Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

Complexity of optimizing over the integers

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

In the first part of this paper, we present a unified framework for analyzing the algorithmic complexity of any optimization problem, whether it be continuous or discrete in nature. This helps to formalize notions like “input”, “size” and “complexity” in the context of general mathematical optimization, avoiding context dependent definitions which is one of the sources of difference in the treatment of complexity within continuous and discrete optimization. In the second part of the paper, we employ the language developed in the first part to study information theoretic and algorithmic complexity of mixed-integer convex optimization, which contains as a special case continuous convex optimization on the one hand and pure integer optimization on the other. We strive for the maximum possible generality in our exposition. We hope that this paper contains material that both continuous optimizers and discrete optimizers find new and interesting, even though almost all of the material presented is common knowledge in one or the other community. We see the main merit of this paper as bringing together all of this information under one unifying umbrella with the hope that this will act as yet another catalyst for more interaction across the continuous-discrete divide. In fact, our motivation behind Part I of the paper is to provide a common language for both communities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. There is also a factor of \(\log (d)\) which shows up due to technical reasons of using \(\Vert \cdot \Vert _2\) instead of \(\Vert \cdot \Vert _\infty \).

  2. A subtlety here is to make sure that one has access to a separation oracle for the lower dimensional subproblems. This is not hard to implement given access to a separation oracle for C: given a point in the new space, one maps back to \(\mathbb R^n\times \mathbb R^d\) and queries the separation oracle there.

  3. There is a slight discrepancy because of the use of the \(\Vert \cdot \Vert _\infty \)-norm for the information complexity bound (see Theorem 4.2), and the use of \(\Vert \cdot \Vert _2\)-norm here. This adds a \(\log (d)\) factor to the complexity of the ellipsoid algorithm, compared to the information complexity bound. We are not aware of any work that resolves this discrepancy.

  4. There is a technical problem that arises here between the notions of algorithm and proof. We have omitted all discussions of cutting plane and branch-and-bound proofs here, which are powerful tools to prove unconditional lower bounds on these algorithms. The precise statement is that any branch-and-bound proof based on variable disjunctions can be replaced by a lift-and-project cutting plane proof of the same size. See [10] for details.

  5. “Lift-and-project cuts” here mean disjunctive cutting planes based on the variable disjunctions (typically the phrase “lift-and-project” is reserved for 0/1 problems).

  6. The same caveat as in point 1. regarding algorithms versus proofs applies.

References

  1. Abramson, Fred G.: Effective computation over the real numbers. In 12th Annual Symposium on Switching and Automata Theory (SWAT 1971), pages 33–37. IEEE Computer Society, (1971)

  2. Aggarwal, Divesh, Dadush, Daniel, Regev, Oded, Stephens-Davidowitz, Noah: Solving the shortest vector problem in 2\(^n\) time using discrete gaussian sampling. In Proceedings of the forty-seventh annual ACM Symposium on Theory of computing (STOC), pages 733–742. ACM, (2015)

  3. Aggarwal, Divesh, Dadush, Daniel, Stephens-Davidowitz, Noah: Solving the closest vector problem in 2\(^n\) time–the discrete gaussian strikes again! In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pages 563–582. IEEE, (2015)

  4. Anstreicher, Kurt M.: On Vaidya’s volumetric cutting plane method for convex programming. Math. Oper. Res. 22(1), 63–89 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Artmann, Stephan, Eisenbrand, Friedrich, Glanzer, Christoph, Oertel, Timm, Vempala, Santosh, Weismantel, Robert: A note on non-degenerate integer programs with small sub-determinants. Oper. Res. Lett. 44(5), 635–639 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Artmann, Stephan, Weismantel, Robert, Zenklusen, Rico: A strongly polynomial algorithm for bimodular integer linear programming. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 1206–1219. ACM, (2017)

  7. Averkov, Gennadiy, Weismantel, Robert: Transversal numbers over subsets of linear spaces. Adv. Geom. 12(1), 19–28 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Banaszczyk, Wojciech: Inequalities for convex bodies and polar reciprocal lattices in \({R}^n\) II: Application of K-convexity. Discrete & Computational Geometry 16(3), 305–311 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Banaszczyk, Wojciech, Litvak, Alexander E., Pajor, Alain, Szarek, Stanislaw J.: The flatness theorem for nonsymmetric convex bodies via the local theory of Banach spaces. Math. Oper. Res. 24(3), 728–750 (1999)

    MathSciNet  MATH  Google Scholar 

  10. Basu, Amitabh, Conforti, Michele, Di Summa, Marco, Jiang, Hongyi: Complexity of cutting plane and branch-and-bound algorithms for mixed-integer optimization. To appear in Mathematical Programming, (2019)

  11. Basu, Amitabh, Conforti, Michele, Di Summa, Marco, Jiang, Hongyi: Complexity of cutting plane and branch-and-bound algorithms for mixed-integer optimization–II. To appear in Combinatorica, (2020)

  12. Basu, Amitabh, Conforti, Michele, Di Summa, Marco, Jiang, Hongyi: Split cuts in the plane. SIAM J. Optim. 31(1), 331–347 (2021)

    MathSciNet  MATH  Google Scholar 

  13. Basu, Amitabh, Jiang, Hongyi: Enumerating integer points in polytopes with bounded subdeterminants. SIAM J. Discret. Math. 36(1), 449–460 (2022)

    MathSciNet  MATH  Google Scholar 

  14. Basu, Amitabh, Oertel, Timm: Centerpoints: A link between optimization and convex geometry. SIAM J. Optim. 27(2), 866–889 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Basu, Saugata, Pollack, Richard, Roy, Marie-Françoise.: Algorithms in real algebraic geometry. Springer Science & Business Media (2006)

  16. Beame, Paul, Fleming, Noah, Impagliazzo, Russell, Kolokolova, Antonina, Pankratov, Denis, Pitassi, Toniann, Robere, Robert: Stabbing Planes. In Anna R. Karlin, editor, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018), volume 94 of Leibniz International Proceedings in Informatics (LIPIcs), pages 10:1–10:20, Dagstuhl, Germany, (2018). Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik

  17. Beeson, Michael J.: Foundations of constructive mathematics: Metamathematical studies, vol. 6. Springer Science & Business Media, (2012)

  18. Bell, David E.: A theorem concerning the integer lattice. Stud. Appl. Math. 56(2), 187–188 (1977)

    MathSciNet  MATH  Google Scholar 

  19. Bertsimas, Dimitris, Weismantel, Robert: Optimization Over Integers. Dynamic Ideas, Belmont, MA (2005)

    Google Scholar 

  20. Daniel Bienstock, Alberto Del Pia, and Robert Hildebrand. Complexity, exactness, and rationality in polynomial optimization. In International Conference on Integer Programming and Combinatorial Optimization, pages 58–72. Springer, (2021)

  21. Bishop, Errett: Foundations of constructive analysis, volume 5. McGraw-Hill New York, (1967)

  22. Bixby, Robert E.: A brief history of linear and mixed-integer programming computation. Documenta Mathematica, pages 107–121, (2012)

  23. Bixby, Robert E., Fenelon, Mary, Gu, Zonghao, Rothberg, Ed, Wunderling, Roland: Mixed integer programming: A progress report. In The Sharpest Cut, pages 309–325. MPS-SIAM Series on Optimization, Philadelphia, PA, (2004)

  24. Blum, Lenore, Shub, Mike, Smale, Steve: On a theory of computation and complexity over the real numbers: W-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc 21(1), 1–46 (1989)

    MathSciNet  MATH  Google Scholar 

  25. Bockmayr, Alexander, Eisenbrand, Friedrich, Hartmann, Mark, Schulz, Andreas S.: On the Chvátal rank of polytopes in the 0/1 cube. Discret. Appl. Math. 98(1–2), 21–27 (1999)

    MATH  Google Scholar 

  26. Bonet, Maria, Pitassi, Toniann, Raz, Ran: Lower bounds for cutting planes proofs with small coefficients. The Journal of Symbolic Logic 62(3), 708–728 (1997)

    MathSciNet  MATH  Google Scholar 

  27. Borodin, Allan, Munro, Ian: The computational complexity of algebraic and numeric problems. American Elsevier, New York (1975)

    MATH  Google Scholar 

  28. Braun, Gábor., Guzmán, Cristóbal, Pokutta, Sebastian: Lower bounds on the oracle complexity of nonsmooth convex optimization via information theory. IEEE Trans. Inf. Theory 63(7), 4709–4724 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Bubeck, Sébastien: Convex optimization: Algorithms and complexity. arXiv preprint arXiv:1405.4980, (2014)

  30. Buss, Samuel R., Clote, Peter: Cutting planes, connectivity, and threshold logic. Arch. Math. Logic 35(1), 33–62 (1996)

    MathSciNet  MATH  Google Scholar 

  31. Carmon, Yair: The Complexity of Optimization beyond Convexity. PhD thesis, Stanford University, August (2020)

  32. Chvátal, Va.šek: Hard knapsack problems. Oper. Res. 28(6), 1402–1411 (1980)

    MathSciNet  MATH  Google Scholar 

  33. Chvátal, Va.šek: Cutting-plane proofs and the stability number of a graph, Report Number 84326-OR. Universität Bonn, Bonn, Institut für Ökonometrie und Operations Research (1984)

  34. Chvátal, Va.šek, Cook, William J., Hartmann, Mark: On cutting-plane proofs in combinatorial optimization. Linear Algebra Appl. 114, 455–499 (1989)

    MathSciNet  MATH  Google Scholar 

  35. Clote, Peter: Cutting planes and constant depth frege proofs. In Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, pages 296–307, (1992)

  36. Cook, William J., Coullard, Collette R., Turán, Gy.: On the complexity of cutting-plane proofs. Discret. Appl. Math. 18(1), 25–38 (1987)

    MathSciNet  MATH  Google Scholar 

  37. Cook, William J., Dash, Sanjeeb: On the matrix-cut rank of polyhedra. Math. Oper. Res. 26(1), 19–30 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Cook, William J., Hartmann, Mark: On the complexity of branch and cut methods for the traveling salesman problem. Polyhedral Combinatorics 1, 75–82 (1990)

    MathSciNet  MATH  Google Scholar 

  39. Cook, William J., Kannan, Ravindran, Schrijver, Alexander: Chvátal closures for mixed integer programming problems. Math. Program. 47, 155–174 (1990)

    MATH  Google Scholar 

  40. Cunningham, William H., Geelen, Jim: On integer programming and the branch-width of the constraint matrix. In International Conference on Integer Programming and Combinatorial Optimization, pages 158–166. Springer, (2007)

  41. Dadush, Daniel: Integer programming, lattice algorithms, and deterministic volume estimation. ProQuest LLC, Ann Arbor, MI, (2012). Thesis (Ph.D.)–Georgia Institute of Technology

  42. Dadush, Daniel, Tiwari, Samarth: On the complexity of branching proofs. arXiv preprint arXiv:2006.04124, (2020)

  43. Dash, Sanjeeb: An exponential lower bound on the length of some classes of branch-and-cut proofs. In International Conference on Integer Programming and Combinatorial Optimization (IPCO), pages 145–160. Springer, (2002)

  44. Dash, Sanjeeb: Exponential lower bounds on the lengths of some classes of branch-and-cut proofs. Math. Oper. Res. 30(3), 678–700 (2005)

    MathSciNet  MATH  Google Scholar 

  45. Dash, Sanjeeb: On the complexity of cutting-plane proofs using split cuts. Oper. Res. Lett. 38(2), 109–114 (2010)

    MathSciNet  MATH  Google Scholar 

  46. Dash, Sanjeeb, Dobbs, Neil B., Günlük, Oktay, Nowicki, Tomasz J., Świrszcz, Grzegorz M.: Lattice-free sets, multi-branch split disjunctions, and mixed-integer programming. Math. Program. 145(1–2), 483–508 (2014)

    MathSciNet  MATH  Google Scholar 

  47. Dash, Sanjeeb, Günlük, Oktay: On t-branch split cuts for mixed-integer programs. Math. Program. 141(1–2), 591–599 (2013)

    MathSciNet  MATH  Google Scholar 

  48. De Loera, Jesús A.: Raymond Hemmecke, and Matthias Köppe. Algebraic and geometric ideas in the theory of discrete optimization. SIAM, (2012)

  49. Del Pia, Alberto: On approximation algorithms for concave mixed-integer quadratic programming. Math. Program. 172(1), 3–16 (2018)

    MathSciNet  MATH  Google Scholar 

  50. Del Pia, Alberto: Subdeterminants and concave integer quadratic programming. SIAM J. Optim. 29(4), 3154–3173 (2019)

    MathSciNet  MATH  Google Scholar 

  51. Del Pia, Alberto, Dey, Santanu S., Molinaro, Marco: Mixed-integer quadratic programming is in np. Math. Program. 162(1), 225–240 (2017)

    MathSciNet  MATH  Google Scholar 

  52. Del Pia, Alberto, Di Gregorio, Silvia: On the complexity of binary polynomial optimization over acyclic hypergraphs. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2684–2699. SIAM, (2022)

  53. Del Pia, Alberto, Hildebrand, Robert, Weismantel, Robert, Zemmer, Kevin: Minimizing cubic and homogeneous polynomials over integers in the plane. Math. Oper. Res. 41(2), 511–530 (2016)

    MathSciNet  MATH  Google Scholar 

  54. Del Pia, Alberto, Khajavirad, Aida: A polyhedral study of binary polynomial programs. Math. Oper. Res. 42(2), 389–410 (2017)

    MathSciNet  MATH  Google Scholar 

  55. Del Pia, Alberto, Khajavirad, Aida: The multilinear polytope for acyclic hypergraphs. SIAM J. Optim. 28(2), 1049–1076 (2018)

    MathSciNet  MATH  Google Scholar 

  56. Del Pia, Alberto, Khajavirad, Aida: The running intersection relaxation of the multilinear polytope. Math. Oper. Res. 46(3), 1008–1037 (2021)

    MathSciNet  MATH  Google Scholar 

  57. Del Pia, Alberto, Khajavirad, Aida, Sahinidis, Nikolaos V.: On the impact of running intersection inequalities for globally solving polynomial optimization problems. Math. Program. Comput. 12(2), 165–191 (2020)

    MathSciNet  MATH  Google Scholar 

  58. Del Pia, Alberto, Walter, Matthias: Simple odd \(\beta \)-cycle inequalities for binary polynomial optimization. to appear in Proceedings of IPCO (2022)

  59. Del Pia, Alberto, Weismantel, Robert: Integer quadratic programming in the plane. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 840–846, (2014)

  60. Dey, Santanu S., Dubey, Yatharth, Molinaro, Marco: Branch-and-bound solves random binary packing IPs in polytime. arXiv preprint arXiv:2007.15192, (2020)

  61. Dey, Santanu S., Dubey, Yatharth, Molinaro, Marco: Lower bounds on the size of general branch-and-bound trees. arXiv preprint arXiv:2103.09807, (2021)

  62. Doignon, J.-P.: Convexity in cristallographical lattices. J. Geometry 3, 71–85 (1973)

    MathSciNet  MATH  Google Scholar 

  63. Downey, Rodney G., Fellows, Michael R.: Parameterized Complexity. Springer Science & Business Media. Berlin/Heidelberg, Germany (2012)

    MATH  Google Scholar 

  64. Eisenbrand, Friedrich: Integer programming and algorithmic geometry of numbers. In M. Jünger, T. Liebling, D. Naddef, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey, editors, 50 Years of Integer Programming 1958–2008. Springer-Verlag, (2010)

  65. Eisenbrand, Friedrich, Hunkenschröder, Christoph, Klein, Kim-Manuel, Kouteckỳ, Martin, Levin, Asaf, Onn, Shmuel: An algorithmic theory of integer programming. arXiv preprint arXiv:1904.01361, (2019)

  66. Eisenbrand, Friedrich, Schulz, Andreas S.: Bounds on the Chvátal rank of polytopes in the 0/1-cube. Combinatorica 23(2), 245–261 (2003)

    MathSciNet  MATH  Google Scholar 

  67. Fleming, Noah, Göös, Mika, Impagliazzo, Russell, Pitassi, Toniann, Robere, Robert, Tan, Li-Yang, Wigderson, Avi: On the power and limitations of branch and cut. arXiv preprint arXiv:2102.05019, (2021)

  68. Fomin, Fedor V., Panolan, Fahad, Ramanujan, M. S., Saurabh, Saket: Fine-grained complexity of integer programming: The case of bounded branch-width and rank. arXiv preprint arXiv:1607.05342, (2016)

  69. Frank, András, Tardos, Éva.: An application of simultaneous diophantine approximation in combinatorial optimization. Combinatorica 7(1), 49–65 (1987)

    MathSciNet  MATH  Google Scholar 

  70. Friedman, Harvey: Algorithmic procedures, generalized turing algorithms, and elementary recursion theory. In Studies in Logic and the Foundations of Mathematics, volume 61, pages 361–389. Elsevier, (1971)

  71. Garey, Michael R., Johnson, David S.: Computers and intractability. W. H. Freeman and Co., San Francisco, Calif., 1979. A guide to the theory of NP-completeness, A Series of Books in the Mathematical Sciences

  72. Goerdt, Andreas: Cutting plane versus frege proof systems. In International Workshop on Computer Science Logic, pages 174–194. Springer, (1990)

  73. Goerdt, Andreas: The cutting plane proof system with bounded degree of falsity. In International Workshop on Computer Science Logic, pages 119–133. Springer, (1991)

  74. Gribanov, Dmitry V., Chirkov, Aleksandr Y.: The width and integer optimization on simplices with bounded minors of the constraint matrices. Optimization Letters 10(6), 1179–1189 (2016)

    MathSciNet  MATH  Google Scholar 

  75. Gribanov, Dmitry V., Malyshev, Dmitriy S., Pardalos, Panos M.: A note on the parametric integer programming in the average case: sparsity, proximity, and fpt-algorithms. arXiv preprint arXiv:2002.01307, (2020)

  76. Gribanov, Dmitry V., Veselov, Sergey I.: On integer programming with bounded determinants. Optimization Letters 10(6), 1169–1177 (2016)

    MathSciNet  MATH  Google Scholar 

  77. Grigoriev, Dima, Hirsch, Edward A., Pasechnik, Dmitrii V.: Complexity of semi-algebraic proofs. In Annual Symposium on Theoretical Aspects of Computer Science (STACS), pages 419–430. Springer, (2002)

  78. Grötschel, Martin, Lovász, László., Schrijver, Alexander: Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics: Study and Research Texts, vol. 2. Springer-Verlag, Berlin (1988)

  79. Grünbaum, Branko: Partitions of mass-distributions and of convex bodies by hyperplanes. Pacific J. Math. 10, 1257–1261 (1960)

    MathSciNet  MATH  Google Scholar 

  80. Heinz, Sebastian: Complexity of integer quasiconvex polynomial optimization. J. Complex. 21(4), 543–556 (2005)

    MathSciNet  MATH  Google Scholar 

  81. Helly, Eduard: Über mengen konvexer körper mit gemeinschaftlichen punkte. Jahresber. Deutsch. Math.-Verein. 32, 175–176 (1923)

    MATH  Google Scholar 

  82. Hilbert, David: Mathematische probleme. Nachrichten der Königliche Gesellschaft zur Wissenschaften zu Göttingen, Mathematische-physikalischen Klasse, vol. 3 (1900)

  83. Hildebrand, Robert, Köppe, Matthias: A new lenstra-type algorithm for quasiconvex polynomial integer minimization with complexity \(2^{O(n\log n)}\). Discret. Optim. 10(1), 69–84 (2013)

    MATH  Google Scholar 

  84. Hildebrand, Robert, Weismantel, Robert, Zemmer, Kevin: An fptas for minimizing indefinite quadratic forms over integers in polyhedra. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1715–1723. SIAM, (2016)

  85. Hoffman, Alan J.: Binding constraints and Helly numbers. Ann. N. Y. Acad. Sci. 319, 284–288 (1979)

    MathSciNet  MATH  Google Scholar 

  86. Impagliazzo, Russell, Pitassi, Toniann, Urquhart, Alasdair: Upper and lower bounds for tree-like cutting planes proofs. In Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science, pages 220–228. IEEE, (1994)

  87. Jansen, Klaus, Rohwedder, Lars: On integer programming, discrepancy, and convolution. arXiv preprint arXiv:1803.04744, (2018)

  88. Jeroslow, Robert G.: Trivial integer programs unsolvable by branch-and-bound. Math. Program. 6(1), 105–109 (1974)

    MathSciNet  MATH  Google Scholar 

  89. Jiang, Haotian, Lee, Yin Tat, Song, Zhao, Wong, Sam Chiu-wai: An improved cutting plane method for convex optimization, convex-concave games, and its applications. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 944–953, (2020)

  90. Kannan, Ravindran: Minkowski’s convex body theorem and integer programming. Math. Oper. Res. 12(3), 415–440 (1987)

    MathSciNet  MATH  Google Scholar 

  91. Khachiyan, Leonid: A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR 244, 1093–1096 (1979)

    MathSciNet  MATH  Google Scholar 

  92. Khachiyan, Leonid, Porkolab, Lorant: Integer optimization on convex semialgebraic sets. Discrete & Computational Geometry 23(2), 207–224 (2000)

    MathSciNet  MATH  Google Scholar 

  93. Knop, Dusan, Pilipczuk, Michal, Wrochna, Marcin: Tight complexity lower bounds for integer linear programming with few constraints. ACM Transactions on Computation Theory (TOCT) 12(3), 1–19 (2020)

    MathSciNet  MATH  Google Scholar 

  94. Ko, Ker-I.: Complexity theory of real functions. Boston Inc., Birkhauser (1991)

    MATH  Google Scholar 

  95. Köppe, Matthias: On the complexity of nonlinear mixed-integer optimization. In Mixed Integer Nonlinear Programming, pages 533–557. Springer, (2012)

  96. Krajíček, Jan: Discretely ordered modules as a first-order extension of the cutting planes proof system. The Journal of Symbolic Logic 63(4), 1582–1596 (1998)

    MathSciNet  MATH  Google Scholar 

  97. Lee, Yin Tat, Sidford, Aaron, Wong, Sam Chiu-wai: A faster cutting plane method and its implications for combinatorial and convex optimization. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1049–1065. IEEE, (2015)

  98. Lenstra, Hendrik W., Jr.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)

    MathSciNet  MATH  Google Scholar 

  99. Li, Yanjun, Richard, Jean-Philippe P.: Cook, Kannan and Schrijver’s example revisited. Discret. Optim. 5(4), 724–734 (2008)

    MathSciNet  MATH  Google Scholar 

  100. Lodi, Andrea: Mixed integer programming computation. In 50 Years of Integer Programming 1958-2008, pages 619–645. Springer, (2010)

  101. Lovász, László: An algorithmic theory of numbers, graphs, and convexity, volume 50. SIAM, (1986)

  102. Margulies, Susan, Ma, Jing, Hicks, Illya V.: The cunningham-geelen method in practice: branch-decompositions and integer programming. INFORMS J. Comput. 25(4), 599–610 (2013)

    MathSciNet  Google Scholar 

  103. Matiyasevich, Yuri: Hilbert’s 10th problem. MIT Press, Cambridge, Massachusetts, USA (1993)

    Google Scholar 

  104. Micciancio, Daniele, Voulgaris, Panagiotis: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations [extended abstract]. In Proceedings of the 2010 ACM International Symposium on Theory of Computing (STOC), pages 351–358. ACM, New York, (2010)

  105. Micciancio, Daniele, Voulgaris, Panagiotis: A deterministic single exponential time algorithm for most lattice problems based on Voronoi cell computations. SIAM J. Comput. 42(3), 1364–1391 (2013)

    MathSciNet  MATH  Google Scholar 

  106. Naderi, Mohammad Javad, Buchanan, Austin, Walteros, Jose L: Worst-case analysis of clique MIPs. http://www.optimization-online.org/DB_HTML/2021/01/8198.html, (2021)

  107. Nemirovski, Arkadii: Efficient methods in convex programming. Lecture notes, (1994)

  108. Nemirovski, Arkadii S., Yudin, David B.: Problem Complexity and Method Efficiency in Optimization. John Wiley, Hoboken, New Jersey, USA (1983)

    Google Scholar 

  109. Nesterov, Yurii E.: Introductory Lectures on Convex Optimization. Applied Optimization, vol. 87. Kluwer Academic Publishers, Boston (2004)

  110. Nesterov, Yurii E.: Lectures on Convex Optimization, vol. 137. Springer, Berlin (2018)

    MATH  Google Scholar 

  111. Oertel, Timm: Integer Convex Minimization in Low Dimensions. PhD thesis, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 22288, (2014)

  112. Oertel, Timm, Wagner, Christian, Weismantel, Robert: Integer convex minimization by mixed integer linear optimization. Oper. Res. Lett. 42(6–7), 424–428 (2014)

    MathSciNet  MATH  Google Scholar 

  113. Onn, Shmuel: Nonlinear discrete optimization. Zurich Lectures in Advanced Mathematics, European Mathematical Society, (2010)

  114. Paat, Joseph, Schlöter, Miriam, Weismantel, Robert: The integrality number of an integer program. Mathematical Programming, pages 1–21, (2021)

  115. Papadimitriou, Christos H.: On the complexity of integer programming. Journal of the ACM (JACM) 28(4), 765–768 (1981)

    MathSciNet  MATH  Google Scholar 

  116. Pour-El, Marian Boykan, Richards, Ian: Computability and noncomputability in classical analysis. Trans. Am. Math. Soc. 275(2), 539–560 (1983)

    MathSciNet  MATH  Google Scholar 

  117. Pudlák, Pavel: Lower bounds for resolution and cutting plane proofs and monotone computations. The Journal of Symbolic Logic 62(3), 981–998 (1997)

    MathSciNet  MATH  Google Scholar 

  118. Pudlák, Pavel: On the complexity of the propositional calculus. London Mathematical Society Lecture Note Series, pages 197–218, (1999)

  119. Razborov, Alexander A.: On the width of semialgebraic proofs and algorithms. Math. Oper. Res. 42(4), 1106–1134 (2017)

    MathSciNet  MATH  Google Scholar 

  120. Rothvoß, Thomas, Sanità, Laura: 0/1 polytopes with quadratic Chvátal rank. In International Conference on Integer Programming and Combinatorial Optimization (IPCO), pages 349–361. Springer, (2013)

  121. Rudelson, Mark: Distances between non-symmetric convex bodies and the \({MM}^*\)-estimate. Positivity 4(2), 161–178 (2000)

    MathSciNet  MATH  Google Scholar 

  122. Scarf, Herbert E.: An observation on the structure of production sets with indivisibilities. Proc. Natl. Acad. Sci. 74(9), 3637–3641 (1977)

    MathSciNet  MATH  Google Scholar 

  123. Schrijver, Alexander: Theory of Linear and Integer Programming. John Wiley and Sons, New York (1986)

    MATH  Google Scholar 

  124. Steinitz, Ernst: Bedingt konvergente reihen und konvexe systeme. Journal für die reine und angewandte Mathematik, (1913)

  125. Traub, Joseph Frederick, Wasilkowski, Grzegorz Włodzimierz., Woźniakowski, Henryk: Information, uncertainty, complexity. Addison-Wesley Reading, MA (1983)

    MATH  Google Scholar 

  126. Turing, Alan Mathison: On computable numbers, with an application to the “entscheidungsproblem’’. a correction. Proc. Lond. Math. Soc. 43(2), 544–546 (1937)

    MathSciNet  MATH  Google Scholar 

  127. Vaidya, Pravin M.: A new algorithm for minimizing convex functions over convex sets. Math. Program. 73(3), 291–341 (1996)

    MathSciNet  MATH  Google Scholar 

  128. Veselov, Sergey I., Chirkov, Aleksandr J.: Integer program with bimodular matrix. Discret. Optim. 6(2), 220–222 (2009)

    MathSciNet  MATH  Google Scholar 

  129. Yudin, David B., Nemirovskii, Arkadii S.: Informational complexity and efficient methods for the solution of convex extremal problems. Matekon 13(2), 22–45 (1976)

    MATH  Google Scholar 

Download references

Acknowledgements

The author gratefully acknowledges support from Air Force Office of Scientific Research (AFOSR) grant FA95502010341 and National Science Foundation (NSF) grants CCF2006587.

The author benefited greatly from discussions with Daniel Dadush at CWI, Amsterdam and Timm Oertel at FAU, Erlangen-Nürmberg. Comments from two anonymous referees helped the author significantly to consolidate the material, improve its presentation and make tighter connections to the existing literature on the complexity of optimization.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, USA

    Amitabh Basu

Authors
  1. Amitabh Basu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amitabh Basu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Basu, A. Complexity of optimizing over the integers. Math. Program. 200, 739–780 (2023). https://doi.org/10.1007/s10107-022-01862-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-022-01862-z

Mathematics Subject Classification

Search

Navigation