  We present a general theory of exact penalty functions with vectorial
(multidimensional) penalty parameter for optimization problems in infinite
dimensional spaces. In comparison with the scalar case, the use of vectorial
penalty parameters provides much more flexibility, allows one to adaptively and
independently take into account the violation of each constraint during an
optimization process, and often leads to a better overall performance of an
optimization method using an exact penalty function. We obtain sufficient
conditions for the local and global exactness of penalty functions with
vectorial penalty parameters and study convergence of global exact penalty
methods with several different penalty updating strategies. In particular, we
present a new algorithmic approach to an analysis of the global exactness of
penalty functions, which contains a novel characterisation of the global
exactness property in terms of behaviour of sequences generated by certain
optimization methods.
