Mathematical Programming Computation

, Volume 8, Issue 2, pp 191–214 | Cite as

CBLIB 2014: a benchmark library for conic mixed-integer and continuous optimization

  • Henrik A. Friberg
Full Length Paper


The Conic Benchmark Library is an ongoing community-driven project aiming to challenge commercial and open source solvers on mainstream cone support. In this paper, 121 mixed-integer and continuous second-order cone problem instances have been selected from 11 categories as representative for the instances available online. Since current file formats were found incapable, we embrace the new Conic Benchmark Format as standard for conic optimization. Tools are provided to aid integration of this format with other software packages.


Problem instances Conic programming Mixed integer programming 

Mathematics Subject Classification

90C90 90C25 90C11 



The author owe a special thanks to Erling D. Andersen and Mathias Stolpe who supervised the work on CBLIB, and to Ambros Gleixner and Thorsten Koch for hosting and maintaining the benchmarking library at Zuse Institute Berlin. Another great thank you goes to the anonymous peer reviewers for their valuable feedback, and to the many who has contributed instances or given feedback to drive the benchmark library forward. The author, Henrik A. Friberg, was funded by MOSEK ApS and the Danish Ministry of Higher Education and Science through the Industrial PhD project “Combinatorial Optimization over Second-Order Cones and Industrial Applications”.


  1. 1.
    Alizadeh, F., Goldfarb, G.: Second-order cone programming. Math. Progr. 51, 3–51 (2003)MathSciNetGoogle Scholar
  2. 2.
    Andersen, K.D., Christiansen, E., Overton, M.L.: Computing limit loads by minimizing a sum of norms. SIAM J. Sci. Comput. 19(3), 1046–1062 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anjos, M.F., Burer, S.: On handling free variables in interior-point methods for conic linear optimization. SIAM J. Optim. 18(4), 1310–1325 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ben-tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms, and engineering applications, vol. 2 of MPS-SIAM series on optimization. SIAM. ISBN: 978-0-89871-491-3 (2001)Google Scholar
  5. 5.
    Benjamin, A.T.: Sensible rules for remembering duals–the SOB method. SIAM Rev. 37(1), 85–87 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Berthold, T., Heinz, S., Vigerske, S.: Extending a CIP framework to solve MIQCPs. In: Mixed Integer Nonlinear Programming, volume 154 of the IMA Volumes in Mathematics and its Applications, pp. 427–444. Springer, New York. ISBN: 978-1-4614-1926-6 (2012)Google Scholar
  7. 7.
    Bonami, P., Kilinc, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. Mix. Int. Nonlinear Progr. 154, 1–39 (2012)CrossRefzbMATHGoogle Scholar
  8. 8.
    Borchers, B.: SDPLIB 1.2, a library of semidefinite programming test problems. Optim. Methods Softw. 11(1), 683–690 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press. (2004)
  10. 10.
    Chinneck, J.W.: Feasibility and infeasibility in optimization. In: International Series in Operations Research and Management Science, vol. 118, Springer, US, Boston, MA. ISBN: 978-0-387-74931-0 (2008)Google Scholar
  11. 11.
    Christiansen, E., Andersen, K.D.: Computation of collapse states with von Mises type yield condition. Int. J. Numer. Meth. Eng. 46(8), 1185–1202 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coleman, J.O., Vanderbei, R.J.: Random-process formulation of computationally efficient performance measures for wideband arrays in the far field. In: Proceedings of the 42nd Midwest Symposium on Circuits and Systems, vol. 2, pp. 761–764 (1999)Google Scholar
  13. 13.
    Coleman, J.O., Scholnik, D.P., Brandriss, J.J.: A specification language for the optimal design of exotic FIR filters with second-order cone programs. Conference Record of the Thirty-Sixth Asilomar Conference on Signals, Systems and Computers 1, 341–345 (2002)Google Scholar
  14. 14.
    Conn, A.R., Gould, N.I.M., Toint, P.L.: Testing a class of methods for solving minimization problems with simple bounds on the variables. Math. Comput. 50(182), 399–430 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    de Klerk, E., Sotirov, R.: A new library of structured semidefinite programming instances. Optim. Methods Softw. 24(6), 959–971 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Drewes, S.: Mixed integer second order cone programming. PhD thesis, Department of Mathematics, Technical University Darmstadt. (2009)
  17. 17.
    Elhedhli, S.: Service system design with immobile servers, stochastic demand, and congestion. Manuf Serv Oper Manag 8(1), 92–97 (2006)Google Scholar
  18. 18.
    Fair Isaac Corporation. Xpress-optimizer reference manual, Release 20.00. Technical report XPRESS2013. (2009)
  19. 19.
    Friberg, H.A.: The conic benchmark format: version 1 - technical reference manual. Technical Report E-0047, Department of Wind Energy, Technical University of Denmark. (2014)
  20. 20.
    Gay, D.M.: Electronic mail distribution of linear programming test problems. Math. Progr. Soc. COAL Newslett. 13, 10–12 (1985)Google Scholar
  21. 21.
    Glineur, F.: Conic optimization: an elegant framework for convex optimization. Belg. J. Oper. Res. Stat. Comput. Sci. 41, 5–28 (2001)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Góez, J.C.: Mixed integer second order cone optimization disjunctive conic cuts: theory and experiments. PhD thesis, Lehigh University (2013)Google Scholar
  23. 23.
    Gruber, G., Rendl, F.: Computational experience with ill-posed problems in semidefinite programming. Comput. Optim. Appl. 21(2), 201–212 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Günlük, O., Lee, J., Weismantel, R.: MINLP strengthening for separable convex quadratic transportation-cost UFL. Technical report, IBM Research Report RC24213.$File/rc24213.pdf (2007)
  25. 25.
    Gurobi Optimization, Inc. Gurobi Optimizer Reference Manual - Version 5.6. Technical report. (2013)
  26. 26.
    Härter, V., Jansson, C., Lange, M.: VSDP: A Matlab toolbox for verified semidefinite-quadratic-linear programming. Technical report, Institute for Reliable Computing, Hamburg University of Technology., (2012)
  27. 27.
    IBM Corporation. IBM ILOG CPLEX optimization studio V12.6.0 documentation. Technical report. (2014)
  28. 28.
    Iwane, H., Yanami, H., Anai, H., Yokoyama, K.: An effective implementation of symbolic-numeric cylindrical algebraic decomposition for quantifier elimination. Theor. Comput. Sci. 479, 43–69 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Jansson, C.: Guaranteed accuracy for conic programming problems in vector lattices. Technical report, Institute for Reliable Computing, Technical University Hamburg.arXiv:0707.4366v1 (2007)
  30. 30.
    Kobayashi, K., Kim, S., Kojima, M.: Sparse second order cone programming formulations for convex optimization problems. J. Oper. Res. Soc. Jpn 51(3), 241–264 (2008)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Koch, T.: The final NETLIB-LP results. Oper Res Lett 32(2), 138–142 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Koch, T., Achterberg, T., Andersen, E.D., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H.D., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010. Math Program Comput 3(2), 103–163 (2011)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Markowitz, H.: Portfolio selection. J. Financ. 7(1), 77–91 (1952)Google Scholar
  35. 35.
    Mars, S., Schewe, L.: An SDP-package for SCIP. Technical Report August, TU Darmstadt. (2012)
  36. 36.
    Mittelmann, H.D.: MISOCP and large SOCP benchmark. (2014)
  37. 37.
    MOSEK ApS. The MOSEK C optimizer API manual, Version 7.0. technical report. (2013)
  38. 38.
    Nesterov, Y., Nemirovskii, A.: Interior-point polynomial algorithms in convex programming, vol. 13. SIAM, Philadelphia. ISBN: 978-0-89871-319-0 (1994)Google Scholar
  39. 39.
    Nesterov, Y., Todd, M.J., Ye, Y.: Infeasible-start primal-dual methods and infeasibility detectors for nonlinear programming problems. Math. Progr. 84(2), 227–267 (1999)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Pataki, G., Schmieta, S.H.: The DIMACS library of semidefinite-quadratic-linear programs. Technical report, Computational Optimization Research Center, Columbia University. (2002)
  41. 41.
    Permenter, F., Parrilo, P.A.: Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone, (2014).arXiv:1408.4685
  42. 42.
    Permenter, F., Friberg, H.A., Andersen, E.D.: Solving conic optimization problems via self-dual embedding and facial reduction: a unified approach. (2015)
  43. 43.
    Perregaard, M.: Advances in convex quadratic integer optimization with xpress (2014). In: Presented at INFORMS Annual Meeting 2014Google Scholar
  44. 44.
    Pólik, I.: Conic optimization software. In: Cochran, J.J., Cox Jr, L.A., Keskinocak, P., Kharoufeh, J.P., Smith, J.C. (eds.) Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2011)Google Scholar
  45. 45.
    Pólik, I., Terlaky, T.: New stopping criteria for detecting infeasibility in conic optimization. Optim. Lett. 3(2), 187–198 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Renegar, J.: Incorporating condition measures into the complexity theory of linear programming. SIAM J. Optim. 5(3), 506–524Google Scholar
  47. 47.
    Sagnol, G.: PICOS: a python interface for conic optimization solvers. (2015)
  48. 48.
    Skajaa, A., Andersen, E.D., Ye, Y.: Warmstarting the homogeneous and self-dual interior point method for linear and conic quadratic problems. Math. Progr. Comput. 5(1), 1–25 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Skutella, M.: Convex quadratic and semidefinite programming relaxations in scheduling. J. ACM 48(2), 206–242 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Sturm, J.F.: Using sedumi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11–12, 625–653 (1999)Google Scholar
  51. 51.
    The MathWorks, Inc. MATLAB Primer, R2013b. Technical report. (2013)
  52. 52.
    Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Progr. 95(2), 189–217 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Vielma, J.P., Ahmed, S., Nemhauser, G.L.: A lifted linear programming branch-and-bound algorithm for mixed integer conic quadratic programs. INFORMS J. Comput. 20, 438–450 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Waki, H., Muramatsu, M.: Facial reduction algorithms for conic optimization problems. J. Optim. Theor Appl. 158(1), 188–215 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Waki, H., Nakata, M., Muramatsu, M.: Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization. Comput Optim Appl 53(3), 823–844 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Wiegele, A.: Biq Mac library—a collection of max-cut and quadratic 0–1 programming instances of medium size Quadratic 0–1 Programming problems. Technical report, Alpen-Adria-Universität Klagenfurt. (2007)
  57. 57.
    Yamashita, M., Fujisawa, K., Nakata, K., Nakata, M., Fukuda, M., Kobayashi, K., Goto, K.: A high-performance software package for semidefinite programs: SDPA 7. Technical report. (2010)
  58. 58.
    Ziegler, H.: Solving certain singly constrained convex optimization problems in production planning. Oper Res. Lett. 1(6), 246–252 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and The Mathematical Programming Society 2015

Authors and Affiliations

  1. 1.Department of Wind EnergyTechnical University of DenmarkRoskildeDenmark
  2. 2.MOSEK ApSCopenhagenDenmark

Personalised recommendations