Solving Linear Programming with Constraints Unknown

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


What is the value of input information in solving linear programming? The celebrated ellipsoid algorithm tells us that the full information of input constraints is not necessary; the algorithm works as long as there exists an oracle that, on a proposed candidate solution, returns a violation in the form of a separating hyperplane. Can linear programming still be efficiently solved if the returned violation is in other formats?

Motivated by some real-world scenarios, we study this question in a trial-and-error framework: there is an oracle that, upon a proposed solution, returns the index of a violated constraint (with the content of the constraint still hidden). When more than one constraint is violated, two variants in the model are investigated. (1) The oracle returns the index of a “most violated” constraint, measured by the Euclidean distance of the proposed solution and the half-spaces defined by the constraints. In this case, the LP can be efficiently solved (under a mild condition of non-degeneracy). (2) The oracle returns the index of an arbitrary (i.e., worst-case) violated constraint. In this case, we give an algorithm with running time exponential in the number of variables. We then show that the exponential dependence on n is unfortunately necessary even for the query complexity. These results put together shed light on the amount of information that one needs in order to solve a linear program efficiently.

The proofs of the results employ a variety of geometric techniques, including the weighted spherical Voronoi diagram and the furthest Voronoi diagram.


  1. 1.
    Ash, P.F., Bolker, E.D.: Recognizing dirichlet tessellations. Geometriae Dedicata 19(2), 175–206 (1985)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bei, X., Chen, N., Zhang, S.: On the complexity of trial and error. In: Proceedings of the 45th ACM Symposium on Theory of Computing, pp. 31–40 (2013)Google Scholar
  3. 3.
    Bei, X., Chen, N., Zhang, S.: Solving linear programming with constraints unknown (2013). arXiv:1304.1247
  4. 4.
    Chiang, M., Hande, P., Lan, T., Tan, C.-W.: Power control in wireless cellular networks. Foundations and Trends in Networking 2(4), 381–533 (2007)CrossRefGoogle Scholar
  5. 5.
    Clarkson, K.L.: Las vegas algorithms for linear and integer programming when the dimension is small. Journal of the ACM 42(2), 488–499 (1995)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Doherty, L., Pister, K., Ghaoui, L.E.: Convex position estimation in wireless sensor networks. In: Proceedings of the Twentieth IEEE Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), pp. 1655–1663 (2001)Google Scholar
  7. 7.
    Dyer, M., Megiddo, N., Welzl, E.: Linear programming. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry. CRC Press (2004)Google Scholar
  8. 8.
    Dyer, M.E.: On a multidimensional search technique and its application to the euclidean one-centre problem. SIAM Journal on Computing 15(3), 725–738 (1986)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Foschini, G.J., Miljanic, Z.: A simple distributed autonomous power control algorithm and its convergence. IEEE Transactions on Vehicular Technology 42(3), 641–646 (1993)CrossRefGoogle Scholar
  10. 10.
    Gentile, C.: Distributed sensor location through linear programming with triangle inequality constraints. In: Proceedings of IEEE Conference on Communications, pp. 3192–3196 (2005)Google Scholar
  11. 11.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric methods in combinatorial optimization. In: Progress in Combinatorial Optimization, pp. 167–183 (1984)Google Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer (1988)Google Scholar
  13. 13.
    Hartvigsen, D.: Recognizing voronoi diagrams with linear programming. INFORMS Journal on Computing 4(4), 369–374 (1992)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Ivanyos, G., Kulkarni, R., Qiao, Y., Santha, M., Sundaram, A.: On the complexity of trial and error for constraint satisfaction problems. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 663–675. Springer, Heidelberg (2014) Google Scholar
  15. 15.
    Kalai, G.: A subexponential randomized simplem algorithm. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 475–482 (1992)Google Scholar
  16. 16.
    Khachiyan, L.: A polynomial algorithm in linear programming. Doklady Akademii Nauk SSSR 244, 1093–1096 (1979)MATHMathSciNetGoogle Scholar
  17. 17.
    Kovalev, M.: A property of convex sets and its application. Matematicheskie Zametki, pp. 89–99. English translation: Mathematical Notes, 44, 537–543 (1988)Google Scholar
  18. 18.
    Matousek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16(4/5), 498–516 (1996)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Megiddo, N.: Linear programming in linear time when the dimension is fixed. Journal of the ACM 31(1), 114–127 (1984)MATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Montgomery, D.: Design and Analysis of Experiments, 7 edn. Wiley (2008)Google Scholar
  21. 21.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover Publications (1998)Google Scholar
  22. 22.
    Papadimitriou, C.H., Yannakakis, M.: Linear programming without the matrix. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 121–129 (1993)Google Scholar
  23. 23.
    Ryzhov, I.O., Powell, W.B.: Information collection for linear programs with uncertain objective coefficients. SIAM Journal on Optimization 22(4), 1344–1368 (2012)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Sandgren, L.: On convex cones. Mathematica Scandinavica 2, 19–28 (1954)MATHMathSciNetGoogle Scholar
  25. 25.
    Yudin, D.B., Nemirovskii, A.S.: Informational complexity and efficient methods for the solution of convex extremal problems. Ekonomika i Matematicheskie Metody, 12, 357–369 (1976). English translation: Matekon 13(3), 25–45 (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.Nanyang Technological UniversitySingaporeSingapore
  3. 3.The Chinese University of Hong KongShatinHong Kong

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