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Monograph

 

book     

Vector Optimization with Infimum and Supremum. Springer, 20111st Edition., 2011, X, 206 p. 26 illus. Hardcover, ISBN 978-3-642-18350-8 The theory of Vector Optimization is developed by a systematic usage of infimum and supremum. In order to get existence and appropriate properties of the infimum, the image space of the vector optimization problem is embedded into a larger space, which is a subset of the power set, in fact, the space of self-infimal sets. Solution concepts, existence and duality results, and algorithms for the linear case are established based on this idea. The main advantage of thisapproach is the high degree of analogy to corresponding results of Scalar Optimization. This book contains a self-contained exposition of Linear Vector Optimization with new approaches to solution concepts and duality theory. Practically relevant algorithms to solve linear problems are derived.

 

An implementation of the algorithms provided in the book can be found under Bensolve.   


Theses

Habilitation thesis: Vector Optimization with Infimum and Supremum (an extended version is published by Springer, see above)

Ph.D. thesis: Optimization with set relations

Master's thesis: Minimal point theorems in uniform spaces and related assertions

 

Book contribution

Duality in Vector Optimization with Infimum and Supremum.

Chapter 3 in:  Ansari, Q.H.; Yao, J.-C. (Eds.): Recent Developments in Vector Optimization. Springer, 2011

1st Edition., 2011, XXIV, 547 p.

 

Peer-reviewed articles

 

Preprints of all recent articles are available at 

http://arxiv.org/a/lohne_a_1.

 

2024

  • Polyhedral approximation of spectrahedral shadows via homogenization (with Daniel Dörfler)
  • Convex sets approximable as the sum of a compact set and a cone (with Daniel Dörfler)
    • to appear in: Journal of Nonlinear and Variational Analysis, 2024
    • arXiv preprint (January 18, 2024)
  • A polyhedral approximation algorithm for recession cones of spectrahedral shadows (with Daniel Dörfler)
    • to appear in: Journal of Nonlinear and Variational Analysis, 2024
    • arXiv preprint (June 30, 2022)

2023

  • A solution method for arbitrary polyhedral convex set optimization problems
  • The natural ordering cone of a polyhedral convex set-valued objective mapping
  • Existence of solutions for polyhedral convex set optimization problems
  • Finite representation of quantile sets for multivariate data via vector linear programming (with Benjamin Weißing)
  • Approximate Vertex Enumeration
  • On unbounded polyhedral convex set optimization problems (with Niklas Hey)
    • to appear in: Minimax Theory and its Applications
    • arXiv preprint (November 03, 2021)

2022

  • A Benson-Type Algorithm for Bounded Convex Vector Optimization Problems with Vertex Selection (with Daniel Dörfler, Christopher Schneider and Benjamin Weißing)

2021

  • On the approximation error for approximating convex bodies using multiobjective optimization (with Fangyuan Zhao and Lizhen Shao)

2020

  • Solving polyhedral d.c. optimization problems via concave minimization (with Simeon vom Dahl)

2019

  • Solving bilevel problems with polyhedral constraint set (with Daniel Dörfler, Alexandra Rittmann and Benjamin Weißing)
    • published in: Journal of Applied and Numerical Optimization 1 No.3 (2019), 243-251
    • Link to the article
  • Computing Convex Hulls of Trajectories (with Daniel Ciripoi, Nidhi Kaihnsa and Bernd Sturmfels)
  • Using Benson's Algorithm for Regularization Parameter Tracking (with Joachim Giesen, Sören Laue, Christopher Schneider)
    • published in: Proceedings of the 33d AAAI Conference on Artificial Intelligence (AAAI), (2019)
    • Link to the article
  • Calculus of convex polyhedra and polyhedral convex functions by utilizing a multiple objective linear programming solver (with Daniel Ciripoi and Benjamin Weißing)

2018

  • On parallelizing Benson's algorithm: limits and opportunities (with Martin Bücker, Benjamin Weißing and Gerhard Zumbusch)
    • published in: Computational science and its applications—ICCSA 2018. Part II, 653–668, Lecture Notes in Comput. Sci., 10961, Springer, Cham, 2018.
    • Link to the article
  • Geometric duality and parametric duality for multiple objective linear programs are equivalent (with Daniel Dörfler)
    • published in: Journal of Nonlinear and Convex Analysis 19 No.7 (2018), 1181-1188
    • arXiv preprint (March 15, 2018)
  • A vector linear programming approach for certain global optimization problems (with Daniel Ciripoi and Benjamin Weißing)
  • A set optimization approach to zero-sum matrix games with multi-dimensional payoffs (with Andreas Hamel)

2017

  • Solving DC programs with polyhedral component utilizing a multiple objective linear programming solver (with Andrea Wagner)
  • The vector linear program solver Bensolve -- notes on theoretical background (with Benjamin Weißing)
    • published in: European Journal of Operational Research 260 No. 3 (2017), 807-813, DOI: 10.1016/j.ejor.2016.02.039
    • Link to the article
    • arXiv preprint (October 16, 2015)

2016

  • Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming (with Benjamin Weißing)
    • published in: Mathematical Methods of Operations Research  84 No.2 (2016), 411-426, DOI: 10.1007/s00186-016-0554-0
    • Link to the article
    • arXiv preprint (July 1, 2015)

2015

  • Set optimization - A rather short introduction (with Andreas H. Hamel, Frank Heyde, Birgit Rudloff and Carola Schrage)
    • published in: A. H. Hamel, F. Heyde, A. Löhne, B. Rudloff and C. Schrage (eds.), Set Optimization and Applications in Finance - The State of the Art. From Set-Relations to Set-Valued Risk Measures. Springer Proceedings in Mathematics & Statistics, Vol. 151, Springer, 2015
    • Link to article
    • arXiv preprint (April 23, 2014)
  • On the dual of the solvency cone (with Birgit Rudloff)

2014

  • Projection of polyhedral cones and linear vector optimization
  • Primal and Dual Approximation Algorithms for Convex Vector Optimization Problems (with Birgit Rudloff and Firdevs Ulus)
  • An algorithm for calculating the set of superhedging portfolios and strategies in markets with transaction costs (with Birgit Rudloff)
    • published in: International Journal of Theoretical and Applied Finance 17 No. 2 (2014), [33 pages]
    • arXiv preprint (July 28, 2011)
  • Lagrange duality in set optimization (with Andreas Hamel)
  • Benson type algorithms for linear vector optimization and applications (with Andreas H. Hamel, Birgit Rudloff)

2013

  • Lagrange duality, stability and subdifferentials in vector optimization (with Elvira Hernandez, Luis Rodriguez-Marin, Christiane Tammer)
  • An Algorithm to Solve Polyhedral Convex Set Optimization Problems (with Carola Schrage)

2012

  • A dual variant of Benson's outer approximation algorithm (with Matthias Ehrgott und Lizhen Shao)
    • published in: Journal of Global Optimization 52 No. 4 (2012), 757-778
    • Link to article
    • Report of the Institute of Mathematics No. 12, 2007: download pdf

2011

  • Solution concepts in vector optimization. A fresh look at an old story (with Frank Heyde)
    • published in Optimization 60 No. 12 (2011), 1421-1440
    • Link to article
    • Preprint: Report of the Institute of Mathematics No. 19, 2008: download pdf

2010

  • On totally Fenchel unstable functions in finite dimensional spaces (with Radu Ian Bot)
    • published in: Math. Programming 123 No. 1 (2010), 25-30
    • Link to article
    • Report of the Institute of Mathematics No. 01, 2008: download pdf

2009

  • Set-valued duality theory for multiple objective linear programs and application to mathematical finance (with Frank Heyde and Christiane Tammer)
    • published in: Mathematical Methods of Operations Research 69 No. 1 (2009), 159-179
    • Link to article
    • Report of the Institute of Mathematics No. 04>, 2007: download pdf

2008

  • A chartacterization of maximal monotone operators
    • published in: Journal of Set-valued Analysis 16 No. 5/6 (2008), 663-700
    • Report of the Institute of Mathematics No. 06, 2007: download pdf
  • On semicontinuity of convex-valued multifunctions and Cesari's property (Q)
  • Geometric duality in multiple objective linear programming (with Frank Heyde)
    • published in: SIAM Journal of Optimization 19 No. 2 (2008), 836-845
    • Link to article
    • Report of the Institute of Mathematics No. 15, 2006: download pdf

2007

  • A new approach to duality in vector optimization (with Christiane Tammer)
    • published in: Optimization 56 No. 1/2 (2007), 221-239
    • Link to article
    • Report of the Institute of Optimization and Stochastics No. 08, 2005: download pdf

2006

  • On convergence of closed convex sets (with Constantin Zǎlinescu)
    • published in: Journal of Mathematical Analysis and Applications 319 No. 2 (2006), 617-634
    • Report of the Institute of Optimization and Stochastics No. 02, 2005: download pdf
  • On convex functions with values in conlinear spaces
    • published in: Journal of Nonlinear and Convex Analysis 7 No. 1 (2006), 115-122
    • Report of the Institute of Optimization and Stochastics No. 07, 2003: download pdf
  • Minimal element theorems and Ekeland's principle with set relations (with Andreas Hamel) 
    • published in: Journal of Nonlinear and Convex Analysis 7 No. 1 (2006), 19-37
    • Report of the Institute of Optimization and Stochastics No. 11, 2002: download pdf
      (former title: Minimal set theorems)

2005

  • Optimization with set relations: Conjugate duality
    • published in: Optimization 54 No. 3 (2005), 265-282 
    • Link to article
    • Report of the Institute of Optimization and Stochastics No. 17, 2004: download pdf

2004

  • Closing the duality gap in linear vector optimization (with A. Hamel, F. Heyde, C. Tammer, K. Winkler)

2003

  • Minimal point theorems in uniform spaces (with Andreas Hamel)
    • published in: Agarval R. P., O'Regan, D. (eds.),
      Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday,
      Kluwer Academic Publishers, Dordrecht, 2003 
    • Report of the Institute of Optimization and Stochastics No. 09, 2002: download pdf

Further publications

  • The attainment of the solution of the dual program in vertices for vectorial linear programs (with Frank Heyde and Christiane Tammer)
    • published in: Barichard, Vincent (ed.) et al., Multiobjective programming and goal programming. Theoretical results and practical applications. Selected papers based on the presentations at the international conference on multiobjective programming and goal programming (MOP/GP 2006), Tours, France, June 12-14, 2006. Berlin: Springer. Lecture Notes in Economics and Mathematical Systems 618, 13-24 (2009) 
    • Report of the Institute of Mathematics No. 16, 2006: download pdf
  • On conjugate duality in optimization with set relations
    • published in: Geldermann, J.; Treitz, M. (eds.):
      Entscheidungstheorie und -praxis in industrieller Produktion und Umweltforschung,
      Shaker, Aachen, 2004
    • Overview of: Optimization with set relations: Conjugate duality
  • Necessary and sufficient conditions for approximate saddle points (with W.W. Breckner, A. Hamel, C.Tammer)
    • published in: Geldermann, J.; Treitz, M. (eds.):
      Entscheidungstheorie und -praxis in industrieller Produktion und Umweltforschung,
      Shaker, Aachen, 2004