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A generalized nonconvex duality with zero gap and applications

  • Phan Thien Thach
I Optimality And Duality
  • 113 Downloads
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 180)

Abstract

In a quasiconvex minimization over a convex set although a local minimum may not global, we obtain a convex type duality scheme. In convex type problems, the optimality criterion at a feasible solution z is of the form
$$0 \in A(z)$$
(11)
where A(z) is a convex set depending on z in the dual space. In nonconvex type problems the global optimality criterion at a feasible solution z is of the form
$$C(z) \subset {\rm A}(z)$$
(12)
where C(z), A(z) are convex sets depending on z in the dual space (see Thach 10). Of course criterion (12) becomes criterion (11) when C(z) is reduced to {0}. Thus, in some senses, criterion (12) is a generalization of criterion (11). But the duality obtained from (12) is quite different from the duality obtained from (11). The duality obtained from (12) is of the form max=−min (max=max or min=min), whereas the duality obtained from (11) is of the form min=max (max=−max or min=−min). By the duality we can reduce a convex type optimization problem to solving a system of inequations but we could not do the similar thing for a nonconvex type optimization problems.

Keywords

Quasiconvex Function Duality Scheme Convex Type Reverse Convex Outer Approximation Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© International Federation for Information Processing 1992

Authors and Affiliations

  • Phan Thien Thach
    • 2
    • 1
  1. 1.Institute of MathematicsHanoiVietnam
  2. 2.AvH-Foundation at Trier UniversityVietnam

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