Tight and Compact Sample Average Approximation for Joint Chance-Constrained Problems with Applications to Optimal Power Flow.

In this paper, we tackle the resolution of chance-constrained problems reformulated via sample average approximation. The resulting data-driven deterministic reformulation takes the form of a large-scale mixed-integer program (MIP) cursed with Big-Ms. We introduce an exact resolution method for the MIP that combines the addition of a set of valid inequalities to tighten the linear relaxation bound with coefficient strengthening and constraint screening algorithms to improve its Big-Ms and considerably reduce its size. The proposed valid inequalities are based on the notion of k-envelopes and can be computed off-line using polynomial-time algorithms and added to the MIP program all at once. Furthermore, they are equally useful to boost the strengthening of the Big-Ms and the screening rate of superfluous constraints. We apply our procedures to a probabilistically constrained version of the DC optimal power flow problem with uncertain demand. The chance constraint requires that the probability of violating any of the power system’s constraints be lower than some positive threshold. In a series of numerical experiments that involve five power systems of different size, we show the efficiency of the proposed methodology and compare it with some of the best performing convex inner approximations currently available in the literature.This work was supported in part by the European Research Council under the EU Horizon 2020 research and innovation program [Grant 755705], in part by the Spanish Ministry of Science and Innovation [Grant AEI/10.13039/501100011033] through project PID2020-115460GB-I00, and in part by the Junta de Andalucía and the European Regional Development Fund through the research project P20_00153. Á. Porras is also financially supported by the Spanish Ministry of Science, Innovation and Universities through the University Teacher Training Program with fellowship number FPU19/03053

Execution and authentication of function queries

We introduce a new query primitive called Function Query (FQ). An FQ operates on a set of math functions and retrieves the functions whose output with a given input satisfies a query condition (e.g., being among top-k, within a given range). While FQ finds its natural uses in querying a database of math functions, it can also be applied on a database of discrete values. We show that by interpreting the database as a set of user-defined functions, FQ can retrieve the information like existing analytic queries such as top-k query and scalar product query and even more. Our research addresses the challenges of FQ execution and authentication. The former is how to minimize the computation and storage costs in processing an FQ, whereas the latter, how to verify that the result of an FQ returned by a potentially untrustworthy server is indeed correct. Our solutions are inspired from the observations that 1) the intersections of a set of continuous functions partition their domain into a number of subdomains, and 2) in each of these subdomains, the functions can be sorted based on their output. We prove the correctness of the proposed techniques and evaluate their performance through analysis, prototyping, and experiments using both synthetic and real-world data. In all settings, our techniques exhibit excellent performance. In addition to FQ, our research has developed another query primitive called Improvement Query, which we also include in this dissertation