The p-center problem consists in selecting p facilities from a set of possible sites and allocating a set of clients to them in such a way that the maximum distance between a client and the facility to which it is allocated is minimized. This paper proposes a new scalable exact solution algorithm based on client clustering and an iterative distance rounding procedure. The client clustering enables to initialize and update a subset of clients for which the p-center problem is iteratively solved. The rounding drastically reduces the number of distinct distances considered at each iteration. Our algorithm is tested on 396 benchmark instances with up to 1.9 million clients and facilities. We outperform the two state-of-the-art exact methods considered when p is not very small (i.e., p > 5).

Future energy projections and their inherent uncertainty play a key role in the design of photovoltaic-battery energy storage systems (PV-BESS) for household use. In this study, both stochastic and robust optimization techniques are simultaneously integrated into a Hybrid Adaptive Robust-Stochastic Optimization (HARSO) model. Uncertainty in future PV generation is addressed using a stochastic approach, while uncertainty in power demand is handled through robust optimization. To solve the tri-level structure emerging from the hybrid approach, a Column-and-Constraint Generation (CCG) algorithm is implemented. The model also accounts for battery degradation by considering multiple commercially available battery chemistries, enabling a more realistic evaluation of long-term system costs and performance. To demonstrate its applicability, the model is applied to a case study involving the optimal design of a PV-BESS system for a household in Spain. The empirical analysis includes both first-life (FL) and second-life (SL) batteries with different chemistries, providing a comprehensive evaluation of design alternatives under uncertainty. Results indicate that the optimal solution is highly dependent on the level of robustness considered, leading to a shift in design strategy. Under less conservative settings, robustness is achieved by increasing battery capacity, while higher levels of conservatism favor expanding PV capacity to meet demand.

The proportional odds cumulative logit model (POCLM) is a standard regression model for an ordinal response. Ordinality of predictors can be incorporated by monotonicity constraints for the corresponding parameters. It is shown that estimators defined by optimization, such as maximum likelihood estimators, for an unconstrained model and for parameters in the interior set of the parameter space of a constrained model are asymptotically equivalent. This is used in order to derive asymptotic confidence regions and tests for the constrained model, involving simple modifications for finite samples. The finite sample coverage probability of the confidence regions is investigated by simulation. Tests concern the effect of individual variables, monotonicity, and a specified monotonicity direction. The methodology is applied on real data related to the assessment of school performance.

This study introduces adaptive robust optimization (ARO) and adaptive robust stochastic optimization (ARSO) approaches to address long- and short-term uncertainties in the optimal sizing and placement of distributed energy resources in distribution networks. ARO models uncertainty using a Budget of Uncertainty (BoU), while ARSO distinguishes long-term (LT) demand (via BoU) and short-term (ST) photovoltaics generation (via scenarios). Adapted Benders cutting plane algorithms are presented to tackle the tri-level optimization challenges. The experiments consider a modified version of the IEEE 33 bus system to test these two approaches and also compare them with traditional robust and stochastic optimization models. The results indicate that distinguishing between LT and ST uncertainties using a hybrid formulation such ARSO yields a solution closer to the optimal solution under perfect information than ARO.