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Global minimization of a difference of two convex functions

  • Hoang Tuy
Chapter
Part of the Mathematical Programming Studies book series (MATHPROGRAMM, volume 30)

Abstract

We study the problem of finding the global minimum of the difference between two convex functions f(x)−g(y), under linear constraints of the form: x∈X, y∈Y, Ax+By+c≤0, where \(X \subset \mathbb{R}^{n_1 } , Y \subset \mathbb{R}^{n_2 }\), are convex polyhedral sets. The proposed solution method consists in converting the problem into a concave minimization problem in \(\mathbb{R}^{n_2 + 1}\) and applying the outer approximation method to the latter problem. Using a special type of separating hyperplanes the same result could also be obtained by applying the generalized Benders’ decomposition method with a proper change of variable in the master problem. As specialized to the indefinite quadratic programming problem, the algorithm is convergent, provided only the set Y 0={yY:(∃xX) Ax+By+c≤0} is bounded.

Keywords

Global minimization difference of two convex functions outer approximation method generalized Benders’ decomposition concave minimization indefinite quadratic programming 

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References

  1. [1]
    E. Balas, “Nonconvex quadratic programming via generalized polars”, SIAM Journal on Applied Mathematics 28 (1975) 335–349.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    L. Bittner, “Some representation theorems for functions and sets and their applications to nonlinear programming”, Numerische Mathematik 16 (1970) 32–51.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    A.V. Cabot and R.L. Francis, “Solving certain nonconvex quadratic minimization problems by ranking extreme points”, Operations Research 18 (1970) 82–86.zbMATHCrossRefGoogle Scholar
  4. [4]
    V.F. Demyanov and A.M. Rubinov, “On quasidifferentiable functionals”, Mathematische Operationsforschung und Statistik, Serie Optimization 14 (1983) 3–21.zbMATHGoogle Scholar
  5. [5]
    V.F. Demyanov and L.N. Polyakova, “Minimization of a quasidifferentiable function on a quasidifferentiable set”, Zurnal Vyċislitelnoi Matematiki i Matematitcheskoi Fiziki 20 (1980) 849–856. (Russian). Translated in U.S.S.R. Computational Mathematics and Mathematical Physics 20 (4) (1981) 34–43.MathSciNetGoogle Scholar
  6. [6]
    J.E. Falk and K.R. Hoffman, “A successive underestimation method for concave minimization problems”, Mathematics of Operations Research 1 (1976) 251–259.zbMATHCrossRefGoogle Scholar
  7. [7]
    J.E. Falk and R.M. Soland, “An algorithm for separable nonconvex programming problems”, Management Science 15 (1969) 550–569.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    A. Geoffrion, “Generalized Benders decompositions”, Journal of Optimization Theory and Applications 10 (1972) 237–260.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    L.A. Istomin, “A modification of H. Tuy’s method for minimizing a concave function over a polytope”, Zurnal Vyčislitelnoi Matematiki i Matematitcheskoi Fiziki 17 (1977) 1592–1597 (Russian). Translated in U.S.S.R. Computational Mathematics and Mathematical Physics 17 (6) (1977) 242–248.zbMATHMathSciNetGoogle Scholar
  10. [10]
    H. Konno, “Maximization of a convex quadratic function under linear constraints”, Mathematical Programming 11 (1976), 117–127.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    P. Kough, “The indefinite quadratic programming problem”, Operations Research 27 (1979) 516–533.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    V.Yu. Lebedev, “Method for maximizing a positive-definite quadratic form on a polytope”, Zurnal Vyčislitelnoi Matematiki i Matematitcheskoi Fiziki 22 (1982) 1344–1351 (Russian). Translated in U.S.S.R. Computational Mathematics and Mathematical Physics 22 (1982) 62–70.zbMATHGoogle Scholar
  13. [13]
    P.A. Meyer, Probabilités et potentiels (Herman, Paris, 1966).Google Scholar
  14. [14]
    B.M. Mukhamediev, “Approximate method of solving concave programming problems”, Zurnal Vyčislitelnoi Matematiki i Matematitcheskoi Fiziki 22 (1982) 727–732 (Russian). Translated in U.S.S.R Computational Mathematics and Mathematical Physics 22 (1982) 238–245.zbMATHMathSciNetGoogle Scholar
  15. [15]
    K. Ritter, “A method for solving maximum problems with a nonconcave quadratic objective function”, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 4 (1966) 340–351.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, NJ, 1970).zbMATHGoogle Scholar
  17. [17]
    J.B. Rosen, “Global minimization of a linearly constrained concave function by partition of feasible domain”, Mathematics of Operations Research 8 (1983) 215–230.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    J.B. Rosen, “Parametric global minimization for large scale problems”, Mathematics of Operations Research (to appear).Google Scholar
  19. [19]
    I. Singer, “Minimization of continuous convex functionals on complements of convex subsets of locally convex spaces”, Mathematische Operationsforschung und Statistik, Serie Optimization 11 (1980) 221–234.zbMATHGoogle Scholar
  20. [20]
    K. Tammer, “Möglichkeiten zur Anwendung der Erkenntnis der parametrischen Optimierung für die Lösung indefiniter quadratischer Optimierungsprobleme”, Mathematische Operations-Forschung und Statistik, Serie Optimization 7 (1976) 209–222.MathSciNetGoogle Scholar
  21. [21]
    Ng.v. Thoai and H. Tuy, “Convergent algorithms for minimizing a concave function”, Mathematics of Operations Research 5 (1980) 556–566.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [22]
    J.F. Toland, “On subdifferential calculus and duality in nonconvex optimization”, in: Analyse non convexe (1977, Pau), Bulletin Société Mathématique de France, Mémoire 60 (1979) pp. 177–183.Google Scholar
  23. [23]
    H. Tuy, “Concave programming under linear constraints”, Doklady Akademii Nauk 159 (1964) 32–35. Translated in Soviet Mathematics Doklady 5 (1964) 1437–1440.Google Scholar
  24. [24]
    H. Tuy, “The Farkas-Minkowski theorem and extremum problems”, in: J. Los and M.W. Los, eds., Mathematical models in economics (Warszawa, 1974) pp. 379–400.Google Scholar
  25. [25]
    H. Tuy, “Global maximization of a convex function over a closed convex, not necessarily bounded set”, Cahiers de Mathématiques de la Décision, 8223, CEREMADE, Université Paris-Dauphine (Paris, 1982).Google Scholar
  26. [26]
    H. Tuy, “Improved algorithm for solving a class of global minimization problems”, Preprint, Hanoi Institute of Mathematics (Hanoi, 1983).Google Scholar
  27. [27]
    H. Tuy, “On outer approximation methods for solving concave minimization problems”, Submitted to Acta Mathematica Vietnamica 1983.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1987

Authors and Affiliations

  • Hoang Tuy
    • 1
  1. 1.Institute of MathematicsHanoiVietnam

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