Global minimization of a difference of two convex functions

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


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.


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


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Copyright information

© The Mathematical Programming Society, Inc. 1987

Authors and Affiliations

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

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