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Monoidal strengthening and unique lifting in MIQCPs
No full textUsing the recently proposed maximal quadratic-free sets and the well-known monoidal strengthening procedure, we show how to improve inter- section cuts for quadratically-constrained optimization problems by exploiting integrality requirements. We provide an explicit construction that allows an efficient implementation of the strengthened cuts along with computational results showing their improvements over the standard intersection cuts. We also show that, in our setting, there is unique lifting which implies that our strengthening procedure is generating the best possible cut coefficients for the integer variables
Online Learning for Scheduling MIP Heuristics
No full textMixed Integer Programming (MIP) is NP-hard, and yet modern solvers often solve large real-world problems within minutes. This success can partially be attributed to heuristics. Since their behavior is highly instance-dependent, relying on hard-coded rules derived from empirical testing on a large heterogeneous corpora of benchmark instances might lead to sub-optimal performance. In this work, we propose an online learning approach that adapts the application of heuristics towards the single instance at hand. We replace the commonly used static heuristic handling with an adaptive framework exploiting past observations about the heuristic’s behavior to make future decisions. In particular, we model the problem of controlling Large Neighborhood Search and Diving – two broad and complex classes of heuristics – as a multi-armed bandit problem. Going beyond existing work in the literature, we control two different classes of heuristics simultaneously by a single learning agent. We verify our approach numerically and show consistent node reductions over the MIPLIB 2017 Benchmark set. For harder instances that take at least 1000 seconds to solve, we observe a speedup of 4%
An efficient solver for multi-objective onshore wind farm siting and network integration
No full textExisting planning approaches for onshore wind farm siting and network integration often do not meet minimum cost solutions or social and environmental considerations. In this paper, we develop an approach for the multi-objective optimization of turbine locations and their network connection using a Quota Steiner tree problem. Applying a novel transformation on a known directed cut formulation, reduction techniques, and heuristics, we design an exact solver that makes large problem instances solvable and outperforms generic MIP solvers. Although our case studies in selected regions of Germany show large trade-offs between the objective criteria of cost and landscape impact, small burdens on one criterion can significantly improve the other criteria. In addition, we demonstrate that contrary to many approaches for exclusive turbine siting, network integration must be simultaneously optimized in order to avoid excessive costs or landscape impacts in the course of a wind farm project. Our novel problem formulation and the developed solver can assist planners in decision making and help optimize wind farms in large regions in the future
Forward and Line-Based Cycle Bases for Periodic Timetabling
No full textThe optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak, and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure, and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances
Early Crop Classification via Multi-Modal Satellite Data Fusion and Temporal Attention
No full textIn this article, we propose a deep learning-based algorithm for the classification of crop types from Sentinel-1 and Sentinel-2 time series data which is based on the celebrated transformer architecture. Crucially, we enable our algorithm to do early classification, i.e., predict crop types at arbitrary time points early in the year with a single trained model (progressive intra-season classification). Such early season predictions are of practical relevance for instance for yield forecasts or the modeling of agricultural water balances, therefore being important for the public as well as the private sector. Furthermore, we improve the mechanism of combining different data sources for the prediction task, allowing for both optical and radar data as inputs (multi-modal data fusion) without the need for temporal interpolation. We can demonstrate the effectiveness of our approach on an extensive data set from three federal states of Germany reaching an average F1 score of 0.92 using data of a complete growing season to predict the eight most important crop types and an F1 score above 0.8 when doing early classification at least one month before harvest time. In carefully chosen experiments, we can show that our model generalizes well in time and space
Accelerated and Sparse Algorithms for Approximate Personalized PageRank and Beyond
No full textIt has recently been shown that ISTA, an unaccelerated optimization method, presents sparse updates for the ℓ1-regularized undirected personalized PageRank problem (Fountoulakis et al., 2019), leading to cheap iteration complexity and providing the same guarantees as the approximate personalized PageRank algorithm (APPR) (Andersen et al., 2006). In this work, we design an accelerated optimization algorithm for this problem that also performs sparse updates, providing an affirmative answer to the COLT 2022 open question of Fountoulakis and Yang (2022). Acceleration provides a reduced dependence on the condition number, while the dependence on the sparsity in our updates differs from the ISTA approach. Further, we design another algorithm by using conjugate directions to achieve an exact solution while exploiting sparsity. Both algorithms lead to faster
convergence for certain parameter regimes. Our findings apply beyond PageRank and work for any quadratic objective whose Hessian is a positive-definite 푀-matrix
Data-driven Distributionally Robust Optimization over Time
No full textStochastic optimization (SO) is a classical approach for optimization under uncertainty that typically requires knowledge about the probability distribution of uncertain parameters. Because the latter is often unknown, distributionally robust optimization (DRO) provides a strong alternative that determines the best guaranteed solution over a set of distributions (ambiguity set). In this work, we present an approach for DRO over time that uses online learning and scenario observations arriving as a data stream to learn more about the uncertainty. Our robust solutions adapt over time and reduce the cost of protection with shrinking ambiguity. For various kinds of ambiguity sets, the robust solutions converge to the SO solution. Our algorithm achieves the optimization and learning goals without solving the DRO problem exactly at any step. We also provide a regret bound for the quality of the online strategy that converges at a rate of O(log T/T−−√), where T is the number of iterations. Furthermore, we illustrate the effectiveness of our procedure by numerical experiments on mixed-integer optimization instances from popular benchmark libraries and give practical examples stemming from telecommunications and routing. Our algorithm is able to solve the DRO over time problem significantly faster than standard reformulations
Strained Small Nitrogen Heterocycles-Azabicyclobutanes and Azirines
No full textSmall ring nitrogen heterocycles, azabicyclobutanes and azirines, were investigated by computational methods in order to address the discrepancy between their regioisomers 1- and 2-azabicyclobutane and 1H- and 2H-azirines. Both 1-azabicyclobutane and 2H-azirine are well known synthetic starting points to larger nitrogen heterocycles whereas 2-azabicyclobutane and 1H-azirine and their derivatives have yet to be reported as isolable compounds. Calculated parameters such as structure, base strength (proton affinities), NICS values and enthalpies of formation from which strain energies are derived are reported. The destabilization of the less stable regioisomers is attributed to homoantiaromaticity in 2-azabicyclobutane and antiaromaticity in 1H-azirine. Two stereoisomers exist for 2-azabicyclobutane with the endo- stereoisomer being more stable. This phenomenon is indicative of the hydrogen bond acceptor properties of the neighboring cyclpropane and the π-bond character of the central bond in 2-azabicyclobutane
Non-Linear Battery Behavior in Electric Vehicle Scheduling Problems
No full textThe currently most popular approach to handle non-linear battery behavior for electric vehicle scheduling is to use a linear spline interpolation of the charge curve. We show that this can lead to approximate models that underestimate the charge duration and overestimate the state of charge, which is not desirable. While the error is of second order with respect to the interpolation step size, the associated mixed-integer linear programs do not scale well with the number of spline segments. It is therefore recommendable to use coarse interpolation grids adapted to the curvature of the charge curve, and to include sufficient safety margins to ensure solutions of approximate models remain feasible subjected to the exact charge curve
