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<p>In last years practitioners and theorists are paying attention on general nonconvex optimization problems due to the presence of modern tools to face such problems. Our focus in this paper is dealing with the infinite dimensional setting. Firstly, associated to two different concepts of interiority, we introduce two notions of almost convexity, and establish some relationships between the function itself and its lower semicontinuous hull. Another important issue in the study of optimization problems is to know when the Fenchel subdifferential of a given nonconvex function at a reference point is nonempty. This property is analyzed in terms of the properness condition introduced, in the context of mathematical economics, by Mas-Colell. Applications to provide necessary and/or sufficient conditions for the fulfillment of strong duality property are given.</p>

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Mas-Colell Properness Condition and Almost Convexity in Nonconvex Optimization

  • 期刊论文

  • 发表日期：

    2025年4月22日

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  • Fabián Flores-Bazán

摘要

In last years practitioners and theorists are paying attention on general nonconvex optimization problems due to the presence of modern tools to face such problems. Our focus in this paper is dealing with the infinite dimensional setting. Firstly, associated to two different concepts of interiority, we introduce two notions of almost convexity, and establish some relationships between the function itself and its lower semicontinuous hull. Another important issue in the study of optimization problems is to know when the Fenchel subdifferential of a given nonconvex function at a reference point is nonempty. This property is analyzed in terms of the properness condition introduced, in the context of mathematical economics, by Mas-Colell. Applications to provide necessary and/or sufficient conditions for the fulfillment of strong duality property are given.