A polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs

We consider the multilinear polytope defined as the convex hull of the set of binary points satisfying a collection of multilinear equations. The complexity of the facial structure of the multilinear polytope is closely related to the acyclicity degree of the underlying hypergraph. We obtain a polynomial-size extended formulation for the multilinear polytope of beta-acyclic hypergraphs, hence characterizing the acyclic hypergraphs for which such a formulation can be constructed

The pseudo-Boolean polytope and polynomial-size extended formulations for binary polynomial optimization

With the goal of obtaining strong relaxations for binary polynomial optimization problems, we introduce the pseudo-Boolean polytope defined as the convex hull of the set of binary points satisfying a collection of equations containing pseudo-Boolean functions. By representing the pseudo-Boolean polytope via a signed hypergraph, we obtain sufficient conditions under which this polytope has a polynomial-size extended formulation. Our new framework unifies and extends all prior results on the existence of polynomial-size extended formulations for the convex hull of the feasible region of binary polynomial optimization problems of degree at least three

Strengthening QC relaxations of optimal power flow problems by exploiting various coordinate changes

Motivated by the potential for improvements in electric power system economics, this dissertation studies the AC optimal power flow (AC OPF) problem. An AC OPF problem optimizes a specified objective function subject to constraints imposed by both the non-linear power flow equations and engineering limits. The difficulty of an AC OPF problem is strongly connected to its feasible space\u27s characteristics. This dissertation first investigates causes of nonconvexities in AC OPF problems. Understanding typical causes of nonconvexities is helpful for improving AC OPF solution methodologies. This dissertation next focuses on solution methods for AC OPF problems that are based on convex relaxations. The quadratic convex (QC) relaxation is one promising approach that constructs convex envelopes around the trigonometric and product terms in the polar representation of the power flow equations. This dissertation proposes several improvements to strengthen QC relaxations of OPF problems. The first group of improvements provides tighter envelopes for the trigonometric functions and product terms in the power flow equations. Methods for obtaining tighter envelopes includes implementing Meyer and Floudas envelopes that yield the convex hull of trilinear monomials. Furthermore, by leveraging a representation of line admittances in polar form, this dissertation proposes tighter envelopes for the trigonometric terms. Another proposed improvement exploits the ability to rotate the base power used in the per unit normalization in order to facilitate the application of tighter trigonometric envelopes. The second group of improvements propose additional constraints based on new variables that represent voltage magnitude differences between connected buses. Using \u27bound tightening\u27 techniques, the bounds on the voltage magnitude difference variables can be significantly tighter than the bounds on the voltage magnitudes themselves, so constraints based on voltage magnitude differences can improve the QC relaxation --Abstract, page iv

Traceability Technology Adoption in Supply Chain Networks

Modern traceability technologies promise to improve supply chain management by simplifying recalls, increasing visibility, or verifying sustainable supplier practices. Initiatives leading the implementation of traceability technologies must choose the least-costly set of firms —or seed set — to target for early adoption. Choosing this seed set is challenging because firms are part of supply chains interlinked in complex networks, yielding an inherent supply chain effect: benefits obtained from traceability are conditional on technology adoption by a subset of firms in a product’s supply chain. We prove that the problem of selecting the least-costly seed set in a supply chain network is hard to solve and even approximate within a polylogarithmic factor. Nevertheless, we provide a novel linear programming-based algorithm to identify the least-costly seed set. The algorithm is fixed-parameter tractable in the supply chain network’s treewidth, which we show to be low in real-world supply chain networks. The algorithm also enables us to derive easily-computable bounds on the cost of selecting an optimal seed set. Finally, we leverage our algorithms to conduct large-scale numerical experiments that provide insights into how the supply chain network structure influences diffusion. These insights can help managers optimize their technology diffusion strategy