Martin Mevissen

Martin Mevissen, Kosuke Yokoyama, Nobuki Takayama

We propose a general algorithm to enumerate all solutions of a zero-dimensional polynomial system with respect to a given cost function. The algorithm is developed and is used to study a polynomial system obtained by discretizing the steady cavity flow problem in two dimensions. The key technique on which our algorithm is based is to solve polynomial optimization problems via sparse semidefinite programming relaxations (SDPR), which has been adopted successfully to solve reaction-diffusion boundary value problems recently. The cost function to be minimized is derived from discretizing the fluid's kinetic energy. The enumeration algorithm's solutions are shown to converge to the minimal kinetic energy solutions for SDPR of increasing order. We demonstrate the algorithm with SDPR of first and second order on polynomial systems for different scenarios of the cavity flow problem and succeed in deriving the $k$ smallest kinetic energy solutions. The question whether these solutions converge to solutions of the steady cavity flow problem is discussed, and we pose a conjecture for the minimal energy solution for increasing Reynolds number.

Claudio Gambella, Jakub Marecek, Martin Mevissen et al.

The unit commitment with transmission constraints in the alternating-current (AC) model is a challenging mixed-integer non-linear optimisation problem. We present an approach based on decomposition of a Mixed-Integer Semidefinite Programming (MISDP) problem into a mixed-integer quadratic (MIQP) master problem and a semidefinite programming (SDP) sub-problem. Between the master problem and the sub-problem, we pass novel classes of cuts. We analyse finite convergence to the optimum of the MISDP and report promising computational results on a test case from the Canary Islands, Spain.

Julien Monteil, Anton Dekusar, Claudio Gambella et al.

The transport literature is dense regarding short-term traffic predictions, up to the scale of 1 hour, yet less dense for long-term traffic predictions. The transport literature is also sparse when it comes to city-scale traffic predictions, mainly because of low data availability. In this work, we report an effort to investigate whether deep learning models can be useful for the long-term large-scale traffic prediction task, while focusing on the scalability of the models. We investigate a city-scale traffic dataset with 14 weeks of speed observations collected every 15 minutes over 1098 segments in the hypercenter of Los Angeles, California. We look at a variety of state-of-the-art machine learning and deep learning predictors for link-based predictions, and investigate how such predictors can scale up to larger areas with clustering, and graph convolutional approaches. We discuss that modelling temporal and spatial features into deep learning predictors can be helpful for long-term predictions, while simpler, not deep learning-based predictors, achieve very satisfactory performance for link-based and short-term forecasting. The trade-off is discussed not only in terms of prediction accuracy vs prediction horizon but also in terms of training time and model sizing.

Wann-Jiun Ma, Jakub Marecek, Martin Mevissen

There has been much recent interest in hierarchies of progressively stronger convexifications of polynomial optimisation problems (POP). These often converge to the global optimum of the POP, asymptotically, but prove challenging to solve beyond the first level in the hierarchy for modest instances. We present a finer-grained variant of the Lasserre hierarchy, together with first-order methods for solving the convexifications, which allow for efficient warm-starting with solutions from lower levels in the hierarchy.

Bissan Ghaddar, Jakub Marecek, Martin Mevissen

Formulating the alternating current optimal power flow (ACOPF) as a polynomial optimization problem makes it possible to solve large instances in practice and to guarantee asymptotic convergence in theory.