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On Lagrange Duality for Several Classes of Nonconvex Optimization Problems

  • Ewa M. Bednarczuk
  • Monika SygaEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

We investigate a general framework for studying Lagrange duality in some classes of nonconvex optimization problems. To this aim we use an abstract convexity theory, namely \(\varPhi \)-convexity theory, which provides tools for investigating nonconvex problems in the spirit of convex analysis (via suitably defined subdifferentials and conjugates). We prove a strong Lagrangian duality theorem for optimization of \(\varPhi _{lsc}\)-convex functions which is based on minimax theorem for general \(\varPhi \)-convex functions. The class of \(\varPhi _{lsc}\)-convex functions contains among others, prox-regular functions, DC functions, weakly convex functions and para-convex functions. An important ingredient of the study is the regularity condition under which our strong Lagrangian duality theorem holds. This condition appears to be weaker than a number of already known regularity conditions, even for convex problems.

Keywords

Abstract convexity \(\varPhi \)-convexity Minimax theorem Lagrangian duality Nonconvex optimization Weakest constraint qualification condition 

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Systems Research Institute, Polish Academy of SciencesWarsawPoland
  2. 2.Warsaw University of TechnologyFaculty of Mathematics and Information ScienceWarsawPoland

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