Tractable Problem Research Articles

Enhancing Production in Robot-Enabled Manufacturing Systems: A Dynamic Model and Moving Horizon Control Strategy for Mobile Robot Assignment

Abstract This paper presents a dynamic mathematical model for a robot-enabled manufacturing system, where mobile robots independently manage workstation tasks. Each robot possesses one or multiple skills, enabling collaborative work at workstations. A real-time robot assignment problem is formulated to maximize production of the system, and a novel control strategy is developed to address this problem. Leveraging system properties derived from the model and moving window downtime prediction, the problem of maximizing system production is transformed into a more tractable control problem focused on identifying and achieving a defined ideal clean configurations. The proposed solution significantly outperforms various benchmarks, including a pure reinforcement learning-based strategy, underscoring the importance of system understanding and its crucial role in enhancing flexibility and productivity in manufacturing systems.

Synthesizing Formal Semantics from Executable Interpreters

Program verification and synthesis frameworks that allow one to customize the language in which one is interested typically require the user to provide a formally defined semantics for the language. Because writing a formal semantics can be a daunting and error-prone task, this requirement stands in the way of such frameworks being adopted by non-expert users. We present an algorithm that can automatically synthesize inductively defined syntax-directed semantics when given (i) a grammar describing the syntax of a language and (ii) an executable (closed-box) interpreter for computing the semantics of programs in the language of the grammar. Our algorithm synthesizes the semantics in the form of Constrained-Horn Clauses (CHCs), a natural, extensible, and formal logical framework for specifying inductively defined relations that has recently received widespread adoption in program verification and synthesis. The key innovation of our synthesis algorithm is a Counterexample-Guided Synthesis (CEGIS) approach that breaks the hard problem of synthesizing a set of constrained Horn clauses into small, tractable expression-synthesis problems that can be dispatched to existing SyGuS synthesizers. Our tool Synantic synthesized inductively-defined formal semantics from 14 interpreters for languages used in program-synthesis applications. When synthesizing formal semantics for one of our benchmarks, Synantic unveiled an inconsistency in the semantics computed by the interpreter for a language of regular expressions; fixing the inconsistency resulted in a more efficient semantics and, for some cases, in a 1.2x speedup for a synthesizer solving synthesis problems over such a language.

Notes on the Practical Application of Nested Sampling: MultiNest, (Non)convergence, and Rectification

Nested sampling is a promising tool for Bayesian statistical analysis because it simultaneously performs parameter estimation and facilitates model comparison. MultiNest is one of the most popular nested sampling implementations, and has been applied to a wide variety of problems in the physical sciences. However, MultiNest results, like those of any sampling tool, can be unreliable, and accompanying convergence tests are a component of any analysis. Using analytically tractable test problems, I illustrate how MultiNest, when applied without rigorously chosen hyperparameters, (1) can produce systematically erroneous estimates of the Bayesian evidence, which are more significantly biased for problems of higher dimensionality; (2) can derive posterior estimates with errors on the order of ~100%; (3) can, particularly when sampling noisy likelihood functions, systematically underestimate posterior widths. Furthermore, I show how MultiNest, thanks to the advantageous speed at which it explores parameter space, can also be used to jump-start Markov chain Monte Carlo sampling or more rigorous nested sampling techniques, potentially accelerating more robust measurements of posterior distributions and Bayesian evidences, and overcoming the challenge of Markov chain Monte Carlo initialization.

Finite algebras with Hom-sets of polynomial size

We provide an internal characterization of those finite algebras (i.e., algebraic structures) A \mathbf {A} such that the number of homomorphisms from any finite algebra X \mathbf {X} to A \mathbf {A} is bounded from above by a polynomial in the size of X \mathbf {X} . Namely, an algebra A \mathbf {A} has this property if, and only if, no subalgebra of A \mathbf {A} has a nontrivial strongly abelian congruence. We also show that the property can be decided in polynomial time for algebras in finite signatures. Moreover, if A \mathbf {A} is such an algebra, the set of all homomorphisms from X \mathbf {X} to A \mathbf {A} can be computed in polynomial time given X \mathbf {X} as input. As an application of our results to the field of computational complexity, we characterize inherently tractable constraint satisfaction problems over fixed finite structures, i.e., those that are tractable and remain tractable after expanding the fixed structure by arbitrary relations or functions.

Predicting the distribution of red king crab bycatch in Bering Sea flatfish trawl fisheries

Declining Bristol Bay red king crab (BBRKC) abundance has triggered recent closures of this iconic Bering Sea fishery and raised interest in bycatch in non-directed fisheries as a possible conservation concern. One particular concern is the effectiveness of static closed areas for bycatch fisheries in an era of climate warming and widespread distribution shifts. However, spatial data for supporting management decisions concerning bycatch is lacking, as fisheries-independent data are collected only in the summer, and the relationship to BBRKC distribution in the fall/winter/spring, when most bycatch occurs, is unknown. We filled this information gap by using fishery-dependent data to build predictive models of BBRKC bycatch distribution in non-pelagic trawl groundfish fisheries in the data-poor seasons. We trained Boosted Regression Tree models for bycatch occurrence and abundance of four BBRKC sex-size/maturity categories, and evaluation metrics indicated good to excellent predictive ability across all models. We found that flatfish directed-fishery CPUE, summer survey CPUE for BBRKC and flatfish, and depth were important predictors for bycatch occurrence and abundance. Physical variables (ice cover and temperature) were generally less important. We also found strong correlations between the mean latitude of observed bycatch and the summer survey for BBRKC, highlighting the ability of summer survey data to predict non-summer bycatch distributions. BBRKC bycatch prediction is a tractable problem, and our results are the first step towards operating models that may be used to evaluate proposed management actions. We also conclude that northward shifts in fishery-independent and -dependent data suggest the possible value of reassessing decades-old static closure areas for managing BBRKC bycatch.

A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation

In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios χ∥/χ⊥⋙1 (with χ∥, χ⊥ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. The implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ∼(NΔt)1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios (χ∥/χ⊥=1010).

Week-ahead dispatching of active distribution networks using hybrid energy storage systems

This paper presents a week-long scheduling approach to address the issues associated with uncertain stochastic generation. Specifically, the method is designed for active distribution networks (ADNs) hosting hybrid energy storages, composed by a hydrogen energy storage system (HESS) and a battery energy storage system (BESS). The inclusion of a pressurized HESS allows to balance energy over longer time periods, as opposed to methods considering only BESSs. To this end, this paper combines linearized models for the electricity grid with linearized models of the HESS to solve a tractable scheduling problem. The proposed optimal schedule consists of an active power trajectory at the grid connection point (GCP), called the dispatch plan, and the unit commitment schedule of a PEM fuel cell and electrolyzer system interfacing the electricity network with the HESS. Additionally, a bilevel model predictive control strategy is proposed, where the upper layer MPC computes a storage target accounting for the full horizon, while the lower layer computes the controllable resource setpoints to minimize the dispatch tracking error in each period. A numerical experiment shows the effectiveness of the proposed scheduling and control to accurately compute and track a dispatch plan over a full week. The results clearly show the benefits of combining a HESS with a BESS especially in periods where the prosumption is highly uncertain. Finally, we discuss the computational challenges associated with the weekly horizon and the use of a HESS that exhibits different dynamics than a BESS and propose an approach to mitigate the computational cost.

Robust Tensor Completion via Capped Frobenius Norm.

Tensor completion (TC) refers to restoring the missing entries in a given tensor by making use of the low-rank structure. Most existing algorithms have excellent performance in Gaussian noise or impulsive noise scenarios. Generally speaking, the Frobenius-norm-based methods achieve excellent performance in additive Gaussian noise, while their recovery severely degrades in impulsive noise. Although the algorithms using the lp -norm ( ) or its variants can attain high restoration accuracy in the presence of gross errors, they are inferior to the Frobenius-norm-based methods when the noise is Gaussian-distributed. Therefore, an approach that is able to perform well in both Gaussian noise and impulsive noise is desired. In this work, we use a capped Frobenius norm to restrain outliers, which corresponds to a form of the truncated least-squares loss function. The upper bound of our capped Frobenius norm is automatically updated using normalized median absolute deviation during iterations. Therefore, it achieves better performance than the lp -norm with outlier-contaminated observations and attains comparable accuracy to the Frobenius norm without tuning parameter in Gaussian noise. We then adopt the half-quadratic theory to convert the nonconvex problem into a tractable multivariable problem, that is, convex optimization with respect to (w.r.t.) each individual variable. To address the resultant task, we exploit the proximal block coordinate descent (PBCD) method and then establish the convergence of the suggested algorithm. Specifically, the objective function value is guaranteed to be convergent while the variable sequence has a subsequence converging to a critical point. Experimental results based on real-world images and videos exhibit the superiority of the devised approach over several state-of-the-art algorithms in terms of recovery performance. MATLAB code is available at https://github.com/Li-X-P/Code-of-Robust-Tensor-Completion.

Resource allocation for relaying cognitive network with energy harvesting

This paper investigates a relay-based cognitive wireless-powered communication network with multiple secondary receivers and a secondary relaying network. In the secondary network, a relay adopts an energy model to harvest energy from radio frequency signals of the primary and secondary sources and subsequently utilizes the harvested energy to cooperate with the source for information transmission to multiple Internet-of-things users over time-division multiple access. The research group formulated a novel optimization problem that jointly optimizes the power and time fraction for energy and information transmission to maximize the sum throughput of the secondary network. The authors first converted the non-convex problem to a more computationally tractable problem and then proposed an iterative algorithm in which closed-form solutions are obtained at each iteration. Thus, simulation results verify and demonstrate the effectiveness of the proposed approach.

Improving interdependent urban power and gas distribution systems resilience through optimal scheduling of mobile emergency supply and repair resources

Urban electric power and natural gas systems, serving as the most critical facilities that provide fundamental support to people's daily life, have been increasingly interdependent in their operation over the past decades. This growing interdependency presents great challenges and demands for enhancing the resilience of the interdependent systems, particularly driven by the recent cold weather events. In this paper, we propose a comprehensive strategy for scheduling various types of emergency resources, including mobile and stationary generators, repair crews, and fuel tankers, with the goal of enhancing the resilience of interdependent power and gas distribution systems in the aftermath of disasters. In particular, the strategy incorporates the routing of mobile resources to orchestrate their dynamic moving behaviors. Fuel supply issue for mobile and stationary distributed generators is also addressed through modeling the fuel exchange processes among generators, fuel tankers, and locations. Furthermore, we present a model of repair process to fully capture the real-world repair activities, allowing varying numbers of repair crews to work in a cooperative way. The proposed strategy is formulated into a tractable mixed-integer second-order cone problem. Finally, case studies are carried out to demonstrate the effectiveness of the strategy in enhancing the resilience of the interdependent power and gas distribution systems.

FoldPAthreader: predicting protein folding pathway using a novel folding force field model derived from known protein universe

Protein folding has become a tractable problem with the significant advances in deep learning-driven protein structure prediction. Here we propose FoldPAthreader, a protein folding pathway prediction method that uses a novel folding force field model by exploring the intrinsic relationship between protein evolution and folding from the known protein universe. Further, the folding force field is used to guide Monte Carlo conformational sampling, driving the protein chain fold into its native state by exploring potential intermediates. On 30 example targets, FoldPAthreader successfully predicts 70% of the proteins whose folding pathway is consistent with biological experimental data.

Getting linear time in graphs of bounded neighborhood diversity

AbstractParameterized complexity, introduced to efficiently solve NP‐hard problems for small values of a fixed parameter, has been recently used as a tool to speed up algorithms for tractable problems. Following this line of research, we design algorithms parameterized by neighborhood diversity () for several graph theoretic problems in : Maximum ‐Matching, Triangle Counting and Listing, Girth, Global Minimum Vertex Cut, and Perfect Graphs Recognition. Such problems are known to admit algorithms parameterized by modular‐width () and consequently—as is a special case of —by . However, the proposed novel algorithms allow for improving the computational complexity from time —where and denote, respectively, the number of vertices and edges in the input graph—to time which is only additive in the size of the input. Then we consider some classical NP‐hard problems (Maximum independent set, Maximum clique, and Minimum dominating set) and show that for several classes of hereditary graphs, they admit linear time algorithms for sufficiently small—nonnecessarily constant—values of the neighborhood diversity parameter.

A new distributionally robust reward-risk model for portfolio optimization

Abstract A new distributionally robust ratio optimization model is proposed under the known first and second moments of the uncertain distributions. In this article, both standard deviation (SD) and conditional value-at-risk (CVaR) are used to measure the risk, avoiding both fat-tail and volatility. The new model can be reduced to a simple distributionally robust model under assumptions on the measurements of reward, CVaR and SD. Furthermore, it can be rewritten as a tractable semi-definite programming problem by the duality theorem under partially known information of the uncertain parameters. Finally, the model is tested on portfolio problems and verified from numerical results that it can give a reasonable decision under only the first and second moments.

Enhancing Make-to-Order Manufacturing Agility: When Flexible Capacity Meets Dynamic Pricing

The rise of online marketplaces has raised customer expectations regarding customization and lead time. It poses significant challenges to manufacturing firms and prompts a move from make-to-stock to a more flexible make-to-order system. Compared to make-to-stock settings, make-to-order systems cannot smooth fluctuations in demand using available stock. While viewing dynamic pricing as a useful strategy to balance supply with demand, many manufacturing firms can also create capacity flexibility. In that scenario, system costs could be cut by managing capacity and demand simultaneously. In this paper, we consider a make-to-order production environment with base and surge capacity as well as the ability to adjust product pricing. Our main focus is on operational decision-making, assuming that the base capacity and surge capacity are fixed, but activating the surge capacity incurs a setup cost. Initially, we propose a stochastic control model to reflect this complex decision problem. However, our initial model leads to an intractable dynamic programming problem. To overcome this, we convert the problem to a more tractable diffusion control problem. This approach helps to reveal the conditions under which utilizing flexible capacity is more advantageous than relying solely on fixed capacity. When flexible capacity is advantageous, we provide a solution to the diffusion control problem that can guide optimal capacity and price adjustments. We discover an interesting interplay between capacity adjustment and dynamic pricing. In particular, we find that the price, which aims at reducing congestion, may not monotonically increase with the congestion level when capacity adjustments incur a fixed cost.

Stacking Revenues From Flexible DERs in Multi-Scale Markets Using Tri-Level Optimization

Rapid proliferation of flexible Distributed Energy Resources (DERs) as a result of Net Zero Emissions objectives entails a profound shift in the paradigm of local and national energy systems. Currently, DERs' simultaneous participation in multiple markets is generally restricted, which undermines their profitability. With the aim of increasing the number of business cases for them, a tri-level optimization problem that seeks the maximisation of revenues from DERs is proposed. The optimization problem considers simultaneous participation of different flexible DERs, such as, Electric Vehicles (EVs), Battery Energy Storage Systems (BESSs) and Heating, Ventilation and Air Conditioning (HVACs), in national and local markets. Markets are cleared sequentially, and the model is recast into a tractable single-level problem using its dual formulation and strong duality condition. Results from a case study based on the IEEE 14 bus transmission network, a realistic distribution network and SimBench dataset demonstrate the effectiveness of the proposed approach in increasing profits compared with a baseline scenario.

Expected values for the accuracy of predicted breeding values accounting for genetic differences between reference and target populations

BackgroundGenetic merit, or breeding values as referred to in livestock and crop breeding programs, is one of the keys to the successful selection of animals in commercial farming systems. The developments in statistical methods during the twentieth century and single nucleotide polymorphism (SNP) chip technologies in the twenty-first century have revolutionized agricultural production, by allowing highly accurate predictions of breeding values for selection candidates at a very early age. Nonetheless, for many breeding populations, realized accuracies of predicted breeding values (PBV) remain below the theoretical maximum, even when the reference population is sufficiently large, and SNPs included in the model are in sufficient linkage disequilibrium (LD) with the quantitative trait locus (QTL). This is particularly noticeable over generations, as we observe the so-called erosion of the effects of SNPs due to recombinations, accompanied by the erosion of the accuracy of prediction. While accurately quantifying the erosion at the individual SNP level is a difficult and unresolved task, quantifying the erosion of the accuracy of prediction is a more tractable problem. In this paper, we describe a method that uses the relationship between reference and target populations to calculate expected values for the accuracies of predicted breeding values for non-phenotyped individuals accounting for erosion. The accuracy of the expected values was evaluated through simulations, and a further evaluation was performed on real data.ResultsUsing simulations, we empirically confirmed that our expected values for the accuracy of PBV accounting for erosion were able to correctly determine the prediction accuracy of breeding values for non-phenotyped individuals. When comparing the expected to the realized accuracies of PBV with real data, only one out of the four traits evaluated presented accuracies that were significantly higher than the expected, approaching h2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{{{\ ext{h}}}^{2}}$$\\end{document}.ConclusionsWe defined an index of genetic correlation between reference and target populations, which summarizes the expected overall erosion due to differences in allele frequencies and LD patterns between populations. We used this correlation along with a trait’s heritability to derive expected values for the accuracy (R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{R}}$$\\end{document}) of PBV accounting for the erosion, and demonstrated that our derived ER|erosion\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext{E}}\\left[{\ ext{R}}|{\ ext{erosion}}\\right]$$\\end{document} is a reliable metric.

A Free Streamline Model for Fully Separated Turbomachine Flows

Abstract This paper describes the use of the free streamline approach to determine flows in fully separated cascades, and to thereby provide a rational basis for estimating the performance of axial turbomachinery at severely off-design conditions. On the assumption of two-dimensional, incompressible flow, the free streamlines that bound the separated region are determined using the hodograph method, in which the methods of complex analysis are used to define a tractable numerical problem. The resulting jet-wake flow profile at cascade exit is then presumed to undergo mixing at constant-area, from which performance parameters for the cascade can be determined. The proposed method is verified by comparison against flat plate cascade solutions and by checks against momentum conservation. Validation of the free streamline approach is carried out by comparison against several cascade configurations for which experimental data and simulation results are available. In general, the predictions based upon the present theoretical model are found to be good, especially for negative incidences such as those that occur in the compressor section of an engine operating at sub-idle conditions. This suggests that the present approach can be used in the estimation of turbomachinery performance maps far from the design point, especially during early design phases.

Optimal phase-based control of strongly perturbed limit cycle oscillators using phase reduction techniques.

Phase reduction is a well-established technique for analysis and control of weakly perturbed limit cycle oscillators. However, its accuracy is diminished in a strongly perturbed setting where information about the amplitude dynamics must also be considered. In this paper, we consider phase-based control of general limit cycle oscillators in both weakly and strongly perturbed regimes. For use at the strongly perturbed end of the continuum, we propose a strategy for optimal phase control of general limit cycle oscillators that uses an adaptive phase-amplitude reduced order model in conjunction with dynamic programming. This strategy can accommodate large magnitude inputs at the expense of requiring additional dimensions in the reduced order equations, thereby increasing the computational complexity. We apply this strategy to two biologically motivated prototype problems and provide direct comparisons to two related phase-based control algorithms. In situations where other commonly used strategies fail due to the application of large magnitude inputs, the adaptive phase-amplitude reduction provides a viable reduced order model while still yielding a computationally tractable control problem. These results highlight the need for discernment in reduced order model selection for limit cycle oscillators to balance the trade-off between accuracy and dimensionality.

A high-resolution handheld millimeter-wave imaging system with phase error estimation and compensation

Despite the enormous potential of millimeter-wave (mmWave) imaging, the high cost of large-scale antenna arrays or stringent prerequisites of the synthetic aperture radar (SAR) principle impedes its widespread application. Here, we report a portable, affordable, and high-resolution 3D mmWave imaging system by overcoming the destructive motion error of handheld SAR imaging. This is achieved by revealing two important phenomenons: spatial asymmetry of motion errors in different directions, and local similarity of phase errors exhibited by different targets, based on which we formulate the challenging phase error estimation problem as a tractable point spread function optimization problem. Experiments demonstrate that our approach can recover high-fidelity 3D mmWave images from severely distorted signals and augment the aperture size by over 50 times. Since our system does not rely on costly massive antennas or bulky motion controllers, it can be applied for diverse applications including security inspection, autonomous driving, and medical monitoring.

A Bayesian Approach Toward Robust Multidimensional Ellipsoid-Specific Fitting.

This work presents a novel and effective method for fitting multidimensional ellipsoids (i.e., ellipsoids embedded in [Formula: see text]) to scattered data in the contamination of noise and outliers. Unlike conventional algebraic or geometric fitting paradigms that assume each measurement point is a noisy version of its nearest point on the ellipsoid, we approach the problem as a Bayesian parameter estimate process and maximize the posterior probability of a certain ellipsoidal solution given the data. We establish a more robust correlation between these points based on the predictive distribution within the Bayesian framework, i.e., considering each model point as a potential source for generating each measurement. Concretely, we incorporate a uniform prior distribution to constrain the search for primitive parameters within an ellipsoidal domain, ensuring ellipsoid-specific results regardless of inputs. We then establish the connection between measurement point and model data via Bayes' rule to enhance the method's robustness against noise. Due to independent of spatial dimensions, the proposed method not only delivers high-quality fittings to challenging elongated ellipsoids but also generalizes well to multidimensional spaces. To address outlier disturbances, often overlooked by previous approaches, we further introduce a uniform distribution on top of the predictive distribution to significantly enhance the algorithm's robustness against outliers. Thanks to the uniform prior, our maximum a posterior probability coincides with a more tractable maximum likelihood estimation problem, which is subsequently solved by a numerically stable Expectation Maximization (EM) framework. Moreover, we introduce an ε-accelerated technique to expedite the convergence of EM considerably. We also investigate the relationship between our algorithm and conventional least-squares-based ones, during which we theoretically prove our method's superior robustness. To the best of our knowledge, this is the first comprehensive method capable of performing multidimensional ellipsoid-specific fitting within the Bayesian optimization paradigm under diverse disturbances. We evaluate it across lower and higher dimensional spaces in the presence of heavy noise, outliers, and substantial variations in axis ratios. Also, we apply it to a wide range of practical applications such as microscopy cell counting, 3D reconstruction, geometric shape approximation, and magnetometer calibration tasks. In all these test contexts, our method consistently delivers flexible, robust, ellipsoid-specific performance, and achieves the state-of-the-art results.