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We propose a transformer architecture for time series forecasting with a focus on time series tokenisation and apply it to a real-world prediction problem from the pricing domain. Our architecture aims to learn effective representations at many scales across all available data simultaneously. The model contains a number of novel modules: a differentiated form of time series patching which employs multiple resolutions, a multiple-resolution module for time-varying known variables, a mixer-based module for capturing cross-series information, and a novel output head with favourable scaling to account for the increased number of tokens. We present an application of this model to a real world prediction problem faced by the markdown team at a very large retailer. On the experiments conducted our model outperforms in-house models and the selected existing deep learning architectures.

Decision-focused learning has emerged as a promising approach for decision making under uncertainty by training the upstream predictive aspect of the pipeline with respect to the quality of the downstream decisions. Most existing work has focused on single stage problems. Many real-world decision problems are more appropriately modelled using multistage optimisation as contextual information such as prices or demand is revealed over time and decisions now have a bearing on future decisions. We propose decision-focused forecasting, a multiple-implicitlayer model which in its training accounts for the intertemporal decision effects of forecasts using differentiable optimisation. The recursive model reflects a fully differentiable multistage optimisation approach. We present an analysis of the gradients produced by this model showing the adjustments made to account for the state-path caused by forecasting. We demonstrate an application of the model to an energy storage arbitrage task and report that our model outperforms existing approaches., Comment: Under review. Preprint

We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices $xx^{\mathrm{H}}$, where the elements of $x \in \mathbb{C}^n$ are $m$th unit roots. These polytopes have applications in ${\text{MAX-3-CUT}}$, digital communication technology, angular synchronization and more generally, complex quadratic programming. For ${m=2}$, the complex cut polytope corresponds to the well-known cut polytope. We generalize valid cuts for this polytope to cuts for any complex cut polytope with finite $m>2$ and provide a framework to compare them. Further, we consider a second semidefinite lifting of the complex cut polytope for $m=\infty$. This lifting is proven to be equivalent to other complex Lasserre-type liftings of the same order proposed in the literature, while being of smaller size. Our theoretical findings are supported by numerical experiments on various optimization problems., Comment: 29 pages (including references) + 6 pages appendix