Continuous Relaxation Research Articles

Almost exact recovery in noisy semi-supervised learning

Abstract Graph-based semi-supervised learning methods combine the graph structure and labeled data to classify unlabeled data. In this work, we study the effect of a noisy oracle on classification. In particular, we derive the maximum a posteriori (MAP) estimator for clustering a degree corrected stochastic block model when a noisy oracle reveals a fraction of the labels. We then propose an algorithm derived from a continuous relaxation of the MAP, and we establish its consistency. Numerical experiments show that our approach achieves promising performance on synthetic and real data sets, even in the case of very noisy labeled data.

Characterization of myocardial ischemia-reperfusion injury in a pig model with novel CW T1r and RAFF2 MRI methods in 1.5T

Abstract Introduction Myocardial ischemia-reperfusion injury (IRI) occurs after myocardial infarction when reperfusion aggravates the initial damage in myocardium by vast infiltration of inflammatory cells and elevated oxidative stress in myocardium that may lead to cardiac remodeling, heart failure and even death [1]. Magnetic resonance imaging (MRI) can accurately determine the anatomy of the heart and to characterize myocardial tissue. Novel endogenous rotating frame MRI methods, including continuous wave (CW) T1r and relaxation along a fictitious field with rank 2 (RAFF2), have been shown to characterize MI and other CVD in mouse and human myocardium with high sensitivity and specificity [2,3]. CW T1r and RAFF2 contrasts are occurring during radio frequency (RF) pulse (in kHz range) making them to be sensitive to slow molecular motions [2]. Comparing to conventional endogenous T1 and T2 methods, these are occurring after the RF pulse (in Larmor frequency (MHz) range) and therefore, these are non-selectively sensitive to different correlation times [4]. Golden standard to image MI is to use contrast agent (CA) in late gadolinium enhancement (LGE) method [2,4]. Using CA has drawbacks and therefore, there is an urgent need for optimizing the endogenous MRI methods in clinical practice. Purpose To characterize myocardial IRI in a pig model and to determine the sensitivity of both novel and conventional cardiac MRI methods. Methods IRI operation was done in 7 pig (30-35 kg) by occluding the LAD with a balloon catheter for 50 minutes. Imaging was done with 1.5T (MAGNETOM Aera) at 3 days and 28 days after the operation. Pig hearts were imaged with conventional native T1, T2, LGE, and with novel CW T1r and RAFF2 methods in both time points. These were imaged before CA injection and LGE was done 25 minutes after CA injection. Relaxation time mappings were done in a pixel-by-pixel manner with monoexponentially fitting. Region-of-interest (ROI) analysis was done to determine the relaxation times in IRI and remote areas. LGE was used to confirm the IRI areas in the myocardium. Relative relaxation time difference (RRTD)-values were calculated with the function of (((Relaxation Time of IRI area) – (Relaxation Time of Remote Area))/ (Relaxation Time of Remote Area)) to determine the contrast difference between IRI and remote areas. Results Relaxation times were elevated in IRI areas compared to remote areas (Figure 1A). Novel rotating frame RRTD-values were significantly higher than RRTD-values in conventional methods. This means that novel methods have larger contrast difference between IRI and remote areas compared to conventional methods, which indicates that novel methods are sensitive to characterize IRI area (Figure 1B). The location of MI areas was similar in both novel rotating frame and conventional methods compared to LGE (Figure 2). Conclusion Novel rotating frame relaxation time methods are sensitive to characterize the IRI area in pig model.Relaxation times, RRTD-valuesRelaxation times maps and LGE

Approach for multi-valued integer programming in multi-material topology optimization: Random discrete steepest descent (RDSD) algorithm

The present study models the multi-material topology optimization problems as the multi-valued integer programming (MVIP) or named as combinatorial optimization. By extending classical convex analysis and convex programming to discrete point-set functions, the discrete convex analysis and discrete steepest descent (DSD) algorithm are introduced. To overcome combinatorial complexity of the DSD algorithm, we employ the sequential approximate integer programming (SAIP) to explicitly and linearly approximate the implicit objective and constraint functions. Considering the multiple potential changed directions for multi-valued design variables, the random discrete steepest descent (RDSD) algorithm is proposed, where a random strategy is implemented to select a definitive direction of change. To analytically calculate multi-material discrete variable sensitivities, topological derivatives with material contrast is applied. In all, the MVIP is finally transferred as the linear 0–1 programming that can be efficiently solved by the canonical relaxation algorithm (CRA). Explicit nonlinear examples demonstrate that the RDSD algorithm owns nearly three orders of magnitude improvement compared with the commercial software (GUROBI). The proposed approach, without using any continuous variable relaxation and interpolation penalization schemes, successfully solves the minimum compliance problem, strength-related problem, and frequency-related optimization problems. Given the algorithm efficiency, mathematical generality and merits over other algorithms, the proposed RDSD algorithm is meaningful for other structural and topology optimization problems involving multi-valued discrete design variables.

Direct Identification of the Continuous Relaxation Time and Frequency Spectra of Viscoelastic Materials

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined.

Effect of dynamic bond concentration on the mechanical properties of vitrimers.

The presence of dynamic covalent bonds allows vitrimers to undergo topology alterations and display self-healing properties. Herein, we study the influence of varying the concentration of dynamic bonds on the macroscopic properties of hybrid vitrimer networks by subjecting them to triaxial stretching tests using molecular simulations. Results show that the presence of dynamic bonds allows for continuous stress relaxation in the hybrid networks leading to delayed craze development and higher stretching as compared to permanently crosslinked networks. The work highlights the ability of glassy vitrimer networks to relax tensile stress during deformation successfully.

Multi-parameter viscoelastic material model for denture adhesives based on time-temperature superposition and multiple linear regression analysis

BackgroundRestorative solutions designed for edentulous patients such as dentures and their accompanying denture adhesives operate in the complex and dynamic environment represented by human oral physiology. Developing material models accounting for the viscoelastic behavior of denture adhesives can facilitate their further optimization within that unique physiological environment. This study aims to statistically quantify the degree of significance of three physiological variables - namely: temperature, adhesive swelling, and pH - on denture adhesive mechanical behavior. Further, based on these statistical significance estimations, a previously-developed viscoelastic material modelling approach for such denture adhesives is further expanded and developed to capture these variables’ effects on mechanical behavior.MethodsIn this study a comparable version of Denture adhesive Corega Comfort was analysed rheologically using the steady state frequency sweep tests. The experimentally derived rheological storage and loss modulus values for the selected physiological variables were statistically analyzed using multi parameter linear regression analysis and the Pearson’s coefficient technique to understand the significance of each individual parameter on the relaxation spectrum of the denture adhesive. Subsequently, the parameters are incorporated into a viscoelastic material model based on Prony series discretization and time-temperature superposition, and the mathematical relationship for the loss modulus is deduced.ResultsThe results of this study clearly indicated that the variation in both the storage and loss modulus values can be accurately predicted using the oral cavity physiological parameters of temperature, swelling ratio, and pH with an adjusted R2 value of 0.85. The R2 value from the multi-parameter regression analysis indicated that the predictor variables can estimate the loss and storage modulus with a reasonable accuracy for at least 85% of the rheologically determined continuous relaxation spectrum with a confidence level of 98%. The Pearson’s coefficient for the independent variables indicated that temperature and swelling have a strong influence on the loss modulus, whereas pH had a weak influence. Based on statistical analysis, these mathematical relationships were further developed in this study.ConclusionsThis multi-parameter viscoelastic material model is intended to facilitate future detailed numerical investigations performed with implementation of denture adhesives using the finite element method.

Robust Recovery of Optimally Smoothed Polymer Relaxation Spectrum from Stress Relaxation Test Measurements.

The relaxation spectrum is a fundamental viscoelastic characteristic from which other material functions used to describe the rheological properties of polymers can be determined. The spectrum is recovered from relaxation stress or oscillatory shear data. Since the problem of the relaxation spectrum identification is ill-posed, in the known methods, different mechanisms are built-in to obtain a smooth enough and noise-robust relaxation spectrum model. The regularization of the original problem and/or limit of the set of admissible solutions are the most commonly used remedies. Here, the problem of determining an optimally smoothed continuous relaxation time spectrum is directly stated and solved for the first time, assuming that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. The relaxation time spectrum model that reproduces the relaxation modulus measurements and is the best smoothed in the class of continuous square-integrable functions is sought. Based on the Hilbert projection theorem, the best-smoothed relaxation spectrum model is found to be described by a finite sum of specific exponential-hyperbolic basis functions. For noise-corrupted measurements, a quadratic with respect to the Lagrange multipliers term is introduced into the Lagrangian functional of the model's best smoothing problem. As a result, a small model error of the relaxation modulus model is obtained, which increases the model's robustness. The necessary and sufficient optimality conditions are derived whose unique solution yields a direct analytical formula of the best-smoothed relaxation spectrum model. The related model of the relaxation modulus is given. A computational identification algorithm using the singular value decomposition is presented, which can be easily implemented in any computing environment. The approximation error, model smoothness, noise robustness, and identifiability of the polymer real spectrum are studied analytically. It is demonstrated by numerical studies that the algorithm proposed can be successfully applied for the identification of one- and two-mode Gaussian-like relaxation spectra. The applicability of this approach to determining the Baumgaertel, Schausberger, and Winter spectrum is also examined, and it is shown that it is well approximated for higher frequencies and, in particular, in the neighborhood of the local maximum. However, the comparison of the asymptotic properties of the best-smoothed spectrum model and the BSW model a priori excludes a good approximation of the spectrum in the close neighborhood of zero-relaxation time.

Index tracking via reparameterizable subset sampling in neural networks

Index tracking aims at replicating the benchmark index performance by constructing a tracking portfolio consisting of partial component stocks of the index. This is an NP-hard question because of the cardinality constraint which limits the number of component stocks in the tracking portfolios. This paper proposes a new method to solve this problem in an end-to-end way by using a continuous subset sampling relaxation which can provide differentiable gradients calculation in neural networks. The proposed approach is tested with the FTSE-100 index data and the S&P100 index data, and compared with widely used index tracking methods, such as the forward and backward selection and optimization solvers. According to the empirical results, our proposed approach achieves the best tracking performance for different evaluation criteria.

Influence of non-pharmaceutical interventions on epidemiological characteristics of Mycoplasma pneumoniae infection in children during and after the COVID-19 epidemic in Ningbo, China.

Since the outbreak of COVID-19, China has implemented a series of non-pharmaceutical interventions (NPIs), effectively containing the spread of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) as well as various respiratory pathogens. With the continuous relaxation of restrictions, China has entered a new phase of the post-pandemic era. However, the epidemiological differences of Mycoplasma pneumoniae (MP) between the two phases in Ningbo and even in China remain unclear. Data of children aged 0-14 years who visited the Ningbo Medical Center LiHuiLi Hospital due to acute respiratory tract infections from January 2020 to December 2023 were collected. PCR was used to detect 13 respiratory pathogens and the macrolide-resistance of Mycoplasma pneumoniae. Among 10,206 children, 2,360 were infected with MP (23.12%). Among the total, the MP positive rate during the NPI phase (6.35%) was significantly lower than that during the non-NPI phase (34.28%), while the macrolide resistance rate increased from 62.5% (NPI phase) to 81.1% (non-NPI phase). The rate of MP co-infection increased from 11.2% (NPI phase) to 30.3% (non-NPI phase). MP infection exhibited obvious seasonality, with the highest prevalence in autumn (30.0%) followed by summer (23.6%). There were differences in MP positivity rates among different age groups, with the highest among school-age children at 39.5%. During the NPI phase, all age groups were less susceptible to MP, while during the non-NPI phase, the susceptible age for MP was 4-12 years, with 8 years being the most susceptible. The susceptible age for MP co-infection was 0-6 years. MP exhibited antagonistic effects against numerous pathogens. Compared to MP single infection, the proportion of pneumonia was higher in MP co-infection cases. The removal of NPIs significantly impacted the spread of MP, altering population characteristics including age, seasonality, macrolide resistance, and MP co-infection rates.

Neural discovery of balance-aware polarized communities

Signed graphs are a model to depict friendly (positive) or antagonistic (negative) interactions (edges) among users (nodes). 2-Polarized-Communities (2pc) is a well-established combinatorial-optimization problem whose goal is to find two polarized communities from a signed graph, i.e., two subsets of nodes (disjoint, but not necessarily covering the entire node set) which exhibit a high number of both intra-community positive edges and negative inter-community edges. The state of the art in 2pc suffers from the limitations that (i) existing methods rely on a single (optimal) solution to a continuous relaxation of the problem in order to produce the ultimate discrete solution via rounding, and (ii) 2pc objective function comes with no control on size balance among communities. In this paper, we provide advances to the 2pc problem by addressing both these limitations, with a twofold contribution. First, we devise a novel neural approach that allows for soundly and elegantly explore a variety of suboptimal solutions to the relaxed 2pc problem, so as to pick the one that leads to the best discrete solution after rounding. Second, we introduce a generalization of 2pc objective function – termed γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}-polarity – which fosters size balance among communities, and we incorporate it into the proposed machine-learning framework. Extensive experiments attest high accuracy of our approach, its superiority over the state of the art, and capability of function γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}-polarity to discover high-quality size-balanced communities.

Unmanned aerial vehicle variable radius set covering problem for emergency wireless network

The utilization of unmanned aerial vehicles (UAVs) has garnered increasing interest across various disciplines. The idea of smart cities led to the exploration of using UAVs in the public service industry, with a focus on integrating all aspects of the system through information and physical items. In this paper, we introduce and model the operational problem of using UAVs to construct an emergency wireless network in a disaster situation using the set covering approach. Unique to our study is the consideration that UAVs do not require predetermined positioning or altitude specifications. Consequently, the problem described in this paper can be classified as a planar variable radius covering problem, which involves a nonconvex continuous relaxation feasible set. We introduce the application of the Dantzig–Wolfe decomposition alongside a branch-and-price algorithm to tackle this problem. Additionally, the pricing subproblem’s solvable equivalent formulation is ingeniously derived from existing theories on the minimum covering circle. For large-scale instances, we propose a heuristic derived from an extended formulation. Comparative computational experiments demonstrate that our proposed algorithms significantly surpass the efficiency of the benchmark genetic algorithm previously reported in the literature.

Continuous Exact Relaxation and Alternating Proximal Gradient Algorithm for Partial Sparse and Partial Group Sparse Optimization Problems

Continuous Exact Relaxation and Alternating Proximal Gradient Algorithm for Partial Sparse and Partial Group Sparse Optimization Problems

Second‐order cone programming models for the unitary weighted Weber problem and for the minimum sum of the squares clustering problem

AbstractIn this work, new mixed integer nonlinear optimization models are proposed for two clustering problems: the unitary weighted Weber problem and the minimum sum of squares clustering. The proposed formulations are convex quadratic models with linear and second‐order cone constraints that can be efficiently solved by interior point algorithms. Their continuous relaxation is convex and differentiable. The numerical experiments show the proposed models are more efficient than some classical models for these problems known in the literature.

Finite-Time Continuous Nonsingular Terminal Modified Adaptive-Gain Super-Twisting Control: Application to a 2-DOF Planar Robot Manipulator System.

This article proposes finite-time continuous nonsingular terminal modified adaptive-gain super-twisting control (FT-CNT-MAG-STC) for the second-order disturbed systems. Compared with existing sliding-mode controllers, the noteworthy improvements of the proposed framework are the fast finite-time convergence, continuous control signal, ease-of-implementation feature, and relaxation of the assumption related to the information on the bounds of the disturbance and its derivative. In the proposed framework, a fast nonsingular terminal sliding surface is first developed such that the singularity problem is avoided and the convergence rate is improved. Then, we design a continuous modified super-twisting algorithm with an adaptive gain, under which the need for the bounds information of the disturbance and its derivative is relaxed and the performance of the closed-loop system is enhanced. Rigorous analysis is provided to prove the finite-time convergence of the system states to small regions containing the origin. We apply the proposed framework to the position control for a 2-DOF planar robot manipulator system. Various experimental results are illustrated to evaluate the effectiveness of the designed position controller.

Supervised Contrastive Discrete Hashing for cross-modal retrieval

Cross-modal hashing aims at mapping data points from varied modalities to a shared Hamming space for providing swift and adaptive search services. The recent progress in contrastive learning has significantly enhanced the retrieval performance of cross-modal hashing, owing to its strong ability in powerful representations learning. However, most of them face two primary challenges: (1) they select negative samples for anchors randomly in each mini-batch, which may bring false negative samples that destroy the intrinsic semantic similarity in the learned hash codes; (2) the continuous relaxation strategy is commonly used to generate discrete hash codes, which results in quantization errors and consequently impairing the quality of hash codes. To alleviate these issues, we propose a deep supervised cross-modal hashing scheme, termed Supervised Contrastive Discrete Hashing (SCDH). Specifically, we extend the unsupervised contrastive method to the supervised setting to construct the positive and negative data points with class labels, avoiding the performance decreasing caused by false negative data. Besides, multiple positive and negative samples for each anchor can be chosen from the generated global dictionary bank by class labels, which can better capture the global data structure. Moreover, discrete hash codes can be generated directly, enhancing the feature representational capacity of hash codes. Comprehensive experiments conducted on two public datasets affirm the superior performance of the proposed SCDH over several other cutting-edge methods.

Experimental study of rotational relaxation for D2(1,12) in collisions with N2.

The rotational relaxation behavior of D2(1,12) in a D2-N2 mixture was investigated using coherent anti-Stokes Raman scattering (CARS) technique. The rovibrational level v = 1 and J = 12 of D2 was selectively excited through stimulated Raman pumping while monitoring the temporal evolution of population for D2(1, J ≤ 12) molecules using time-resolved CARS spectroscopy. The results demonstrate that the rotational relaxation processes of D2(1,12) encompass both multi-quantum relaxation and continuous single-quantum relaxation. When α, the molar ratio of N2, is less than 0.5, D2(1,12) predominantly undergoes a single quantum relaxation process transition. However, when α ≥ 0.5, the multi-quantum relaxation mechanism gradually predominates. The total rotational relaxation rate coefficients of D2(1,12) collisions with N2 and D2 at 295K were determined to be 3.974 × 10-14 and 1.179 × 10-14cm3s-1, respectively. The temperature dependence of rotational relaxation rate of D2(1,12) was investigated within the temperature range of 295-453K. With increasing temperature, the dominant relaxation process exhibited an accelerated behavior, while the minor relaxation process remained largely unaffected. The rotational temperature of the D2molecule at various N2 molar ratios was determined through the utilization of Boltzmann plots. The rotational temperature undergoes a rapid decline within 2 μs, corresponding to the near-resonant rotation-vibration relaxation process of D2(1,12) collisions with N2. The system reaches a quasi-equilibrium state when the delay time is 3 μs. The findings of this study can serve as a valuable empirical basis for further validation of the kinetic theory and simulation.

Mathematical programming for simultaneous feature selection and outlier detection under l1 norm

The goal of simultaneous feature selection and outlier detection is to determine a sparse linear regression vector by fitting a dataset possibly affected by the presence of outliers.The problem is well-known in the literature. In its basic version it covers a wide range of tasks in data analysis. Simultaneously performing feature selection and outlier detection strongly improves the application potential of regression models in more general settings, where data governance is a concern. To trigger this potential, flexible training models are needed, with more parameters under control of decision makers.The use of mathematical programming, although pertinent, is scarce in this context and mostly focusing on the least-squares setting. Instead we consider the least absolute deviation criterion, proposing two mixed-integer linear programs, one adapted from existing studies, and the other obtained from a disjunctive programming argument. We show theoretically and computationally that the disjunctive-based formulation is better in terms of both continuous relaxation quality and integer optimality convergence.We experimentally benchmark against existing methodologies from the literature. We identify the characteristics of contamination patterns, in which mathematical programming is better than state-of-the-art algorithms in combining prediction quality, sparsity and robustness against outliers. Additionally, the mathematical programming approaches allow the decision maker to directly control parameters like the number of features or outliers to tolerate, those based on least absolute deviations performing best. On real world datasets, where privacy is a concern, our approach compares well to state-of-the-art methods in terms of accuracy, being at the same time more flexible.

IS-DARTS: Stabilizing DARTS through Precise Measurement on Candidate Importance

Among existing Neural Architecture Search methods, DARTS is known for its efficiency and simplicity. This approach applies continuous relaxation of network representation to construct a weight-sharing supernet and enables the identification of excellent subnets in just a few GPU days. However, performance collapse in DARTS results in deteriorating architectures filled with parameter-free operations and remains a great challenge to the robustness. To resolve this problem, we reveal that the fundamental reason is the biased estimation of the candidate importance in the search space through theoretical and experimental analysis, and more precisely select operations via information-based measurements. Furthermore, we demonstrate that the excessive concern over the supernet and inefficient utilization of data in bi-level optimization also account for suboptimal results. We adopt a more realistic objective focusing on the performance of subnets and simplify it with the help of the informationbased measurements. Finally, we explain theoretically why progressively shrinking the width of the supernet is necessary and reduce the approximation error of optimal weights in DARTS. Our proposed method, named IS-DARTS, comprehensively improves DARTS and resolves the aforementioned problems. Extensive experiments on NAS-Bench-201 and DARTS-based search space demonstrate the effectiveness of IS-DARTS.

Linear Lexicographic Optimization and Preferential Bidding System

Some airlines use the preferential bidding system to construct the schedules of their pilots. In this system, the pilots bid on the different activities and the schedules that lexicographically maximize the scores of the pilots according to their seniority are selected. A sequential approach to solve this maximization problem is natural: The problem is first solved with the bids of the most senior pilot, and then it is solved with those of the second most senior without decreasing the score of the most senior, and so on. The literature admits that the structure of the problem somehow imposes such an approach. The problem can be modeled as an integer linear lexicographic program. We propose a new efficient method, which relies on column generation for solving its continuous relaxation and returns proven optimality gaps. To design this column generation, we prove that bounded linear lexicographic programs admit “primal-dual” feasible bases, and we show how to compute such bases efficiently. Another contribution on which our method relies is the extension of standard tools for resource-constrained longest path problems to their lexicographic versions. This is useful in our context because the generation of new columns is modeled as a lexicographic resource-constrained longest path problem. Numerical experiments show that this new method is already able to solve to proven optimality industrial instances provided by Air France, with up to 150 pilots. By adding a last ingredient in the resolution of the longest path problems, which exploits the specificity of the preferential bidding system, the method achieves for these instances computational times that are compatible with operational constraints. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0372 .

Prompt Optimization in Large Language Models

Prompt optimization is a crucial task for improving the performance of large language models for downstream tasks. In this paper, a prompt is a sequence of n-grams selected from a vocabulary. Consequently, the aim is to select the optimal prompt concerning a certain performance metric. Prompt optimization can be considered as a combinatorial optimization problem, with the number of possible prompts (i.e., the combinatorial search space) given by the size of the vocabulary (i.e., all the possible n-grams) raised to the power of the length of the prompt. Exhaustive search is impractical; thus, an efficient search strategy is needed. We propose a Bayesian Optimization method performed over a continuous relaxation of the combinatorial search space. Bayesian Optimization is the dominant approach in black-box optimization for its sample efficiency, along with its modular structure and versatility. We use BoTorch, a library for Bayesian Optimization research built on top of PyTorch. Specifically, we focus on Hard Prompt Tuning, which directly searches for an optimal prompt to be added to the text input without requiring access to the Large Language Model, using it as a black-box (such as for GPT-4 which is available as a Model as a Service). Albeit preliminary and based on “vanilla” Bayesian Optimization algorithms, our experiments with RoBERTa as a large language model, on six benchmark datasets, show good performances when compared against other state-of-the-art black-box prompt optimization methods and enable an analysis of the trade-off between the size of the search space, accuracy, and wall-clock time.