The bound quiver of a split extension

In this paper, we give a sufficient (which is also necessary under a compatibility hypothesis) condition on a set of arrows in the quiver of an algebra A so that A is a split extension of A/M, where M is the ideal of A generated by the classes of these arrows. We also compare the notion of split extension with that of semiconvex extension of algebras.

UNIQUENESS OF THE EXTENSION OF 2-HOMOGENEOUS POLYNOMIALS

Homogeneous polynomials of degree 2 on the complex Banach space c(0)(l(n)(2)) are shown to have unique norm-preserving extension to the bidual space. This is done by using M-projections and extends the analogous result for c(0) proved by P.-K. Lin.

On Approximate KKT Condition and its Extension to Continuous Variational Inequalities

In this work, we introduce a necessary sequential Approximate-Karush-Kuhn-Tucker (AKKT) condition for a point to be a solution of a continuous variational inequality, and we prove its relation with the Approximate Gradient Projection condition (AGP) of Garciga-Otero and Svaiter. We also prove that a slight variation of the AKKT condition is sufficient for a convex problem, either for variational inequalities or optimization. Sequential necessary conditions are more suitable to iterative methods than usual punctual conditions relying on constraint qualifications. The AKKT property holds at a solution independently of the fulfillment of a constraint qualification, but when a weak one holds, we can guarantee the validity of the KKT conditions.