We consider the problem of characterizing the convex hull of the graph of a bilinear function f on the n-dimensional unit cube [0, 1] n . Extended formulations for this convex hull are obtained by taking subsets of the facets of the Boolean Quadric Polytope (BQP). Extending existing results, we propose a systematic study of properties of f that guarantee that certain classes of BQP facets are sufficient for an extended formulation. We use a modification of Zuckerberg's geometric method for proving convex hull characterizations [Geometric proofs for convex hull defining formulations, Operations Research Letters 44 (2016), 625-629] to prove some initial results in this direction. In particular, we provide small-sized extended formulations for bilinear functions whose corresponding graph is either a cycle with arbitrary edge weights or a clique or an almost clique with unit edge weights.

We investigate how well the graph of a bilinear function b : [0, 1] n → R can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most c times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor c cannot be bounded by a constant independent of n. More precisely, we show that for a random bilinear function b we have asymptotically almost surely c √ n/4. On the other hand, we prove that c 600 √ n, which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions b for which the McCormick relaxation is equal to the convex hull.

The pooling problem is a nonconvex nonlinear programming problem with numerous applications. The nonlinearities of the problem arise from bilinear constraints that capture the blending of raw materials. Bilinear constraints are well-studied and significant progress has been made in solving large instances of the pooling problem to global optimality. This is due in no small part to reformulations of the problem. Recently, Alfaki and Haugland proposed a multi-commodity flow formulation of the pooling problem based on input commodities. The authors proved that the new formulation has a stronger linear relaxation than previously known formulations. They also provided computational results which show that the new formulation outperforms previously known formulations when used in a global optimization solver. In this paper, we generalize their ideas and propose new multi-commodity flow formulations based on output, input and output and (input, output)-commodities. We prove the equivalence of formulations, and we study the partial order of formulations with respect to the strength of their LP relaxations. In an extensive computational study, we evaluate the performance of the new formulations. We study the trade-off between disaggregating commodities and therefore increasing the size of formulations versus strengthening the relaxed linear programs and improving the computational performance of the nonlinear programs. We provide computational results which show that output commodities often outperform input commodities, and that disaggregating commodities further only marginally strengthens the linear relaxations. In fact, smaller formulations often show a significantly better performance when used in a global optimization solver.

Answering a question of Haugland, we show that the pooling problem with one pool and a bounded number of inputs can be solved in polynomial time by solving a polynomial number of linear programs of polynomial size. We also give an overview of known complexity results and remaining open problems to further characterize the border between (strongly) NP-hard and polynomially solvable cases of the pooling problem.Comment: updated reference

Energy storage (ES) is acknowledged to play an important role in modern energy technologies due to its potential to reduce operational costs, enhance the resilience, and level energy load for energy systems. Efficient ES management can achieve cost savings, also known as energy arbitrage, by charging at off-peak prices and discharging at peak prices. This arbitrage can be further boosted by allowing the ES to be shared by multiple users/buildings. However, since energy arbitrage relies on the variation of energy prices, it is hard to achieve this arbitrage if the prices are uncertain. To address this challenge, we present a robust optimization approach to fairly and efficiently operate an ES shared between two users under price uncertainty. This sharing strategy is formulated as a biobjective mixed integer bilinear programming model. To facilitate solution efficiency, we propose a binary formulation for piecewise McCormick relaxations to approximate the bilinear model by a tractable linear model. A computational study demonstrates the effectiveness of our robust sharing strategy for managing ES sharing under price uncertainty. Also, it shows that the proposed binary formulation for piecewise McCormick relaxations reduces the runtime by around 80% compared to the traditional unary formulation.