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.
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Bednarczuk, E.M., Syga, M. (2020). On Lagrange Duality for Several Classes of Nonconvex Optimization Problems. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_18
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