Characterization of minimal elements in minimization problems with constraints

LetX be a compact Hausdorff space andC(X) be the set of all continuous functions defined onX. LetVC(X), and consider the problem of minimizing sup_xXW[x,v(x)], withvV. The functionW is a generalized weight function and can be chosen such that certain constraints are included.The notions of critical point and extremal signature are used to formulate characterization theorems for a minimal element inV. It is shown that these theorems hold only under certain conditions ofV andW. The results obtained are applied to the problem of the Chebyshev approximation with constraints and to the problem of optimization with strictly quasiconvex constraints.The work of the second author was supported in part by the Alexander von Humboldt Stiftung and the DAAD.