Open Access

In this paper, we develop distributed and parallel general-purpose solvers for combinatorial optimization through the framework of domain-independent dynamic programming (DIDP), a model-based paradigm based on dynamic programming. In particular, we parallelize heuristic state space search algorithms to develop such solvers. Benefiting from the general-purpose nature of DIDP, we apply our solvers to four problem classes: the traveling salesperson problem with time windows (TSPTW), the type1 simple assembly line balancing problem (SALBP-1), the one-to-one multi-commodity pickup and delivery traveling salesperson problem (m-PDTSP), and the type2 assembly line balancing problem with sequence-dependent setup times (SUALBP-2). We demonstrate the scalability of our solvers using up to 192 TB of RAM and 49,152 CPU cores. Using the developed solvers, we close 14 open instances of TSPTW, 49 of m-PDTSP, and 152 of SUALBP-2.

Shifting towards renewable energy sources and reducing carbon emissions necessitate sophisticated energy system planning, optimization, and extension. Energy systems optimization models (ESOMs) often form the basis for political and operational decision-making. ESOMs are frequently formulated as linear (LPs) and mixed-integer linear (MIP) problems. MIPs allow continuous and discrete decision variables. Consequently, they are substantially more expressive than LPs but also more challenging to solve. The ever-growing size and complexity of ESOMs take a toll on the computational time of state-of-the-art commercial solvers. Indeed, for large-scale ESOMs, solving the LP relaxation -- the basis of modern MIP solution algorithms -- can be very costly. These time requirements can render ESOM MIPs impractical for real-world applications. This article considers a set of large-scale decarbonization-focused unit commitment models with expansion decisions based on the REMix framework (up to 83 million variables and 900,000 discrete decision variables). For these particular instances, the solution to the LP relaxation and the MIP optimum lie close. Based on this observation, we investigate the application of relaxation-enforced neighborhood search (RENS), machine learning guided rounding, and a fix-and-propagate (FP) heuristic as a standalone solution method. Our approach generated feasible solutions 20 to 100 times faster than GUROBI, achieving comparable solution quality with primal-dual gaps as low as 1% and up to 35%. This enabled us to solve numerous scenarios without lowering the quality of our models. For some instances that Gurobi could not solve within two days, our FP method provided feasible solutions in under one hour.

The 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.

Basking sharks are thought to be one of the most efficient filter-feeding fish in terms of the throughput of water filtered through their gills. Details about the underlying morphology of their branchial region have not been studied due to various challenges in acquiring real-world data. The present thesis aims to facilitate this, by developing a mathematical shape model which constructs the 3D structure of the head skeleton of a basking shark using annotated landmarks on a single 2D image. This is an ill-posed problem as estimating the depth of a 3D object from a single 2D view is, in general, not possible. To reduce this ambiguity, we create a set of pre-defined training shapes in 3D from CT scans of basking sharks. First, the damaged structures of the sharks in the scans are corrected via solving a set of optimization problems, before using them as accurate 3D representations of the object. Then, two approaches are employed for the 2D-to-3D shape fitting problem–an Active Shape Model approach and a Kendall’s Shape Space approach. The former represents a shape as a point on a high-dimensional Euclidean space, whereas the latter represents a shape as an equivalence class of points in this Euclidean space. Kendall’s shape space approach is a novel technique that has not yet been applied in this context, and a comprehensive comparison of the two approaches suggests this approach to be superior for the problem at hand. This can be credited to an improved interpolation of the training shapes.

二次割当問題は線形緩和が弱いことが知られ，強化のため多様な緩和手法が考案されているが，その一つである二重非負値計画緩和（ DNN 緩和）及びその解法として近年研究が進んでいるニュートン・ブラケット法を紹介し，それらに基づく分枝限定法の実装及び数値実験結果について報告する．