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We study a single-commodity Robust Network Design problem (RND) in which an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. In each scenario, a subset of the nodes is exchanging flow. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. Previously conducted computational investigations on the problem motivated the study of the complexity of some special cases and we present complexity results on them, including hypercubes. In turn, these results lead to the definition of new instances (random graphs with {-1,0,1} balances) that are computationally hard for the natural flow formulation. These instances are then solved by means of a new heuristic algorithm for RND, which consists of three phases. In the first phase the graph representing the network is reduced by heuristically deleting a subset of the arcs, and a feasible solution is built. The second phase consists of a neighborhood search on the reduced graph based on a Mixed-Integer (Linear) Programming (MIP) flow model. Finally, the third phase applies a proximity search approach to further improve the solution, taking into account the original graph. The heuristic is tested on the new instances, and the comparison with the solutions obtained by Cplex on a natural flow formulation shows the effectiveness of the proposed method.

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We describe our experience in obtaining significant computational improvements in the solution of large stochastic unit commitment problems. The model we use is a stochastic version of a planning model used by the California Independent System Operator, covering the entire WECC western regional grid. We solve daily hour-timestep stochastic unit commitment problems using a new progressive hedging approach that features linear subproblems and guided solves for finding feasible solutions. For stochastic problems with 5 scenarios, the algorithm produces near-optimal solutions with a 6 times improvement in serial solution time, and over 20 times improvement when run in parallel; for previously unsolvable stochastic problems, we obtain near-optimal solutions within a couple of hours. We note that although this algorithm is demonstrated for stochastic unit commitment problems, the algorithm itself is suitable for application to generic stochastic optimization problems.

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