Nearest points and delta convex functions in Banach spaces

Given a closed set $C$ in a Banach space $(X, \|\cdot\|)$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_C(x) =\|x-z\|$, where $d_C$ is the distance of $x$ from $C$. We shortly survey the problem of studying how large is the set of points in $X$ which have nearest points in $C$. We then discuss the topic of delta-convex functions and how it is related to finding nearest points.Comment: To appear in Bull. Aust. Math. So

Outer Approximation Algorithms for DC Programs and Beyond

We consider the well-known Canonical DC (CDC) optimization problem, relying on an alternative equivalent formulation based on a polar characterization of the constraint, and a novel generalization of this problem, which we name Single Reverse Polar problem (SRP). We study the theoretical properties of the new class of (SRP) problems, and contrast them with those of (CDC)problems. We introduce of the concept of ``approximate oracle'' for the optimality conditions of (CDC) and (SRP), and make a thorough study of the impact of approximations in the optimality conditions onto the quality of the approximate optimal solutions, that is the feasible solutions which satisfy them. Afterwards, we develop very general hierarchies of convergence conditions, similar but not identical for (CDC) and (SRP), starting from very abstract ones and moving towards more readily implementable ones. Six and three different sets of conditions are proposed for (CDC) and (SRP), respectively. As a result, we propose very general algorithmic schemes, based on approximate oracles and the developed hierarchies, giving rise to many different implementable algorithms, which can be proven to generate an approximate optimal value in a finite number of steps, where the error can be managed and controlled. Among them, six different implementable algorithms for (CDC) problems, four of which are new and can't be reduced to the original cutting plane algorithm for (CDC) and its modifications; the connections of our results with the existing algorithms in the literature are outlined. Also, three cutting plane algorithms for solving (SRP) problems are proposed, which seem to be new and cannot be reduced to each other