Lagrangian Duality and Cone Convexlike Functions
Journal of Optimization Theory and Applications , Volume 134 - Issue 2 p. 207- 222
In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called image space approach initiated by Giannessi.
|Econometric Institute Reprint Series
|Journal of Optimization Theory and Applications
|Erasmus Research Institute of Management
Frenk, H., & Kassay, G. (2007). Lagrangian Duality and Cone Convexlike Functions. Journal of Optimization Theory and Applications, 134(2), 207–222. doi:10.1007/s10957-007-9221-1