Continuity of Optimal Solution Functions and their Conditions on Objective Functions

When an optimization problem has a set of parameters besides decision variables, the mapping from the parameters to the corresponding optimal solutions is determined accordingly. This mapping, either single-valued or set-valued, is called an optimal solution function in the literature of parametric optimization and a generalized inverse mapping in the literature of inverse problems. We present a pair of relationships between classes of objective functions and optimal solution functions in the form of two theorems. The first theorem states that any continuous quasi-convex objective function results in an upper semicontinuous optimal solution function. This is a variant of the Berge maximum theorem, wherein we replace the condition of compact-valuedness of the feasible solution function with other assumptions. The second theorem states that any optimal solution function with a certain continuity can be realized by a continuous convex objective function if the feasible solutions form an affine subspace. This is a variant of Komiya's inverse of Berge's theorem; we strengthened the assertion of convexity by assuming the costs to be nonparameterized, though we only consider the case of linear equation constraints.