in optimization,
    Applicable Analysis, Special Issue on Variational Inequalities, Optimization and Related Topics, Vol. 91 (10), 1831-1846, 2012.
  • Optimal elements in vector optimization with a variable ordering structure,
    Journal of Optimization Theory and Applications, Vol. 151 (2), 217-240, 2011. [pdf]


  • Optimality Conditions for Set Optimization using a Directional Derivative based on Generalized Steiner Sets (with Robert Baier, Tobias Gerlach), 40p., Optimization. See also OptimizationOnline, DOI 10.1080/02331934.2020.18126052, 2020.
  • On classes of set optimization problems which are reducible to vector optimization problems and its impact on numerical test instances (with Tobias Gerlach), Chapter 10 in "Variational Analysis and Set Optimization", Khan, A., Köbis, E., Tammer Chr. (Eds.), CRC Press (Taylor & Francis Group), p. 265-290, 2019. See also [pdf],   OptimizationOnline
  • Set Approach for Set Optimization with Variable Ordering Structures Part II: Scalarization Approaches (with Maria Pilecka),
    Journal of Optimization Theory and Applications, Vol. 171(3), 947-963, 2016. Preprint. Online version
  • Set Approach for Set Optimization with Variable Ordering Structures Part I: Set Relations and Relationship to Vector Approach (with Maria Pilecka),
    Journal of Optimization Theory and Applications, Vol. 171(3), 931-946, 2016. Preprint. Online version
  • Vector and Set Optimization (with Johannes Jahn)
    Multiple Criteria Decision Analysis State of the Art Surveys, Greco, S. et al. (Eds.), 695 - 737, Springer, Heidelberg, 2016

Global scalar-valued optimization, copositive programming and MINLPs

  • Nonconvex Constrained Optimization by a Filtering Branch and Bound
    (with Kathrin Klamroth and Julia Niebling),  Journal of Global Optimization, Doi 10.1007/s10898-020-00956-2, 2020. [pdf]
  • Optimality conditions in discrete-continuous nonlinear optimization (with Johannes Jahn), OptimizationOnline, Minimax Theory and its Applications, 2020.
  • An algorithm for computing Fréchet means on the sphere (with Thomas Hotz and Johannes Wieditz),
    Optimization Letters, 13, 15231533 DOI: 10.1007/s11590-019-01415-y, see also  OptimizationOnline, 2019.
  • Global Optimization Techniques for Copositivity Testing (with I.M. Bomze, C. Brás und J. Júdice)  (Extended Abstract).
    In: Copositivity and Complete Positivity, Mathematisches Forschungsinstitut Oberwolfach, Report No. 52/2017 , Doi: 10.4171/OWR/2017/52, 14–16, 2017.
  • A modification of the alphaBB method for box-constrained optimization and an application to inverse kinematics (with Tobias Gerlach and Susanne Sumi)
    EURO Journal on Computational Optimization, Vol. 4(1), 93-121, 2016.  [pdf]
  • Copositivity tests based on the Linear Complementarity Problem
    (with Carmo Brás and Joaquim Júdice) , Doi 10.1007/s10589-015-9772-2,
    Computational Optimization and Applications, 62(2), Pages 461-493, 2016 [pdf]
  • Copositive programming and copositivity tests (with C. Brás und J. Júdice) (Extended Abstract).
    In: Mixed-integer Nonlinear Optimization: A Hatchery for Modern Mathematics, Mathematisches Forschungsinstitut Oberwolfach, Report No. 46/2015, Doi:
    10.4171/OWR/2015/4, 2723–2725, 2015.
  • Copositivity detection by difference-of-convex decomposition and ω-subdivision (with Immanuel Bomze),
    Mathematical Programming Ser. A, Vol. 138, 1-2, p. 365-400, 2013.
  • Erratum to: "On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets''
    (with Peter Dickinson, Janez Povh) ,  
    Optimization Letters, Vol. 7(6), p.1387-1397, 2013. (OptimizationOnline)
  • On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets (with Janez Povh),
    Optimization Letters, Vol. 7(6), p.1373-1386, 2013.
    [pdf] Please see  erratum above.
  • On the set-semidefinite representation of nonconvex quadratic programs with cone constraints (with Janez Povh),
    Croatian Operational Research Review Vol. 1, 26-39, 2011.
  • Foundations of Set-Semidefinite Optimization (with Johannes Jahn), Chapter 18 in:
    Nonlinear Analysis and Variational Problems, Pardalos, P., Rassias, Th.M. and Khan, A.A. (Eds.), 259-284, Springer, 2009.
  • Set-Semidefinite Optimization (with Johannes Jahn),   
    Journal of Convex Analysis, Vol. 15, Number 4, 767-801, 2008.


(for additional applications, being part of other papers published in mathematical journals, see also above)

  • Expensive multi-objective optimization of electromagnetic mixing in a liquid metal (with Sebastian Prinz, Jana Thomann, Thomas Boeck, Jörg Schumacher), Optimization and Engineering, see also OptimizationOnline , DOI: 10.1007/s11081-020-09561-4, 2020.
  • Coordinate transformation and its uncertainty under consideration of a non orthogonal coordinate base (with Florian Fern, Roland Füßl, Eberhard Manske and Michael Kühnel),  Measurement Science and Technology, DOI 10.1088/1361-6501/aba3f5, pdf, 2020.
  • Maximum electromagnetic drag configurations for a translating conducting cylinder with distant magnetic dipoles (with T. Boeck, D.Terzijska),
    Journal of Engineering Mathematics, 108(1),  123-141, 2018.
  • Vector Optimization in Medical Engineering, Chapter in:  Mathematics Without Boundaries, Pardalos, P.M. and Rassias, T.M. (Eds.), ISBN 978-1-4939-1123-3, Springer, pp. 181-215, 2014.
  • Multi-objective optimization in the Lorentz force velocimetry framework (with Dzulia Terzijska and Margherita Porcelli),
    in: Book of digests & program / OIPE, International Workshop on Optimization and Inverse Problems in Electromagnetism 13, Delft pp. 81-82, 2014.
  • Local specific absorption rate control for parallel transmission by virtual observation points (with Matthias Gebhardt),
    Magnetic Resonance in Medicine, Vol. 66(5), 1468–1476, 2011.

Didactical publications

  • Optimierung mit mehreren konkurrierenden Zielen (with Martin Gugat),Optimierung mit mehreren konkurrierenden Zielen (with Martin Gugat), WISU-das Wirtschaftsstudium 4, 571-576, 588, 2010. WISU-das Wirtschaftsstudium 4, 571-576, 588, 2010.



  • Method and Device for Determining a Magnetic Resonance System Control Sequence (with Dirk Diehl,  Matthias Gebhardt, Jochen Gierling, Johannes Jahn and Dieter Ritter), US Patent  US2012286778  (A1), angemeldet März 2012, veröffentlicht Nov. 2012.
  • Method for Determining Sensitivity Matrices for Hotspots (with Matthias Gebhardt), Chinese Patent Application, CN201110055568.X Filed 8.3.2011,  Patentklassen (IPC) G01R 33/561; G01R 33/341; A61B 5/055 and United States Patent Application, US201113045832 20110311, Filed 11.3.2011, veröffentlicht 15.09.2011.
  • Verfahren und Einrichtung zur Ermittlung einer Magnetresonanzsystem-Ansteuersequenz (with Dirk Diehl, Jochen Gierling, Matthias Gebhardt , Dieter Ritter and Johannes Jahn), Patent application WO2012119673 bzw.  DE102011005174 (A1) in March 2011.
  • Verfahren zur Bestimmung von Sensitivitätsmatrizen für kritische Hotspots (with Matthias Gebhardt), Deutsches Patent DE 10 2010 011 588.6, Priorität 12.03.2010 veröffentlicht 15.09.2011, erteilt 05.04.2012, Patentklassen (IPC) G01R 33/561; G01R 33/36.

Preprints and Submissions

Preprints and Submissions

  • An approximation algorithm for multie-objective optimization problems using a box-coverage (with Leo Warnow), OptimizationOnline, 2020.
  • A Decision Space Algorithm for Multiobjective Convex Quadratic Integer Optimization (with Marianna de Santis), OptimizationOnline, 2020.
  • Ordering Structures in Vector Optimization and Applications in Medical Engineering, Preprint-Series of the Institute for Mathematics, Technische Universität Ilmenau, Germany, 2013 [pdf] . Content to appear as a book chapter in "Applications of Nonlinear Analysis".
  • Solving Nonlinear Multiobjective Bilevel Optimization Problems with Coupled Upper Level Constraints, Preprint No. 320, Preprint-Series of the Institute of Applied Mathematics, Univ. Erlangen-Nürnberg, Germany, 2007 [Abstract]. Content published in the book "Adaptive Scalarization Methods in Multiobjective Optimization".



  • Variable Ordering Structures in Vector Optimization, Habilitation thesis, Univ. Erlangen-Nürnberg, Germany, 2011.

  • Parametergesteuerte Lösung nichtlinearer multikriterieller Optimierungsprobleme (Parameter Controlled Solving of Nonlinear Multi-Objective Optimization Problems), PhD-Thesis, Univ. Erlangen-Nürnberg, Germany, 2006. [pdf]

  • Tangentielle Epiableitung mengenwertiger Abbildungen (Contingent Epiderivatives of Set Valued Maps), Master Thesis, Univ. Erlangen-Nürnberg, Germany, 2001.