Skip to main content

Connections Between Continuous and Discrete Extremum Problems, Generalized Systems, andVariational Inequalities

  • Reference work entry
  • First Online:
Book cover Handbook of Combinatorial Optimization

Abstract

This writing aims at surveying what has been done to analyze some connections between continuous and discrete extremum problems and related models, like generalized systems and variational inequalities.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 3,400.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Section 1 is due to Giannessi; Sects. 2, 5, and 6 are due to Antoni; and Sects. 3 and 4 are due to Tardella.

  2. 2.

    In the sense of not necessarily global.

Recommended Reading

  1. J. Abello, S. Butenko, P.M. Pardalos, M.G.C. Resende, Finding independent sets in a graph using continuous multivariable polynomial formulations. J. Global Optim. 21, 111–137 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Antoni, F. Giannessi, On the equivalence, via relaxation-penalization, between vector generalized systems. Acta Vietnam. 22, 567–588 (1997)

    MathSciNet  MATH  Google Scholar 

  3. C. Antoni, Constrained optimization: equivalence theorems for mixed-integer problems via exact penalty. Technical report, Naval Academy, Livorno, Viale Italia 72, 2011

    Google Scholar 

  4. C. Antoni, Duality for generalized systems, in The Proceedings of the Workshop “Equilibrium Problems with Side Constraint. Lagrangean Theory and Duality II”, Scilla, Reggio Calabria, 1996, pp. 11–18

    Google Scholar 

  5. C. Antoni, M. Pedrazzoli, Estimate of penalty parameters. Technical report, Naval Academy, Livorno, Viale Italia 72, 2011

    Google Scholar 

  6. F. Aurenhammer, Voronoi diagrams – a survey of a fundamental geometric data structure. Comput. Surv. 23, 345–405 (1991)

    Article  Google Scholar 

  7. D. Avis, B.K. Bhattacharya, Algorithms for computing d-dimensional Voronoi diagrams and their duals, in Advances in Computing Research, Computational Geometry, vol. 1, ed. by F.P. Preparata (JAI Press, 1983), pp. 159–180

    Google Scholar 

  8. D. Avis, K. Fukuda, Reverse search for enumeration. Discrete Appl. Math. 65, 21–46 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Avriel, W.E. Diewert, S. Schaible, I. Zang, Mathematical Concepts and Methods in Science and Engineering, vol. 36 (Plenum, New York, 1988)

    Google Scholar 

  10. E. Balas, Disjunctive programming. Ann. Discrete Math. 5, 3–51 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. H.P. Benson, Concave minimization: theory, applications and algorithms, in Handbook of Global Optimization, ed. by R. Horst, P.M. Pardalos (Kluwer Academic, Boston, 1995), pp. 43–148

    Chapter  Google Scholar 

  12. E. Blum, W. Oettli, Direct proof of the existence theorem in quadratic programming. Oper. Res. 20, 165–167 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  13. I.M. Bomze, On standard quadratic optimization problems. J. Global Optim. 13, 369–387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. I.M. Bomze, Copositivity aspects of standard quadratic optimization problems, in Optimization, Dynamics, and Economic Analysis (Physica, Heidelberg, 2000), pp. 1–11

    Google Scholar 

  15. I.M. Bomze, Branch-and-bound approaches to standard quadratic optimization problems. J. Global Optim. 22, 17–37 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. I.M. Bomze, E. De Klerk, Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Global Optim. 24, 163–185 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Boros, P.L. Hammer, Pseudo-Boolean optimization. Discrete Appl. Math. 123, 155–225 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. R. Burkard, B. Klinz, R. Rudolf, Perspective of monge properties in optimization. Discrete Appl. Math. 70, 95–161 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. R.E. Burkard, H. Hamacher, J. Tind, On abstract duality in mathematical programming. Report Mathematisches Institut Universitatzukoln - Angewandte Mathematik, 1981

    Google Scholar 

  20. V. Cabot, R.L. Francis, M.A. Stuart, A network flow solution to a rectilinear distance facility location problem. AIIE Trans. 2, 132–141 (1970)

    Article  Google Scholar 

  21. D.-Z. Du, Minimax and its applications, in Handbook of Global Optimization, ed. by R. Horst, P.M. Pardalos (Kluwer Academic, Boston, 1995), 339–367

    Chapter  Google Scholar 

  22. D.-Z. Du, F.K. Hwang, A proof of Gilbert-Pollak conjecture on the Steiner ratio. Algoritmica 7, 121–135 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. D.-Z. Du, P.M. Pardalos (eds.), Minimax and Applications (Kluwer Academic, Boston, 1995)

    MATH  Google Scholar 

  24. D.-Z. Du, P.M. Pardalos, A continuous version of a result of Du and Hwang. J. Global Optim. 5, 127–130 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Dunford, J.T. Schwartz, Linear Operator, Part I (Interscience, New York, 1958)

    Google Scholar 

  26. F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, in Variational Inequalities and Complementarity Problems, ed. by R.W. Cottle, F. Giannessi, J.-L. Lions (Wiley, New York, 1980), pp. 151–186

    Google Scholar 

  27. B. Eaves, On quadratic programming. Manag. Sci. 17, 698–711 (1971)

    Article  MATH  Google Scholar 

  28. E. Erkut, S. Neumann, Analytical models for locating undesirable facilities. Eur. J. Oper. Res. 40, 275–291 (1989)

    Article  MATH  Google Scholar 

  29. P. Favati, F. Tardella, Convexity in nonlinear integer programming. Ricerca Oper. 53, 3–44 (1990)

    Google Scholar 

  30. M. Frank, P. Wolfe, An algorithm for quadratic programming. Naval Res. Logist. Q. 3, 95–110 (1956)

    Article  MathSciNet  Google Scholar 

  31. S. Fujishige, Annals of Discrete Mathematics, vol. 47 (North-Holland, Amsterdam, 1991)

    Google Scholar 

  32. K. Fukuda, T.M. Liebling, F. Margot, Analysis of backtrack algorithms for listing all vertices and all faces of a convex polyhedron. Comput. Geom. 8, 1–12 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. P. Gahinet, P. Apkarian, M. Chilali, Affine parameter-dependent Lyapunov functions and real parametric uncertainty. IEEE Trans. Autom. Control 41, 436–442 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. F. Giannessi, On minty variational principle, in Trend Mathematical Programming, Proceeding of a Conference held in Matrafured, 1996, pp. 161–176

    Google Scholar 

  35. F. Giannessi, On some connections among variational inequalities, combinatorial and continuous optimization. Ann. Oper. Res. 58, 181–200 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Giannessi, F. Niccolucci, Connections between nonlinear and integer programming problems, in Symposia Mathematica, vol. 19 (Academic, New York, 1976), pp. 161–176

    Google Scholar 

  37. F. Giannessi, F. Tardella, Connections between nonlinear programming and discrete optimization, in Handbook of Combinatorial Optimization, vol. 1, ed. by D.Z. Du, P.M. Pardalos (Kluwer Academic, Boston, 1998), pp. 149–188

    Chapter  Google Scholar 

  38. F. Giannessi, E. Tomasin, Non convex quadratic programs, linear complementarity problems and integer linear programs, in Proceedings of the NATO School on “Mathematical Programming in Theory and in Practice”, North Holland, Amsterdam, 1994

    Google Scholar 

  39. L. Gibbons, D. Hearn, M. Ramana, P.M. Pardalos, A continuous characterization of the maximum clique problem. Math. Oper. Res. 22, 754–768 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Gröschel, L. Lovász, A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–187 (1981)

    Article  MathSciNet  Google Scholar 

  41. S.L. Hakimi, Optimum location of switching centers and the absolute centers and medians of a graph. Oper. Res. 12, 450–459 (1964)

    Article  MATH  Google Scholar 

  42. P.L. Hammer, S. Rudeanu, Mthods Boolennes en Recherche Oprationnelle (Dunod, Paris, 1970)

    Google Scholar 

  43. G.Y. Handler, P.B. Mirchandani, Location on Networks: Theory and Algorithms (MIT, Cambridge, 1979)

    MATH  Google Scholar 

  44. J.-B. Hiriart-Urruty, C. Lemarechal, Testing necessary and sufficient conditions for global optimality in the problem of maximizing a convex quadratic function over a convex polyhedron. Technical report, University Paul Sabatier of Toulouse, Seminar of Numerical Analysis, 1990

    Google Scholar 

  45. W.M. Hirsch, A.J. Hoffman, Extreme varieties, concave functions and the fixed charge problem. Comm. Pure Appl. Math. 14, 355–369 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Horst, H. Tuy, Global Optimization. Deterministic Approaches (Springer, Berlin, 1990)

    Google Scholar 

  47. F.K. Hwang, U.G. Rothblum, Directional quasi-convexity, asymmetric Schur-convexity and optimality of consecutive partitions. Math. Oper. Res. 21, 540–554 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  48. S. Iwata, L. Fleisher, S. Fujishige, A strongly polynomial algorithm for minimizing submodular functions. J. Assoc. Comput. Mach. 48, 761–777 (2001)

    Article  MATH  Google Scholar 

  49. B. Kalantari, J.B. Rosen, Penalty for zero-one integer equivalent problem. Math. Program. 24, 229–232 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  50. G. Kéri, On the minimum value of a quadratic function under linear constraints. Stud. Sci. Math. Hung. 6, 193–196 (1971)

    Google Scholar 

  51. V. Klee, On the complexity of d-dimensional Voronoi diagrams. Arkiv Math. 34, 75–80 (1980)

    MathSciNet  MATH  Google Scholar 

  52. K. Kleibohm, Bemerkungen zum Problem der nichtkonvexen Programmierung. Unternehmensforschung 11, 49–60 (1967)

    MathSciNet  MATH  Google Scholar 

  53. M.K. Kozolov, S.P. Tarasov, L.G. Hac̆ijan, Polynomial solvability of convex quadratic programs. Sov. Math. Doklady 20, 1108–1111 (1979)

    Google Scholar 

  54. M. Labbé, D. Peeters, J.F. Thisse, Location on networks, in Handbooks in Operations Research and Management Science, vol. 8, ed. by M.O. Ball, et al. (Elsevier, Amsterdam, 1995), pp. 551–624

    Google Scholar 

  55. C. Larsen, J. Tind, Lagrangean duality for facial programs with applications to integer and complementarity problems. Oper. Res. Lett. (North-Holland) 11, 293–302 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  56. S. Lucidi, F. Rinaldi, Exact penalty functions for nonlinear integer programming problems. J. Optim. Theory Appl. 145, 479–488 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  57. S. Lucidi, F. Rinaldi, An exact penalty global optimization approach for mixed-integer programming problems. Technical report no. 17, DIS, University of Rome, 2010

    Google Scholar 

  58. Z.Q. Luo, S. Zhang, On extensions of the Frank-Wolfe theorems. Comput. Optim. Appl. 13, 87–110 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  59. P.B. Mirchandani, R.L. Francis (eds.), Discrete Location Theory (Wiley, New York, 1990)

    MATH  Google Scholar 

  60. T.S. Motzkin, E.G. Straus, Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17, 533–540 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  61. P.M. Pardalos, Continuous approaches to discrete optimization problems, in Nonlinear Optimization and Applications, ed. by G. Di Pillo, F. Giannessi (Plenum, New York, 1996), pp. 313–328

    Chapter  Google Scholar 

  62. P.M. Pardalos, J.B. Rosen, Constrained Global Optimization: Algorithms and Applications (Springer, Berlin, 1987)

    Book  MATH  Google Scholar 

  63. A.F. Perold, A generalization of the Frank-Wolfe theorem. Math. Program. 18, 215–227 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  64. M. Ragavachari, On the connection between zero-one integer programming and concave programming under linear constraints. Oper. Res. 17, 680–683 (1969)

    Article  Google Scholar 

  65. F. Rinaldi, New results on the equivalence between zero-one programming and continuous concave programming. Opt. Lett. 3, 377–386 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  66. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)

    MATH  Google Scholar 

  67. I.G. Rosenberg, 0 − 1 optimization and non-linear programming. Rev. Française Autom. Inform. Rec. Opér. 6, 95–97 (1972)

    MATH  Google Scholar 

  68. A. Schrijver, A combinatorial algorithm for minimizing submodular functions in strongly polynomial time. J. Combin. Theory Ser. B 80, 346–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  69. A. Schrijver, Combinatorial Optimization. Polyhedra and Efficiency (Springer, Berlin, 2003)

    MATH  Google Scholar 

  70. A. Scozzari, F. Tardella, A clique algorithm for standard quadratic programming. Discrete Appl. Math. 156, 2439–2448 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  71. J. Shi, Y. Yoshitsugu, A D.C. approach to the largest empty sphere problem in higher dimension, in State of the Art in Global Optimization, ed. by C.A. Floudas, P.M. Pardalos (Kluwer Academic, Boston, 1996), pp. 395–411

    Google Scholar 

  72. F. Tardella, On a class of functions attaining their maximum at the vertices of a polyhedron. Discrete Appl. Math. 22, 191–195 (1988–1989)

    Article  MathSciNet  Google Scholar 

  73. F. Tardella, On the equivalence between some discrete and continuous optimization problems. Ann. Oper. Res. 25, 291–300 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  74. F. Tardella, Piecewise concavity and minimax problems, in Approximation and Complexity in Numerical Optimization. Continuous and Discrete Problems, (Series: Nonconvex Optim. Appl. n. 42), ed. by P.M. Pardalos (Kluwer Academic, Boston, 2000), pp. 511–524

    Chapter  Google Scholar 

  75. F. Tardella, Connections between continuous and combinatorial optimization problems through an extension of the fundamental theorem of linear programming. Electron. Note Discrete Math. 17C, 57–262 (2004)

    MathSciNet  Google Scholar 

  76. F. Tardella, The fundamental theorem of linear programming: extensions and applications. Optimization 60, 15–27 (2011)

    Article  MathSciNet  Google Scholar 

  77. H. Tuy, Concave programming under linear constraints. Sov. Math. 5, 1437–1440 (1964)

    Google Scholar 

  78. D.M. Topkis, Minimizing a submodular function on a lattice. Oper. Res. 26, 305–321 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  79. D.M. Topkis, Adjacency on polymatroids. Math. Program. 30, 229–237 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  80. W.I. Zangwill, The piecewise concave function. Manag. Sci. 13, 900–912 (1967)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to Carla Antoni , Franco Giannessi or Fabio Tardella .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this entry

Cite this entry

Antoni, C., Giannessi, F., Tardella, F. (2013). Connections Between Continuous and Discrete Extremum Problems, Generalized Systems, andVariational Inequalities . In: Pardalos, P., Du, DZ., Graham, R. (eds) Handbook of Combinatorial Optimization. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7997-1_54

Download citation

Publish with us

Policies and ethics