  This paper investigates continuity properties of value functions and
solutions for parametric optimization problems. These problems are important in
operations research, control, and economics because optimality equations are
their particular cases. The classic fact, Berge's maximum theorem, gives
sufficient conditions for continuity of value functions and upper
semicontinuity of solution multifunctions. Berge's maximum theorem assumes that
the objective function is continuous and the multifunction of feasible sets is
compact-valued. These assumptions are not satisfied in many applied problems,
which historically has limited the relevance of the theorem. This paper
generalizes Berge's maximum theorem in three directions: (i) the objective
function may not be continuous, (ii) the multifunction of feasible sets may not
be compact-valued, and (iii) necessary and sufficient conditions are provided.
To illustrate the main theorem, this paper provides applications to inventory
control and to the analysis of robust optimization over possibly noncompact
action sets and discontinuous objective functions.
