Diameter Two Properties, Convexity and Smoothness

Graduated Non-Convexity by Smoothness Focusing

Noise-corrupted signals and images can be reconstructed by regularization. If discontinuities must be preserved in the reconstruction, a non-convex solution space is implied. The solution of minimum energy can be approximated by the Graduated Non-Convexity (GNC) algorithm. The GNC approximates the non-convex solution space by a convex solution space, and varies the solution space slowly towards...

RSG: Beating Subgradient Method without Smoothness and Strong Convexity

In this paper, we study the efficiency of a Restarted SubGradient (RSG) method that periodically restarts the standard subgradient method (SG). We show that, when applied to a broad class of convex optimization problems, RSG method can find an ǫ-optimal solution with a low complexity than SG method. In particular, we first show that RSG can reduce the dependence of SG’s iteration complexity on ...

On smoothness properties of optimal value functions at the boundary of their domain under complete convexity

This article studies continuity and directional differentiability properties of optimal value functions, in particular at boundary points of their domain. We extend and complement standard continuity results from [12] for abstract feasible set mappings under complete convexity as well as standard differentiability results from [13] for feasible set mappings in functional form under the Slater c...

Multimodularity, Convexity, and Optimization Properties

We investigate in this paper the properties of multimodular functions. In doing so we give alternative proofs for properties already established by Hajek, and we extend his results. In particular, we show the relation between convexity and multimodularity, which allows us to restrict the study of multimodular functions to convex subsets of Z m. We then obtain general optimization results for av...

Convexity Properties of Graamannians