Outer and Inner Approximation

  • Hoang Tuy
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 22)


One of the’ most popular methods of global optimization consists in approximating a given problem by a sequence of easier relaxed problems constructed in such a way that the sequence of solutions of these relaxed problems converges to a solution of the given problem. This approach, first introduced in convex programming in the late fifties (Cheney and Goldstein (1959), Kelley (1960)) was later extended, under the name of outer approximation, to concave minimization under convex constraints (Hoffman (1981), Thy (1983), Thieu, Tam and Ban (1983), Thy and Horst (1988)) and to more general nonconvex optimization problems (Thy (1983), Mayne and Polak (1984), Thy (1987a)). Referring to the fact that cutting planes are used to eliminate unfit solutions of relaxed problems, this approach is sometimes also called a cutting plane method. It should be noted, however, that in an outer approximation procedure cuts are always conjunctive, i.e. the polyhedron resulting from the cuts is always the intersection of all the cuts performed.


Accumulation Point Global Optimal Solution Outer Approximation Combine Algorithm Good Feasible Solution 
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Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

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

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