Mathematical Programming

, Volume 58, Issue 1–3, pp 415–428 | Cite as

D.c. sets, d.c. functions and nonlinear equations

  • Phan Thien Thach


Ad.c. set is a set which is the difference of two convex sets. We show that any set can be viewed as the image of a d.c. set under an appropriate linear mapping. Using this universality we can convert any problem of finding an element of a given compact set in ℝ n into one of finding an element of a d.c. set. On the basis of this approach a method is developed for solving a system of nonlinear equations—inequations. Unlike Newton-type methods, our method does not require either convexity, differentiability assumptions or an initial approximate solution.

Key words

D.c. sets d.c. functions nonlinear equations—inequations 


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  1. [1]
    B.C. Eaves and W.I. Zangwill, “Generalized cutting plane algorithms,”SIAM Journal on Control 9 (1971) 529–542.Google Scholar
  2. [2]
    J.E. Falk and K.R. Hoffman, “A successive underestimation method for concave minimization problems,”Mathematics of Operations Research 1 (1976) 251–259.Google Scholar
  3. [3]
    R.M. Freund and J.B. Orlin, “On the complexity of four polyhedral set containment problems,”Mathematical Programming 33 (1985) 139–145.Google Scholar
  4. [4]
    J.B. Hiriart-Urruty, “Generalized differentiability, duality and optimization for problems dealing with differences of convex functions,”Lecture Notes in Economics and Mathematical Systems No. 256 (Springer, Berlin, 1984) pp. 37–70.Google Scholar
  5. [5]
    K.L. Hoffman, “A method for globally minimizing concave functions over convex sets,”Mathematical Programming 20 (1981) 22–32.Google Scholar
  6. [6]
    R. Horst, J. de Vries and N.V. Thoai, “On finding new vertices and redundant constraints in cutting plane algorithms for global optimization,”Operations Research Letters 7 (1988) 85–90.Google Scholar
  7. [7]
    L.A. Istomin, “A modification of Hoang Tuy's method for minimizing a concave function over a polytope,” Žurnal Vyčislitel'noi Matematiki i Matematičesko428-2 17 (1977) 1592–1597. [In Russian.]Google Scholar
  8. [8]
    B.M. Mukhamediev, “Approximate method for solving the concave programming problem,” Žurnal Vyčislitel'noi Matematiki i Matematičesko428-4 22 (1982) 727–731 [In Russian.]Google Scholar
  9. [9]
    R.T. Rockafellar, “Favorable classes of Lipschitz continuous functions in subgradient optimization,” Working Paper, IIASA (Laxenburg, 1983).Google Scholar
  10. [10]
    R.T. Rockafellar,,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
  11. [11]
    J.B. Rosen and P.M. Pardalos, “Global minimization of large scale constrained concave quadratic problems by separable programming,”Mathematical Programming 34 (1986) 163–174.Google Scholar
  12. [12]
    P.T. Thach and H. Tuy, ‘Global optimization under Lipschitzian constraints,”Japan Journal of Applied Mathematics 4 (1987) 205–217.Google Scholar
  13. [13]
    P.T. Thach, “Concave minimization under nonconvex constraints with special structure,”Essays on Nonlinear Analysis and Optimization Problems (Hanoi Institute of Mathematics Press, Hanoi, 1987) pp. 121–139.Google Scholar
  14. [14]
    T.V. Thieu, B.T. Tam and V.T. Ban, “An outer approximation method for globally minimizing a concave function over a compact convex set,”Acta Mathematica Vietnamica 8 (1983) 21–40.Google Scholar
  15. [15]
    N.V. Thoai and H. Tuy, “Convergent algorithm for minimizing a concave function,”Mathematics of Operations Research 5 (1980) 556–566.Google Scholar
  16. [16]
    H. Tuy, “Concave programming under linear constraints,”Doklady Akademii Nauk SSR 159 (1964) 32–35. [English translation in:Soviet Mathematics (1964) 1437–1440.]Google Scholar
  17. [17]
    H. Tuy, “On outer approximation methods for solving concave minimization problems,”Acta Mathematica Vietnamica 8 (1983) 3–34.Google Scholar
  18. [18]
    H. Tuy, T.V. Thieu and N.Q. Thai, “A conical algorithm for globally minimizing a concave function over a closed convex set,”Mathematics of Operations Research 10 (1985) 489–514.Google Scholar
  19. [19]
    H. Tuy, “Concave Minimization under linear constraints with special structure,”Optimization 26 (1985) 335–352.Google Scholar
  20. [20]
    H. Tuy, “A general deterministic approach to global optimization via d.c. programming,” in: J.B. Hiriart-Urruty, ed.,Mathematics Studies No. 129 (North-Holland, Amsterdam, 1987) pp. 273–303.Google Scholar
  21. [21]
    N.S. Vassiliev, “Active computing method for finding the global minimum of a concave function,” Žurnal Vyčislitel'noi Matematiki i Matematičesko428-6 23 (1983) 152–156. [In Russian.]Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1993

Authors and Affiliations

  • Phan Thien Thach
    • 1
  1. 1.Institute of MathematicsHanoiViet Nam

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