Open Journal of Mathematical Optimization

Consider the problem of minimizing functions that are Lipschitz and convex, but not necessarily differentiable. We construct a function from this class for which the $Tþ$ iterate of subgradient descent has error $\Omega (log\left(T\right)/\sqrt{T})$. This matches a known upper bound of $O(log\left(T\right)/\sqrt{T})$. We prove analogous results for functions that are additionally strongly convex. There exists such a function for which the error of the $Tþ$ iterate of subgradient descent has error $\Omega (log(T)/T)$, matching a known upper bound of $O(log(T)/T)$. These results resolve a question posed by Shamir (2012).

We consider structured minimization problems subject to smooth inequality constraints and present a flexible algorithm that combines interior point (IP) and proximal gradient schemes. While traditional IP methods cannot cope with nonsmooth objective functions and proximal algorithms cannot handle complicated constraints, their combined usage is shown to successfully compensate the respective shortcomings. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems, thus bridging the gap with previous works that successfully addressed the convex case. Our interior proximal gradient algorithm benefits from warm starting, generates strictly feasible iterates with decreasing objective value, and returns after finitely many iterations a primal-dual pair approximately satisfying suitable optimality conditions. As a byproduct of our analysis of proximal gradient iterations we demonstrate that a slight refinement of traditional backtracking techniques waives the need for upper bounding the stepsize sequence, as required in existing results for the nonconvex setting.