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This archive contains data related to the paper, "A framework for discrete optimization of stellarator coils", published in Nuclear Fusion (https://doi.org/10.1088/1741-4326/adbb02). Included are data plotted in the figures, as well as scripts for reproducing the results in the paper.
Designing magnets for three-dimensional plasma confinement is a key task for advancing the stellarator as a fusion reactor concept. Stellarator magnets must produce an accurate field while leaving adequate room for other components and being reasonably simple to construct and assemble. In this paper, a framework for coil design and optimization is introduced that enables the attainment of sparse magnet solutions with arbitrary restrictions on where coils may be located. The solution space is formulated as a 'wireframe' consisting of a mesh of interconnected wire segments enclosing the plasma. Two methods are developed for optimizing the current distribution on a wireframe: Regularized Constrained Least Squares, which uses a linear least-squares approach to optimize the currents in each segment, and Greedy Stellarator Coil Optimization, a fully discrete procedure in which loops of current are added to the mesh one by one to achieve the desired magnetic field on the plasma boundary. Examples are presented of solutions obtainable with each method, some of which achieve high field accuracy while obeying spatial constraints that permit easy assembly.
Challenging combinatorial optimization problems are ubiquitous in science and engineering. Several quantum methods for optimization have recently been developed, in different settings including both exact and approximate solvers. Addressing this field of research, this manuscript has three distinct purposes. First, we present an intuitive method for synthesizing and analyzing discrete (i.e., integer-based) optimization problems, wherein the problem and corresponding algorithmic primitives are expressed using a discrete quantum intermediate representation (DQIR) that is encoding-independent. This compact representation often allows for more efficient problem compilation, automated analyses of different encoding choices, easier interpretability, more complex runtime procedures, and richer programmability, as compared to previous approaches, which we demonstrate with a number of examples. Second, we perform numerical studies comparing several qubit encodings; the results exhibit a number of preliminary trends that help guide the choice of encoding for a particular set of hardware and a particular problem and algorithm. Our study includes problems related to graph coloring, the traveling salesperson problem, factory/machine scheduling, financial portfolio rebalancing, and integer linear programming. Third, we design low-depth graph-derived partial mixers (GDPMs) up to 16-level quantum variables, demonstrating that compact (binary) encodings are more amenable to QAOA than previously understood. We expect this toolkit of programming abstractions and low-level building blocks to aid in designing quantum algorithms for discrete combinatorial problems.
This paper proposes an endogenous interface pricing model for day-ahead transmission–distribution co-optimization that co-determines the interface locational marginal price (LMP) and the transmission–distribution exchange, ensuring price–dispatch consistency while optimally scheduling discrete distribution controls. The formulation couples a DC optimal power flow (OPF) with a branch-flow AC OPF that schedules distributed energy resources (DERs), tap-changer settings, capacitor banks (CBs), and multi-period energy storage systems (ESSs) under feeder voltage and current limits, and is solved as a mixed-integer second-order cone program (MISOCP). In a T14–D33 system, coordinated device scheduling recovers about 90% of the distribution-to-transmission export achievable in a reference case that ignores distribution network (DN) limits, while satisfying a 1.05 p.u. voltage upper bound. In a T39–D34/D37/D123 system, a sequential decoupled benchmark produces interface LMP distortions up to 12.5% and a 7.28% mismatch in net export energy, whereas the proposed model removes these distortions and the associated settlement mismatches. Second-order cone (SOC) relaxation gaps remain below $10^{-3}$ in all cases.
This paper proposes a framework to determine the optimal active and reactive power dispatch of distributed photovoltaic (PV) generation, switched capacitors, and voltage regulators in multi-phase unbalanced distribution systems. The objectives of the optimal dispatch are minimization of the energy loss, PV real power curtailment, and switching operations of capacitors and voltage regulators, in addition to elimination of voltage violation and reverse power flow. The optimization problem is formulated in rectangular coordinates as a nonlinear, nonconvex problem with discrete variables. A hierarchical twostage framework is proposed to effectively handle those discrete variables and reduce the computational time compared to the unified approach in which all variables are solved simultaneously. The efficacy of the proposed approach and the accuracy of the obtained numerical solution is validated using the unbalanced multi-phase IEEE 34-bus with 15-minute load and PV data.
We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the adjoint system associated with an evolutionary partial differential equation has an infinite-dimensional Hamiltonian structure, which is useful for connecting the fully continuous, semi-discrete, and fully discrete levels. We subsequently address the question of discretize-then-optimize versus optimize-then-discrete for both semi-discretization and time integration, by characterizing the commutativity of discretize-then-optimize methods versus optimize-then-discretize methods uniquely in terms of an adjoint-variational quadratic conservation law. For Galerkin semi-discretizations and one-step time integration methods in particular, we explicitly construct these commuting methods by using structure-preserving discretization techniques.
Quantum computing (QC) has gained popularity due to its unique capabilities that are quite different from that of classical computers in terms of speed and methods of operations. This paper proposes hybrid models and methods that effectively leverage the complementary strengths of deterministic algorithms and QC techniques to overcome combinatorial complexity for solving large-scale mixed-integer programming problems. Four applications, namely the molecular conformation problem, job-shop scheduling problem, manufacturing cell formation problem, and the vehicle routing problem, are specifically addressed. Large-scale instances of these application problems across multiple scales ranging from molecular design to logistics optimization are computationally challenging for deterministic optimization algorithms on classical computers. To address the computational challenges, hybrid QC-based algorithms are proposed and extensive computational experimental results are presented to demonstrate their applicability and efficiency. The proposed QC-based solution strategies enjoy high computational efficiency in terms of solution quality and computation time, by utilizing the unique features of both classical and quantum computers.
Technologies for a quantum/classical hybrid approach to solving optimization problems is disclosed. In the illustrative embodiment, an optimization problem is decomposed into two sub-problems. The first sub-problem is solved on a classical computer, and a result from the first sub-problem is provided to a quantum computer. The quantum computer then solves the second sub-problem based on the result of the first sub-problem from the classical computer. The quantum computer can then provide a result to the classical computer to re-solve the first problem. The iterative calculation is continued until an end condition is met.
Optimizing the controls of quantum systems plays a crucial role in advancing quantum technologies. The time-varying noises in quantum systems and the widespread use of inhomogeneous quantum ensembles raise the need for high-quality quantum controls under uncertainties. In this paper, we consider a stochastic discrete optimization formulation of a discretized binary optimal quantum control problem involving Hamiltonians with predictable uncertainties. We propose a sample-based reformulation that optimizes both risk-neutral and risk-averse measurements of control policies, and solve these with two gradient-based algorithms using sum-up-rounding approaches. Furthermore, we discuss the differentiability of the objective function and prove upper bounds of the gaps between the optimal solutions to binary control problems and their continuous relaxations. We conduct numerical simulations on various sized problem instances based on two applications of quantum pulse optimization; we evaluate different strategies to mitigate the impact of uncertainties in quantum systems. In conclusion, we demonstrate that the controls of our stochastic optimization model achieve significantly higher quality and robustness compared with the controls of a deterministic model.
Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. Then, we show that the high problem density region, which limits QAOA’s performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified.
Critical minerals and rare earth elements play an important role in our climate change initiatives, particularly in applications related with energy storage. Here, we use discrete optimization approaches to design a process for the recovery of Lithium and Cobalt from battery recycling, through membrane separation. Our contribution involves proposing a Generalized Disjunctive Programming (GDP) model for the optimal design of a multistage diafiltration cascade for Li-Co separation. By solving the resulting nonconvex mixed-integer nonlinear program model to global optimality, we investigated scalability and solution quality variations with changes in the number of stages and elements per stage. Results demonstrate the computational tractability of the nonlinear GDP formulation for design of membrane separation processes while opening the door for decom-position strategies for multicomponent separation cascades. Future work aims to extend the GDP formulation to account for stage installation and explore various decomposition techniques to enhance solution efficiency.
Security-constrained unit commitment (SCUC) is a fundamental problem in power systems and electricity markets. In practical settings, SCUC is repeatedly solved via mixed-integer linear programming (MIP), sometimes multiple times per day, with only minor changes in input data. In this work, we propose a number of machine learning techniques to effectively extract information from previously solved instances in order to significantly improve the computational performance of MIP solvers when solving similar instances in the future. Based on statistical data, we predict redundant constraints in the formulation, good initial feasible solutions, and affine subspaces where the optimal solution is likely to lie, leading to a significant reduction in problem size. Computational results on a diverse set of realistic and large-scale instances show that using the proposed techniques, SCUC can be solved on average 4.3 times faster with optimality guarantees and 10.2 times faster without optimality guarantees, with no observed reduction in solution quality. Out-of-distribution experiments provide evidence that the method is somewhat robust against data-set shift. Summary of Contribution. The paper describes a novel computational method, based on a combination of mixed-integer linear programming (MILP) and machine learning (ML), to solve a challenging and fundamental optimization problem in the energy sector. The method advances the state-of-the-art, not only for this particular problem, but also, more generally, in solving discrete optimization problems via ML. We expect that the techniques presented can be readily used by practitioners in the energy sector and adapted, by researchers in other fields, to other challenging operations research problems that are solved routinely.
Traditionally, centralized approaches have predominantly been used for the power system operation and control. With increasing penetration of small-scale distributed energy resources (DERs) in the distribution network, especially independently owned renewable resources, distributed algorithms can serve as a potential alternative for improving scalability, resiliency and addressing privacy concerns. However, the complexity of distributed algorithms significantly increases with the integration of the legacy devices, the operation of which depend on discrete control variables. This paper aims to provide a review of the distributed optimization algorithms incorporating discrete control variables for the power distribution system. While the research in this domain is still at its nascence, an extensive comparison of the approaches in the literature for applying quadratic penalty, branch and bound,ordinal optimization and proximal operator to handle discrete variables in the framework of ADMM and dual decomposition have been addressed. Future research direction in this field have been also provided.
Energy disaggregation is the task of discerning the energy consumption of individual appliances from aggregated measurements, which holds promise for understanding and reducing energy usage. In this paper, we propose PHASED, an optimization approach for energy disaggregation that has two key features: PHASED (i) exploits the structure of power distribution systems to make use of readily available measurements that are neglected by existing methods, and (ii) poses the problem as a minimization of a difference of sub-modular functions. We leverage this form by applying a discrete optimization variant of the majorization-minimization algorithm to iteratively minimize a sequence of global upper bounds of the cost function to obtain high-quality approximate solutions. PHASED improves the disaggregation accuracy of state-of-the-art models by up to 61% and achieves better prediction on heavy load appliances.
Stochastic optimization and simulation models ubiquitously arise in designing and operating complex service/engineering systems. They can be extreme in scale due to high-dimensional data and decisions, and can also involve decisions made sequentially in response to newly revealed data, both causing significant computational challenge. The objective of this research is to explore a unified framework that integrates machine learning with discrete optimization and risk-averse modeling, to improve the efficiency of decomposition paradigms for stochastic optimization and simulations at extreme scale. The models we consider represent a broad class of complex decision-making problems, where 0-1 or continuous decisions are made before and/or after knowing multiple sources of uncertainties that could be correlated. We will employ machine learning methods to dynamically decide and prioritize computational procedures, including cut generation, branching, and bounding of the optimal objective. Furthermore, the research will shed new lights on the traditional decomposition algorithms for extreme-scale computing. Deliverables of the research include new modeling and computational methods for advancing the state-of-the-art research in optimization and simulation, bringing many relevant risk-averse, data-driven optimization problems in practice within the range of tractability. Examples include distributed computing server scheduling and sensor deployment for monitoring critical infrastructures. Success in this effort will enable progress in solving multiple extreme-scale problems in the complex system design and operations arising from DoE missions in energy, environment, and national security.
Here we propose a windowed least-squares (WLS) approach for model reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space–time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as particular instances of the methodology recover Galerkin, least-squares Petrov–Galerkin (LSPG), and space–time LSPG projection. In addition, the approach addresses key deficiencies in existing model reduction techniques, e.g., the dependence of LSPG and space–time LSPG projection on the time discretization and the exponential growth in time exhibited by a posteriori error bounds for both Galerkin and LSPG projection. We consider two types of space–time trial subspaces within the proposed approach: one that reduces only the spatial dimension of the full-order model, and one that reduces both the spatial and temporal dimensions of the full-order model. For each type of trial subspace, we consider two different solution techniques: direct (i.e., discretize then optimize) and indirect (i.e., optimize then discretize). Numerical experiments conducted using trial subspaces characterized by spatial dimension reduction demonstrate that the WLS approach can yield more accurate solutions with lower space–time residuals than Galerkin and LSPG projection.
We study the design of an electromagnetic cloak from multiple materials with an additional constraint on the mass of the cloak. Our problem is an example of a topology optimization problem, and we formulate this problem as a mixed-integer partial-differential equation constrained optimization (MIPDECO) problem, where Maxwell’s equation models the propagation of the wave through the cloak and surrounding medium. We use binary variables to model the assignment of the different materials, and their relevant properties (permittivity and density). The mass constraint adds a nontrivial constraint to this problem. We propose a two-phase strategy to solve this problem. In the first phase, we solve a continuous relaxation, and then propose a new variant of the feasibility pump that exploits the structure of the PDE to obtain an initial integral solution candidate. In the second phase, we use a trust-region approach to improve this incumbent. We also consider a continuation or mesh-sequencing approach to find better solutions faster on consecutively finer meshes. We present detailed numerical results to illustrate the effectiveness of our approaches for constructing multi-material cloaks with a mass constraint.
In this work, we present the Sequential Ensemble Transform (SET) method, an approach for generating approximate samples from a Bayesian posterior distribution. The method explores the posterior distribution by solving a sequence of discrete optimal transport problems to produce a series of transport plans which map prior samples to posterior samples. We prove that the sequence of Dirac mixture distributions produced by the SET method converges weakly to the true posterior as the sample size approaches infinity. Furthermore, our numerical results indicate that, when compared to standard Sequential Monte Carlo (SMC) methods, the SET approach is more robust to the choice of Markov mutation kernels and requires less computational efforts to reach a similar accuracy when used to explore complex posterior distributions. Finally, we describe adaptive schemes that allow to completely automate the use of the SET method.