Sequential Semicontinuity Research Articles

Discontinuous but Well‐Posed Optimization Problems

Using classes of sequentially pseudocontinuous functions, recently introduced by the authors, the aim of the paper is to investigate Tikhonov and parametric well-posedness for optimization problems when the objective functions are not necessarily sequentially lower semicontinuous. Sequential pseudocontinuity is a property more general than sequential semicontinuity and finds motivations in choice theory, since the continuity of preference relations on first countable topological spaces is characterized by the sequential pseudocontinuity of any utility function. Examples show that it is not possible to improve the results with other well-known classes of discontinuous functions.

Localisation of continuity to bounded sets for nonmetrisable vector topologies and its applications to economic equilibrium theory

We present two ‘localisation’ techniques which facilitate verifications of the topological properties of sets, functions and correspondences needed in, e.g., economic equilibrium analysis with infinite-dimensional commodity spaces. The first uses the Krein-Smulian theorem, which shows that weak ∗ upper semicontinuity of a concave function on a dual Banach space is equivalent to bounded weak ∗ u.s. continuity. The second is based on the continuity of lattice operations: for a nondecreasing function on a topological vector lattice, we show that lower semicontinuity on a set bounded from below is equivalent to l.s. continuity on bounded subsets. In the case of L ∞, we use convergence in measure to establish that sequential semicontinuity, lower for the Mackey topology or upper for the weak ∗ one, is equivalent to semicontinuity. This greatly simplifies some arguments; e.g., the Mackey continuity of a concave, nondecreasing integral functional on L + ∞ becomes an immediate consequence of Lebesgue's Dominated Convergence Theorem. Other uses in mathematical economics are also discussed.