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We present a novel algorithm that fuses the existing convex-programming based approach with heuristic information to find optimality guarantees and near-optimal paths for the Shortest Path Problem in the Graph of Convex Sets (SPP-GCS). Our method, inspired by A* initiates a best-first-like procedure from a designated subset of vertices and iteratively expands it until further growth is neither possible nor beneficial. Traditionally, obtaining solutions with bounds for an optimization problem involves solving a relaxation, modifying the relaxed solution to a feasible one, and then comparing the two solutions to establish bounds. However, for SPP-GCS, we demonstrate that reversing this process can be more advantageous, especially with Euclidean travel costs. In other words, we initially employ A* to find a feasible solution for SPP-GCS, then solve a convex relaxation restricted to the vertices explored by A* to obtain a relaxed solution, and finally, compare the solutions to derive bounds. We present numerical results to highlight the advantages of our algorithm over the existing approach in terms of the sizes of the convex programs solved and computation time. 

Although electricity transmission systems are typically very robust, the impacts that arise when they are disrupted motivate methods for analyzing outage risk. For example, N-k interdiction models were developed to characterize disruptions by identifying the sets of k power system components whose failure results in “worst case” outages. While such models have advanced considerably, they generally neglect how failures outside the power system can cause large-scale outages. Specifically, failures in natural gas pipeline networks that provide fuel for gas-fired generators can affect the function of the power grid. In this study, we extend N-k interdiction modeling to gas pipeline networks. We use recently developed convex relaxations for natural gas flow equations to yield tractable formulations for identifying sets of k components whose failure can cause curtailment of natural gas delivery. We then present a novel cutting-plane algorithm to solve these problems. Finally, we use test instances to analyze the performance of the approach in conjunction with simulations of outage effects on electrical power grids.