We present necessary and sufficient conditions for reachable sets of discrete-time systems x(k+1) = F (k, x(k)) to be convex. In particular, the set of states reachable at a given time from a sufficiently small ellipsoid of initial states is always convex if F is smooth enough, and we provide explicit bounds on the size of those ellipsoids. Our results imply that outer discrete approximations w...

In this paper we consider cardinality-constrained convex programs that minimize a convex function subject to a cardinality constraint and other linear constraints. This class of problems has found many applications, including portfolio selection, subset selection and compressed sensing. We propose a successive convex approximation method for this class of problems in which the cardinality funct...

We provide a general framework to construct finite-dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise-...