Open Journal of Mathematical Optimization

Given a nominal combinatorial optimization problem, we consider a robust two-stages variant with polyhedral cost uncertainty, called Decision-Dependent Information Discovery (DDID). In the first stage, DDID selects a subset of uncertain cost coefficients to be observed, and in the second-stage, DDID selects a solution to the nominal problem, where the remaining cost coefficients are still uncertain. Given a compact linear programming formulation for the nominal problem, we provide a mixed-integer linear programming (MILP) formulation for DDID. The MILP is compact if the number of constraints describing the uncertainty polytope other than lower and upper bounds is constant. The proof of this result involves the generalization to any polyhedral uncertainty set of a classical result, showing that solving a robust combinatorial optimization problem with cost uncertainty amounts to solving several times the nominal counterpart. We extend this formulation to more general nominal problems through column generation and constraint generation algorithms. We illustrate our reformulations and algorithms numerically on the selection problem, the orienteering problem, and the spanning tree problem.

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).