Local-global minimum property in unconstrained minimization problems (Q467402)

The author investigates classes of functions which possess the local-global minimum property. A sufficient condition is provided for functions \(f : X\to \mathbb R\), where \(X\) is a metric space, in terms of lower semicontinuity of the lower-level-set function \(L_f\). The class of quasi-connected (connected) functions on topological spaces is investigated, and it is shown that every connected function has the local-global minimum property. If \(X\) is a real normed space and \(f : D\subseteq X\to \mathbb R\) is regularly connected, then a necessary condition for the local optimality in terms of variational inequality turns out to be sufficient for the global one. In the last section, as an application, a classical problem from calculus of variations is considered. In particular, the well-known property that the Euler-Lagrange equation is necessary and sufficient for a solution, under appropriate smoothness conditions, is generalized.