Software for cut-generating functions in the Gomory--Johnson model and beyond

We present software for investigations with cut generating functions in the Gomory-Johnson model and extensions, implemented in the computer algebra system SageMath.Comment: 8 pages, 3 figures; to appear in Proc. International Congress on Mathematical Software 201

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject

Combining problem structure and basis reduction to solve a class of hard integer programs

We consider a hard integer programming problem that is difficult for the standard branch-and-bound approach even for small instances. A reformulation based on lattice basis reduction is known to be more effective. However the step to compute the reduced basis, even if it is found in polynomial time, becomes a bottleneck for small to medium instances. By using the structure of the problem, we show that we can decompose the problem and obtain the basis by taking the kronecker product of two smaller bases easier to compute. Furthermore, if the two small bases are reduced, the kronecker product is also reduced up to a reordering of the vectors. Computational results show the gain from such an approach

Approximation-friendly discrepancy rounding

Rounding linear programs using techniques from discrepancy is a recent approach that has been very successful in certain settings. However this method also has some limitations when compared to approaches such as randomized and iterative rounding. We provide an extension of the discrepancy-based rounding algorithm due to Lovett-Meka that (i) combines the advantages of both randomized and iterated rounding, (ii) makes it applicable to settings with more general combinatorial structure such as matroids. As applications of this approach, we obtain new results for various classical problems such as linear system rounding, degree-bounded matroid basis and low congestion routing

A quantitative Doignon-Bell-Scarf theorem

The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions to systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer k, we prove that there exists a constant c(n,k), depending only on the dimension n and k, such that if a polyhedron {x∈Rn: Ax≤b} contains exactly k integer points, then there exists a subset of the rows, of cardinality no more than c(n,k), defining a polyhedron that contains exactly the same k integer points. In this case c(n,0)=2n as in the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. We work on both upper and lower bounds for the constant c(n,k) and discuss some consequences, including a Clarkson-style algorithm to find the l-th best solution of an integer program with respect to the ordering induced by the objective function

Rescaled coordinate descent methods for linear programming

We propose two simple polynomial-time algorithms to find a positive solution to Ax = 0. Both algorithms iterate between coordinate descent steps similar to von Neumann’s algorithm, and rescaling steps. In both cases, either the updating step leads to a substantial decrease in the norm, or we can infer that the condition measure is small and rescale in order to improve the geometry. We also show how the algorithms can be extended to find a solution of maximum support for the system Ax =0, x ≥ 0. This is an extended abstract. The missing proofs will be provided in the full versio