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Continuous Optimization pp 251–273Cite as

A Numerical Method for Concave Programming Problems

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Part of the book series: Applied Optimization ((APOP,volume 99))

Summary

Concave programming problems constitute one of the most important and fundamental classes of problems in global optimization. Concave minimization problems have a diverse range of direct and indirect applications. Moreover, concave minimization problems are well known to be NP-hard. In this paper, we present three algorithms which are similar to each other for concave minimization problems. In each iteration of the algorithms, linear programming problems with the same constraints as the initial problem are required to solve and a local search method is required to use. Furthermore, the convergence result is given. From the result, we see that the local search method is not necessarily required but we require that some conditions must hold on the constraint.

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References

  1. Bertsekas, D.P.: Nonlinear programming. Athena Scientific, Belmont, Mass. (1995)

    Google Scholar 

  2. Coleman, T.F., Li, Y.: A reflective Newton method for minimizing a quadratic function subject to bounds on some of the variables. SIAM Journal on Optimization, 6, 1040–1058 (1996)

    Article  MathSciNet  Google Scholar 

  3. Dantzig, G.B., Orden, A., Wolfe, P.: The generalized Simplex Method for minimizing a linear form under linear inequality constraints. Pacific Journal Math., 5, 183–195.

    Google Scholar 

  4. Dietrich, H.: Global optimization conditions for certain nonconvex minimization problems. Journal of Global Optimization, 5, (359–370) (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Enkhbat, R.: An algorithm for maximizing a convex function over a simple set. Journal of Global Optimization, 8, 379–391 (1996)

    Article  MathSciNet  Google Scholar 

  6. Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Academic Press, London, UK (1981)

    Google Scholar 

  7. Hiriart-Urruty, J.B.: Prom convex optimization to nonconvex optimization. In: Nonsmooth Optimization and Related Topics, 219–239. Plenum (1989)

    Google Scholar 

  8. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches (second edition). Springer Verlag, Heidelberg (1993)

    Google Scholar 

  9. Horst, R., Pardalos, P.M. (eds): Handbook of Global Optimization. Kluwer Academic, Netherlands (1995)

    Google Scholar 

  10. Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization (second edition). Kluwer Academic, Netherlands (2001)

    Google Scholar 

  11. Khachiyan, L.: A polynomial algorithm in linear programming. Math. Doklady, 20, 191–194 (1979)

    MATH  Google Scholar 

  12. Mehrotra, S.: On the implementation of a Primal-Dual Interior Point Method, SIAM Journal on Optimization, 2, 575–601 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Pardalos, P.M.: Complexity in Numerical Optimization. World Scientific Publishing, River Edge, New Jersey (1993)

    Google Scholar 

  14. Pardalos, P.M., Rosen, J.B.: Constrained Global Optimization: Algorithms and Applications. Lecture Notes in Computer Science 268, Springer-Verlag (1987)

    Google Scholar 

  15. Pardalos, P.M., Rosen, J.B.: Methods for Global Concave Minimization: A Bibliographic Survey. SIAM Review, 28, 367–379 (1986)

    Article  MathSciNet  Google Scholar 

  16. Pardalos, P.M., Schnitger, G.: Checking local optimality in constrained quadratic programming is NP-hard. Operations Research Letters, 7, 33–35 (1988)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  18. Strekalovsky, A.S.: Global optimality conditions for nonconvex optimization. Journal of Global Optimization, 12, 415–434 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Strekalovsky, A.S., Enkhbat, R.: Global maximum of convex functions on an arbitrary set. Dep.in VINITI, Irkutsk, No. 1063, 1–27 (1990)

    Google Scholar 

  20. Thoai, N.V.: On the construction of test problems for concave minimization problem. Journal of Global Optimization, 5, 399–402 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, Gainesville, FL, 32611, USA

    Altannar Chinchuluun & Panos M. Pardalos

  2. Department of Mathematical Modeling School of Mathematics and Computer Science, National University of Mongolia, Ulaanbaatar, Mongolia

    Enkhbat Rentsen

Authors
  1. Altannar Chinchuluun
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  2. Enkhbat Rentsen
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  3. Panos M. Pardalos
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Editor information

Editors and Affiliations

  1. University of New South Wales, Sydney, Australia

    Vaithilingam Jeyakumar

  2. University of Ballarat, Ballarat, Australia

    Alexander Rubinov

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© 2005 Springer Science+Business Media, Inc.

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Chinchuluun, A., Rentsen, E., Pardalos, P.M. (2005). A Numerical Method for Concave Programming Problems. In: Jeyakumar, V., Rubinov, A. (eds) Continuous Optimization. Applied Optimization, vol 99. Springer, Boston, MA. https://doi.org/10.1007/0-387-26771-9_8

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