Functional Inequalities in the Absence of Convexity and Lower Semicontinuity with Applications to Optimization
本文在函数映射不满足凸性和下半连续标准假设时，通过第二共轭条件扩展了函数不等式理论，并应用于推导非凸系统的次微分法则、Farkas引理推广以及无穷非凸优化问题的最优性条件和对偶理论。
In this paper we extend some results in [Dinh, Goberna, López, and Volle, Set-Valued Var. Anal., to appear] to the setting of functional inequalities when the standard assumptions of convexity and lower semicontinuity of the involved mappings are absent. This extension is achieved under certain condition relative to the second conjugate of the involved functions. The main result of this paper, Theorem 1, is applied to derive some subdifferential calculus rules and different generalizations of the Farkas lemma for nonconvex systems, as well as some optimality conditions and duality theory for infinite nonconvex optimization problems. Several examples are given to illustrate the significance of the main results and also to point out the potential of their applications to get various extensions of Farkas-type results and to the study of other classes of problems such as variational inequalities and equilibrium models.