Estimating generation costs from observed electricity market data is essential for market simulation, strategic bidding, and system planning. To that end, we model the relationship between generation costs and production schedules with a latent variable model. Estimating generation costs from observed schedules is then formulated as Bayesian inference. A prior distribution encodes an initial belief on parameters, and the inference consists of updating the belief with the posterior distribution given observations. We use balanced neural posterior estimation (BNPE) to learn this posterior. Validation on the IEEE RTS-96 test system shows that marginal costs are recovered with narrow credible intervals, while start-up costs remain largely unidentifiable from schedules alone. The method is benchmarked against an inverse-optimization algorithm that exhibits larger parameter errors without uncertainty quantification.

Branch-and-bound (B&B) is a widely used algorithm for solving mixed-integer linear programs (MILPs). One of the elements of its efficiency is the branching strategy. While strong branching empirically produces small B&B trees, its high computational cost makes it impractical for real-world applications. Recent advances in machine learning explored imitation learning to replicate the effectiveness of strong branching. However, these approaches are limited to mimicking existing strategies rather than discovering new ones. In this work, we propose lifted branching, a novel framework that iteratively improves an existing branching strategy by using imitation learning to learn from an improved version of the strategy. Lifted branching is designed to generate smaller B&B trees while avoiding the computational overhead of strong branching, making it suitable for offline training and efficient real-time deployment. We demonstrate the effectiveness of our approach through extensive experiments on four different problem classes, showing that our learned lifted branching models can outperform state-of-the-art strategies, such as reliability pseudo-branching and learned strong branching, depending on the problem of interest.

Linear problems often include many parameters that may be uncertain. Sensitivity analysis studies how these parameters impact optimal values. Instead of analyzing the objective function, we shift the focus to the optimal values of the variables. Three types of linear modifications are considered: on the cost vector, the right-hand side, and on the constraint matrix. Several theorems establish properties of these modifications, including conditions for continuity of optimal variable values, as well as local convexity and concavity properties.

We consider the problem of computing the optimal solution and objective of a linear program under linearly changing linear constraints. The problem studied is given by min c^t x s.t Ax + λDx ≤ b where λ belongs to a set of predefined values Λ. Based on the information given by a precomputed basis, we present three efficient LP warm-starting algorithms. Each algorithm is either based on the eigenvalue decomposition, the Schur decomposition, or a tweaked eigenvalue decomposition to evaluate the optimal solution and optimal objective of these problems. The three algorithms have an overall complexity O(pm^2+pmn) where m (resp. n) is the number of constraints (resp. variables) of the original problem and p the number of values in Λ after an initial preprocessing step. We also provide theorems related to the optimality conditions to verify when a basis is still optimal and a local bound on the objective.

The Job Shop Scheduling Problem (JSSP) is a well-known optimization problem in manufacturing, where the goal is to determine the optimal sequence of jobs across different machines to minimize a given objective. In this work, we focus on minimizing the weighted sum of job completion times. We explore the potential of Monte Carlo Tree Search (MCTS), a heuristic-based reinforcement learning technique, to solve large-scale JSSPs, especially those with recirculation. We propose several Markov Decision Process (MDP) formulations to model the JSSP for the MCTS algorithm. In addition, we introduce a new synthetic benchmark derived from real manufacturing data, which captures the computational burden of large, non-rectangular instances often encountered in practice. Our experimental results show that MCTS effectively produces good-quality solutions for large-scale JSSP instances, outperforming our constraint programming approach.

In this practice report, we present a practical experience of a virtual exchange carried out by a Belgian and a Brazilian professor from two research intensive universities in Northwestern Europe and in Latin America, respectively. Particularly, this paper is focused on how we designed a common syllabus in a specific topic of STEM, aiming to explore math and computer science skills of the students, and how we implemented this virtual exchange step by step, despite the students being from different courses and different levels of degree. In addition, we discuss the pros and cons of the process from the instructors’ perspective as well as under students’ views, and provide some suggestions to be applied and tested for future virtual exchanges in this field. Overall, this practice report reinforces our deep reflection about the results and desire to share our STEM syllabus to encourage other colleagues to apply it (or modify it) for different courses.

Understanding how the optimal value of an optimisation problem changes when its input data is modified is an old question in mathematical optimisation. This paper investigates the computation of the optimal values of a family of (possibly mixed-integer) linear optimisation problems in which the constraint matrix is subject to linear perturbations controlled by a scalar parameter that varies within a given interval. This is a largely unresolved question with the additional burden that the resulting value function may be largely irregular. We propose several bounding techniques that provide formal guarantees on the behaviour of the objective value across the entire parameter range. The proposed bounds rely on tools from robust optimisation, Lagrangian relaxation, and ad-hoc reformulations. Each method is assessed in terms of accuracy, precision, and computational performance. Experimental results on a large benchmark set show that the proposed bounding techniques effectively address this class of problems, delivering strong guarantees and good precision. In addition, we introduce a spatial branch-and-bound algorithm that incorporates these bounds to compute an anytime approximation of the value function within a given error tolerance, and we analyse its computational performance.

Mixed-Integer Linear Programs (MILPs) have many practical applications. Most modelling tools for MILPs fall in two broad categories. Indeed, tools such as algebraic modelling languages allow practitioners to compactly encode models using syntax close to mathematical notation but usually lack support for special structures, while other tools instead provide predefined components that can be easily assembled but modifying or adding new components is difficult. In this work, we present the inner workings of the Graph-Based Optimization Modelling Language (GBOML), an open-source modelling tool implemented in Python combining the strengths of both worlds. GBOML natively supports special structures that can be encoded by a hierarchical hypergraph, offers syntax close to mathematical notation and facilitates the modular construction and reuse of time-indexed models. We detail design choices enabling these features and show that they simplify problem encoding, lead to faster instance generation times and sometimes faster solve times. We benchmark the times taken by GBOML, JuMP, Plasmo, Pyomo and AMPL to generate instances of a structured MILP. We find that GBOML outperforms Plasmo and Pyomo, is tied with JuMP but is slower than AMPL. With parallel model generation, GBOML outperforms JuMP and closes the gap with AMPL. GBOML has the smallest memory footprint.

This paper provides a detailed account of the impact of different offshore wind siting strategies on the design of the European power system. To this end, a two-stage method is proposed. In the first stage, a highly-granular siting problem identifies a suitable set of sites where offshore wind plants could be deployed according to a pre-specified criterion. Two siting schemes are analysed and compared within a realistic case study. These schemes essentially select a pre-specified number of sites so as to maximize their aggregate power output and their spatiotemporal complementarity, respectively. In addition, two variants of these siting schemes are provided, wherein the number of sites to be selected is specified on a country-by-country basis rather than Europe-wide. In the second stage, the subset of previously-identified sites is passed to a capacity expansion planning framework that sizes the power generation, transmission and storage assets that should be deployed and operated in order to satisfy pre-specified electricity demand levels at minimum cost. Results show that the complementarity-based siting criterion leads to system designs which are up to 5% cheaper than the ones relying on the power output-based scheme when offshore wind plants are deployed with no consideration for country-based deployment targets. On the contrary, the power output-based scheme leads to system designs which are consistently 2% cheaper than the ones leveraging the complementarity-based siting strategy when such constraints are enforced. The robustness of the reported results is supported by a sensitivity analysis on offshore wind capital expenditure and inter-annual weather variability, respectively.

Motivated by the problem of bounding the number of iterations of the Simplex algorithm we investigate the possible lengths of monotone paths followed by the Simplex method inside the oriented graphs of polyhedra (oriented by the objective function). We consider both the shortest and the longest monotone paths and estimate the monotone diameter and height of polyhedra. Our analysis applies to transportation polytopes, matroid polytopes, matching polytopes, shortest-path polytopes, and the TSP, among others. We begin by showing that combinatorial cubes have monotone and Bland pivot height bounded by their dimension and that in fact all monotone paths of zonotopes are no larger than the number of edge directions of the zonotope. We later use this to show that several polytopes have polynomial-size pivot height, for all pivot rules. In contrast, we show that many well-known combinatorial polytopes have exponentially-long monotone paths. Surprisingly, for some famous pivot rules, e.g., greatest improvement and steepest edge, these same polytopes have polynomial-size simplex paths.

The implementation of district heating networks into cities is a main topic in policy planning that looks for sustainable solutions to reduce CO2 emissions. However, their development into cities is generally limited by a high initial investment cost. The development of optimization methods intended to draft efficient systems using heating consumption profiles into a prescribed geographic area are useful in this purpose. Such tools are already referred to in the scientific literature, yet they are often restricted to limit the computational load. To bridge this gap, the present contribution proposes a multi-period mixed integer linear programming model for the optimal outline and sizing of a district heating network maximizing the net cash flow based on a geographic information system. This methodology targets a large range of problem sizes from small-scale to large-scale heating networks while guaranteeing numerical robustness. For sake of simplicity, the developed model is first applied to a scaled down case study with 3 available heating sources and a neighborhood of 16 streets. The full-scale model is presented afterwards to demonstrate the applicability of the tool for city-scale heating networks with around 2000 streets to potentially connect within a reasonable computational time of around only one hour.

This paper studies the problem of siting renewable power generation assets using large amounts of climatological data while accounting for their spatiotemporal complementarity. The problem is cast as a combinatorial optimisation problem selecting a pre-specified number of sites so as to minimise the number of simultaneous low electricity production events that they experience relative to a pre-specified reference production level. It is shown that the resulting model is closely related to submodular optimisation and can be interpreted as generalising the well-known maximum coverage problem. Both deterministic and randomised algorithms are discussed, including greedy, local search and relaxation-based heuristics as well as combinations of these algorithms. The usefulness of the model and methods is illustrated by a realistic case study inspired by the problem of siting onshore wind power plants in Europe, resulting in instances featuring over ten thousand candidate locations and ten years of hourly-sampled meteorological data. The proposed solution methods are benchmarked against a state-of-the-art mixed-integer programming solver and several algorithms are found to consistently produce better solutions at a fraction of the computational cost. The physical nature of solutions provided by the model is also investigated, and all deployment patterns are found to be unable to supply a constant share of the electricity demand at all times. Finally, a cross-validation analysis shows that, except for an edge case, the model can successfully and reliably identify deployment patterns that perform well on previously unseen climatological data from historical data spanning a small number of weather years.

We propose to use input convex neural networks (ICNN) to build convex approximations of non-convex feasible sets of optimization problems, in the form of a set of linear equalities and inequalities in a lifted space.Our approach may be tailored to yield both inner-and outer-approximations, or to maximize its accuracy in regions closer to the minimum of a given objective function.We illustrate the method on twodimensional toy problems and motivate it by various instances of reliability management problems of large-scale electric power systems.

Spatiotemporal complementarity between variable renewable energy sources (RES) has received a great deal of attention in recent years. However, its value for power systems is still not properly understood. This research gap is tackled in the current work by evaluating the benefits of siting RES assets according to resource complementarity criteria. To this end, a two-stage method is employed. First, the complementarity between RES is assessed and the locations sets that maximize it are selected using an integer programming model. Subsequently, the outcome of the first stage is used within an expansion planning framework which identifies the optimal system design and serves as a basis for assessing the economic value of RES complementarity for power systems. The analysis is conducted on a realistic case study targeting the deployment of 450 GW of offshore wind in Europe. Results show that siting based on RES complementarity is particularly attractive when the power density of wind developments is relatively high and when the inter-annual variability of the underlying resource is accounted for. More specifically, such a siting strategy leads to yearly savings between 0.3 and 1.2 billion EUR compared with conventional schemes seeking to deploy generation capacity at the most productive locations.

Flow cytometry raw data and supplementary table S1.

Standardization, data mining techniques, and comparison to normality are changing the landscape of multiparameter flow cytometry in clinical hematology. On the basis of these principles, a strategy was developed for measurable residual disease (MRD) assessment. Herein, suspicious cell clusters are first identified at diagnosis using a clustering algorithm. Subsequently, automated multidimensional spaces, named "Clouds", are created around these clusters on the basis of density calculations. This step identifies the immunophenotypic pattern of the suspicious cell clusters. Thereafter, using reference samples, the "Abnormality Ratio" (AR) of each Cloud is calculated, and major malignant Clouds are retained, known as "Leukemic Clouds" (L-Clouds). In follow-up samples, MRD is identified when more cells fall into a patient's L-Cloud compared to reference samples (AR concept). This workflow was applied on simulated data and real-life leukemia flow cytometry data. On simulated data, strong patient-dependent positive correlation (R2 = 1) was observed between the AR and spiked-in leukemia cells. On real patient data, AR kinetics was in line with the clinical evolution for five out of six patients. In conclusion, we present a convenient flow cytometry data analysis approach for the follow-up of hematological malignancies. Further evaluation and validation on more patient samples and different flow cytometry panels is required before implementation in clinical practice.

Long-term reservoir management often uses bounds on the reservoir level, between which the operator can work. However, these bounds are not always kept up-to-date with the latest knowledge about the reservoir drainage area, and thus become obsolete. The main difficulty with bounds computation is to correctly take into account the high uncertainty about the inflows to the reservoir. In this article, we propose a methodology to derive minimum bounds while providing formal guarantees about the quality of the obtained solutions. The uncertainty is embedded using either stochastic or robust programming in a model-predictive-control framework. We compare the two paradigms to the existing solution for a case study and find that the obtained solutions vary substantially. By combining the stochastic and the robust approaches, we also assign a confidence level to the solutions obtained by stochastic programming. The proposed methodology is found to be both efficient and easy to implement. It relies on sound mathematical principles, ensuring that a global optimum is reached in all cases.