Formulating the alternating current optimal power flow (ACOPF) as a polynomial optimization problem makes it possible to solve large instances in practice and to guarantee asymptotic convergence in theory.We formulate the ACOPF as a degree-two polynomial program and study two approaches to solving it via convexifications. In the first approach, we tighten the first order relaxation of the non-convex quadratic program by adding valid inequalities. In the second approach, we exploit the structure of the polynomial program by using a sparse variant of Lasserre's hierarchy. This allows us to solve instances of up to 39 buses to global optimality and to provide strong bounds for the Polish network within an hour.

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The minimum k-partition (MkP) problem is the problem of partitioning the set of vertices of a graph into k disjoint subsets so as to minimize the total weight of the edges joining vertices in the same partition. The main contribution of this paper is the design and implementation of a branch-and-cut algorithm based on semidefinite programming (SBC) for the MkP problem. The two key ingredients for this algorithm are: the combination of semidefinite programming with polyhedral results; and a novel iterative clustering heuristic (ICH) that finds feasible solutions for the MkP problem. We compare ICH to the hyperplane rounding techniques of Goemans and Williamson and of Frieze and Jerrum, and the computational results support the conclusion that ICH consistently provides better feasible solutions for the MkP problem. ICH is used in our SBC algorithm to provide feasible solutions at each node of the branch-and-bound tree. The SBC algorithm computes globally optimal solutions for dense graphs with up to 60 vertices, for grid graphs with up to 100 vertices, and for different values of k, providing a fast exact approach for k ≥ 3.Keywords Minimum k-partition · Semidefinite programming · Branch-and-cut · Polyhedral cuts Dedicated to the memory of Peter L. Hammer and in celebration of his outstanding contribution to the field of operations research.

Long Range (LoRa) is a wireless communication standard specifically targeted for resource-constrained Internet of Things (IoT) devices. LoRa is a promising solution for smart city applications as it can provide long-range connectivity with a low energy consumption. The number of LoRa-based networks is growing due to its operation in the unlicensed radio bands and the ease of network deployments. However, the scalability of such networks suffers as the number of deployed devices increases. In particular, the network performance drops due to increased contention and interference in the unlicensed LoRa radio bands. This results in an increased number of dropped messages and, therefore, unreliable network communications. Nevertheless, network performance can be improved by appropriately configuring the radio parameters of each node. To this end, we formulate integer linear programming models to configure LoRa nodes with the optimal parameters that allow all devices to reliably send data with a low energy consumption. We evaluate the performance of our solutions through extensive network simulations considering different types of realistic deployments. We find that our solution consistently achieves a higher delivery ratio (up to 8% higher) than the state of the art with minimal energy consumption. Moreover, the higher delivery ratio is achieved by a large percentage of nodes in each network, thereby resulting in a fair allocation of radio resources. Finally, the optimal network configurations are obtained within a short time, usually much faster than the state of the art. Thus, our solution can be readily used by network operators to determine optimal configurations for their IoT deployments, resulting in improved network reliability.