Bayesian Optimization Approach Research Articles

Optimizing data acquisition: a Bayesian approach for efficient machine learning model training

Abstract Acquiring a substantial number of data points for training accurate machine learning (ML) models is a big challenge in scientific fields where data collection is resource-intensive. Here, we propose a novel approach for constructing a minimal yet highly informative database for training ML models in complex multi-dimensional parameter spaces. To achieve this, we mimic the underlying relation between the output and input parameters using Gaussian process regression (GPR). Using a set of known data, GPR provides predictive means and standard deviation for the unknown data. Given the predicted standard deviation by GPR, we select data points using Bayesian optimization to obtain an efficient database for training ML models. We compare the performance of ML models trained on databases obtained through this method, with databases obtained using traditional approaches. Our results demonstrate that the ML models trained on the database obtained using Bayesian optimization approach consistently outperform the other two databases, achieving high accuracy with a significantly smaller number of data points. Our work contributes to the resource-efficient collection of data in high-dimensional complex parameter spaces, to achieve high precision machine learning predictions.

A Bayesian optimization approach for data‐driven mixed‐integer nonlinear programming problems

AbstractThe optimization of process systems within the field of Chemical Engineering often confronts uncertainties that can exert significant influences on the performance and dependability of the obtained solutions. This research endeavors to investigate the application of Bayesian optimization in the realm of constrained mixed integer nonlinear problems. A comparative analysis was conducted, exploring different surrogate models, and evaluating diverse kernel functions and acquisition functions. Furthermore, a sampling strategy was devised to assess the enhancement achieved by the acquisition function. The findings of this article reveal the superior performance of sparse Gaussian processes in conjunction with computationally inexpensive acquisition functions, thereby highlighting their suitability for addressing mixed integer nonlinear programming problems characterized by noisy functions and stochastic behavior. Consequently, this article presents a computationally efficient approach to effectively tackle the challenges associated with data‐driven mixed integer nonlinear programming problems within the domain of Process System Engineering.

Machine learning for efficient grazing-exit x-ray absorption near edge structure spectroscopy analysis: Bayesian optimization approach

In materials science, traditional techniques for analyzing layered structures are essential for obtaining information about local structure, electronic properties and chemical states. While valuable, these methods often require high vacuum environments and have limited depth profiling capabilities. The grazing exit x-ray absorption near-edge structure (GE-XANES) technique addresses these limitations by providing depth-resolved insight at ambient conditions, facilitating in situ material analysis without special sample preparation. However, GE-XANES is limited by long data acquisition times, which hinders its practicality for various applications. To overcome this, we have incorporated Bayesian optimization (BO) into the GE-XANES data acquisition process. This innovative approach potentially reduces measurement time by a factor of 50. We have used a standard GE-XANES experiment, which serve as reference, to validate the effectiveness and accuracy of the BO-informed experimental setup. Our results show that this optimized approach maintains data quality while significantly improving efficiency, making GE-XANES more accessible to a wider range of materials science applications.

Data-driven approach for intelligent tunnel dust concentration prediction

The precise evaluation of tunnel dust concentration (TDC) stands as a primary concern within engineering practices. However, comprehensive and accurate prediction of TDC becomes increasingly challenging due to the expanding multi-source datasets and cumulative errors. This study employs innovative data-driven ensemble learning methodologies, specifically random forest (RF) and gradient boosting regression tree (GBRT), to delineate the intricate relationships between TDC values and various rock tunnel characteristics. Five diverse variables sourced from multiple origins are examined and employed as inputs within the database, while the TDC values derived from the dust concentration meter serve as the target outputs. Subsequently, a Bayesian optimization approach, the Tree-structured Parzen Estimator (TPE), is introduced to automatically ascertain the optimal hyper-parameters for the ensemble models. A comprehensive comparison is conducted between the two ensemble learning models and a singular machine learning algorithm, the classification and regression tree (CART), concerning predictive accuracy and resilience via 10-fold cross-validation (CV). The findings reveal the superior performance of the hybrid ensemble learning models over the individual ML models. Notably, the TPE-GBRT algorithm adeptly captures the measurement evolution, showcasing the lowest prediction errors and the highest correlation coefficient. These predicted outcomes significantly contribute to enhancing the engineering comprehension of the interrelation between rock tunnel parameters and TDC values.

CFD-based multi-objective Bayesian optimization approach for compact and efficient hydrogen production in steam reforming reactor

This paper introduces a multi-objective Bayesian optimization approach for concurrently optimizing the size and productivity of a steam methane reforming reactor. Seven design variables, resulting in a total of 3,929,310 possible combinations, were explored, leading to three Pareto optimal designs after 100 iterations. Parametric studies were conducted first, and the results of Bayesian optimization were suggested with initial samples obtained via Latin Hypercube Sampling. The Pareto optimal designs revealed significant reduction in size and improvement in productivity. Furthermore, adjusting operating conditions, specifically the steam-to-carbon ratio, showed additional improvements of productivity. This study can contribute to efficiently proposing Pareto optimal designs with potential applications in hydrogen supply for compact and highly efficient systems.

Bayesian optimization for stable properties amid processing fluctuations in sputter deposition

We introduce a Bayesian optimization approach to guide the sputter deposition of molybdenum thin films, aiming to achieve desired residual stress and sheet resistance while minimizing susceptibility to stochastic fluctuations during deposition. Thin films are pivotal in numerous technologies, including semiconductors and optical devices, where their properties are critical. Sputter deposition parameters, such as deposition power, vacuum chamber pressure, and working distance, influence physical properties like residual stress and resistance. Excessive stress and high resistance can impair device performance, necessitating the selection of optimal process parameters. Furthermore, these parameters should ensure the consistency and reliability of thin film properties, assisting in the reproducibility of the devices. However, exploring the multidimensional design space for process optimization is expensive. Bayesian optimization is ideal for optimizing inputs/parameters of general black-box functions without reliance on gradient information. We utilize Bayesian optimization to optimize deposition power and pressure using a custom-built objective function incorporating observed stress and resistance data. Additionally, we integrate prior knowledge of stress variation with pressure into the objective function to prioritize films least affected by stochastic variations. Our findings demonstrate that Bayesian optimization effectively explores the design space and identifies optimal parameter combinations meeting desired stress and resistance specifications.

Optimization tools for Twin-in-the-Loop vehicle control design: analysis and yaw-rate tracking case study

Given the urgent need to simplify the end-of-line tuning of complex vehicle dynamics controllers, the Twin-in-the-Loop Control (TiL-C) approach was recently proposed in the automotive field. In TiL-C, a digital twin is run in real time on-board the vehicle to compute a nominal control action; an additional controller is used to compensate for the mismatch between the simulator and the actual vehicle. As the digital twin is assumed to be the best replica available of the real plant, the key issue in TiL-C becomes the tuning of the compensator, which must be performed relying on data only. In this paper, we investigate the use of different black-box optimization techniques for the calibration of the compensator. More specifically, we compare the initially proposed Bayesian Optimization (BO) approach with Virtual Reference Feedback Tuning (VRFT), a one-shot direct data-driven design method, and with Set Membership Global Optimization (SMGO), a recently proposed black-box optimization method. The analysis will be carried out within a professional multibody simulation environment on a novel TiL-C application case study – the yaw-rate tracking problem – to further prove the TiL-C effectiveness on a challenging problem. Simulations will show that the VRFT approach is capable of providing a well-tuned controller after a single iteration, while 10 to 15 iterations are necessary for refining it with global optimizers. Also, SMGO is shown to reduce the computational effort required by BO significantly.

Enhancing removal of air contaminants in existing aircraft cabins by optimizing supply air direction based on Re-field synergy and Bayesian optimization

There are a large number of airplanes currently being operated, in which the ventilation system needs to be improved to more effectively remove air contaminants. A potential approach is to adjust the supply air directions with the use of simple airflow deflectors. This study proposed a method for optimizing the supply air direction of ventilation in aircraft cabins based on the Re-field synergy index and Bayesian optimization. A validated numerical model was used to calculate the air distribution and air contaminant transport in a single-row single-aisle aircraft cabin to obtain the Re-field synergy values. The Bayesian optimization approach was used to identify the supply air direction which maximizes the Re-field synergy, namely, maximizes the mass transfer effectiveness. Finally, the air contaminant transport in a 7-row single-aisle aircraft cabin with the optimized supply air direction was evaluated to demonstrate the enhancement of ventilation performance. The results show that the proposed method based on the Re-field synergy index and Bayesian optimization can efficiently optimize the supply air direction in order to enhance the air contaminant removal in aircraft cabins. In the 7-row single-aisle aircraft cabin, the optimized supply air direction can reduce the average air contaminant concentration in the breathing zone of the passengers by up to 23 %.

Optimizing seasonally variable photosynthetic parameters based on joint carbon and water flux constraints

Terrestrial biosphere models (TBMs) often adopt the Farquhar biochemical model coupled with the Ball-Berry stomatal conductance (gs) model to simulate ecosystem carbon and water fluxes. The parameters m, representing the sensitivity of gs to the photosynthetic rate, and Vcmax25, representing the leaf photosynthetic capacity, are two pivotal parameters but the two main sources of uncertainties in TBM simulations. The temporal variations of m in TBMs are still elusive, due to the lack of direct observations. It also remains unclear how accurate estimates of m and Vcmax25 can improve the simulations of carbon and water fluxes. In this study, we used a Bayesian parameter optimization approach to estimate seasonally varying m and Vcmax25 from eddy covariance observations in a mixed forest stand at the Borden Forest Research Station located in southern Ontario, Canada and used in-situ observations of m and Vcmax25 for validation. Three strategies were tested for optimizing m and Vcmax25, including the carbon, water, and carbon-water coupling scenarios. m and Vcmax25 optimized from carbon-water coupling constraints shows best correlations with the measured m (R2 = 0.70) and Vcmax25 (R2 = 0.70). By incorporating optimized m and Vcmax25with seasonal variations, we found considerable improvements in the estimated gross primary productivity (GPP) and evapotranspiration (ET) compared with constant m and Vcmax25, with R2 increasing from 0.78 to 0.85 for GPP, from 0.65 to 0.71 for ET and RMSE reducing from 2.579 g C m−2d−1 to 2.038 g C m−2d−1 for GPP, from 1.151 mm d−1 to 0.137 mm d−1 for ET. This study proposes an effective approach to retrieve m and Vcmax25 for TBMs and demonstrates the efficacy of incorporating seasonally variable m and Vcmax25 for reducing the uncertainties in GPP and ET simulations, which supports accurate quantifications of land-atmosphere exchanges.

NS+NDT: Smart integration of Network Simulation in Network Digital Twin, application to IoT networks

Network Digital Twin (NDT) and Network Simulation (NS) are two paradigms leveraging virtual representations of networks to help decision-making. These tools may seem similar or interchangeable and are often confused or opposed. However, they have their respective purposes, use cases and underlying concepts, which differ and are complementary. The goal of this article is to explore and clarify the specificity, the benefits and the limits of these two decision support tools, analyze how they complement each other and can be nicely combined. We argue that a smart integration of NS in NDT, named NS+NDT, can ease, accelerate and strength decisions for network design, deployment, operations, management and evolution. To study and demonstrate this claim, we focus on the domain of Internet of Things (IoT) solutions, where wireless networks are critical for connecting the physical assets to the Internet, but are complicated to configure for meeting the requirements of a specific application. We examine how NS, coupled with NDT, can contribute to support IoT architects and operators decisions throughout the life cycle of an IoT network. We analyze the different steps required to use NS in the context of NDT and examine how this helps remove NS barriers such as credibility and reliability. In particular, we show how NDT data enable to fine tune and customize the energy consumption models, making the simulation results more context-aware and insightful. Then, addressing the often-prohibitive simulation cost for exploring a large parameter space, we propose to associate surrogate modeling to NS+NDT. As surrogate models, we first introduce a simple ML (Machine Learning)-based surrogate model and illustrate this method with two IoT network configuration optimization use cases. Secondly, we propose a Bayesian optimization approach based on Gaussian Processes as surrogate model to further accelerate the (re)configuration decisions. We show how this method enables to select simulation scenarios that converge rapidly to the optimal solution, and allows the NDT to timely perform the adaptation. The contribution of this article is threefold. It provides (i) the first systematic analysis of the differences and potential synergies between NDT and NS; (ii) a synthetic presentation of the integration of NS and associated decision algorithms within a NDT to unlock NS accessibility and credibility throughout the life cycle of an IoT solution; (iii) a proposal for a smart and cost-efficient integration of NS in NDT via surrogate modeling, for reducing evaluation and optimization cost, paving the way to NS-augmented NDT-based dynamic adaptation and real-time optimization of IoT networks.

Multi-Objective Bayesian Optimization with Active Preference Learning

There are a lot of real-world black-box optimization problems that need to optimize multiple criteria simultaneously. However, in a multi-objective optimization (MOO) problem, identifying the whole Pareto front requires the prohibitive search cost, while in many practical scenarios, the decision maker (DM) only needs a specific solution among the set of the Pareto optimal solutions. We propose a Bayesian optimization (BO) approach to identifying the most preferred solution in the MOO with expensive objective functions, in which a Bayesian preference model of the DM is adaptively estimated by an interactive manner based on the two types of supervisions called the pairwise preference and improvement request. To explore the most preferred solution, we define an acquisition function in which the uncertainty both in the objective function and the DM preference is incorporated. Further, to minimize the interaction cost with the DM, we also propose an active learning strategy for the preference estimation. We empirically demonstrate the effectiveness of our proposed method through the benchmark function optimization and the hyper-parameter optimization problems for machine learning models.

Dynamic joint decision of matching parameters and relocation strategies in ride-sourcing systems interacting with traffic congestion

As the ride-sourcing market expands, ride-sourcing fleets have increased urban traffic congestion, and in turn, road congestion is affecting ride-sourcing operations. It is crucial to incorporate this interaction mechanism into the operational decision of ride-sourcing systems. This mechanism can be described in detail by simulation models. Applying simulation-based optimization methods to the real-time operation of ride-sourcing systems is challenging due to the limitations of simulation efficiency. To tackle this issue, this study first proposes a modular simulation model combining time-driven and event-driven mechanisms. The simulation model comprises two layers: the traffic flow module at the bottom, defined by trip-based and multi-region macroscopic fundamental diagrams and cell transmission models, and the ride-sourcing operation module at the top, which considers the interaction between the traffic congestion and ride-sourcing operation and models the random behavior of passengers and drivers. We verify the efficiency and accuracy of the simulation model. The simulation model is utilized as an evaluator of the objective functions and performance measures. Then, we integrate the Bayesian optimization, parallel sampling, and rolling horizon approaches to develop an efficient and effective simulation-based optimization framework for the dynamic joint decision-making of matching radius, matching intervals, and threshold relocation strategies. Six groups of experiments for the traffic scenario in Berlin reveal some interesting findings. (1) The matching rate first increases and then decreases with the matching radius (a concave function), while the closest study does not think so. (2) As the fleet size grows, passenger waiting time reaches its minimum and then rises because of the increased endogenous congestion generated by ride-sourcing fleets. (3) The optimal operation strategy considering endogenous congestion not only leads to an increase in the system revenue and efficiency but also leads to congestion relief, even though the latter was not our initial objective. This benefit is the most pronounced when the fleet size and background traffic are moderate.

Improving Hardenability Modeling: A Bayesian Optimization Approach to Tuning Hyperparameters for Neural Network Regression

This study investigates the application of regression neural networks, particularly the fitrnet model, in predicting the hardness of steels. The experiments involve extensive tuning of hyperparameters using Bayesian optimization and employ 5-fold and 10-fold cross-validation schemes. The trained models are rigorously evaluated, and their performances are compared using various metrics, such as mean square error (MSE), root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R2). The results provide valuable insights into the models’ effectiveness and their ability to generalize to unseen data. In particular, Model 4208 (8-85-141-1) emerges as the top performer with an impressive RMSE of 1.0790 and an R2 of 0.9900. The model, which was trained with different datasets for nearly 40 steel grades, enables the prediction of hardenability curves, but is limited to the range of the training dataset. The research paper contains an illustrative example that demonstrates the practical application of the developed model in determining the hardenability band for a specific steel grade and shows the effectiveness of the model in predicting and optimizing heat treatment results.

Run-indexed time-varying Bayesian optimization with positional encoding for auto-tuning of controllers: Application to a plasma-assisted deposition process with run-to-run drifts

Bayesian optimization (BO) has emerged as a useful paradigm for automatic calibration (aka auto-tuning) of advanced optimization- and learning-based controllers whose closed-loop performance depends on the choice of several tuning parameters in highly nonlinear and nonconvex ways. However, BO approaches to controller auto-tuning commonly rely on the assumption that system dynamics remain constant, which does not hold for systems with time-varying dynamics, for example, due to gradual aging or persistent environmental drifts. This challenge can be further compounded when gradual and persistent system drifts occur over a series of process runs. Existing time-varying BO (TVBO) approaches with spatio-temporal kernels fall short of effectively handling an integer run index, which is imperative for capturing run-to-run changes in the system behavior. To this end, this paper presents a run-indexed TVBO (RI-TVBO) approach that can systematically account for run-to-run process drifts as the system is queried over sequential process runs. The proposed approach relies on adapting the non-stationary Wiener process kernel to accommodate an integer run index, instead of time. This is done via positional encoding that incorporates the integer run index and, thus, enables describing run-to-run variations in system dynamics. The positional embedding vector associated with each run index is then mapped onto a scalar value to leverage the relationships between different process runs within the probabilistic surrogate model of the objective function in RI-TVBO. The performance of RI-TVBO is evaluated for auto-tuning of an offset-free model predictive controller for a low-temperature plasma-assisted process for thin film deposition. Simulation results demonstrate the superior performance of RI-TVBO over standard BO and TVBO under different scenarios of run-to-run process drifts encountered in plasma-assisted deposition processes in semiconductor manufacturing.

A stochastic data-driven Bayesian optimization approach for intensified ethanol–water separation systems

The optimization of advanced intensified distillation operation systems is crucial for achieving sustainability in the chemical industry. In this paper, we present a stochastic data-driven Bayesian optimization approach for the design and operation of ethanol distillation separation systems. The proposed optimal design methodology does not require a first-principles mathematical model to draw optimal operating conditions. The approach involves an initial sampling of input/output system measurements, followed by the construction of an approximate input/output Gaussian process model. The next best sampling point is then drawn using an acquisition function, and the algorithm converges when the optimization function value does not change or when the maximum number of iterations is reached. We address the optimal parametric design of advanced ethanol–water intensified distillation separation systems, assuming a mixture of the ethanol–water system is available, and almost anhydrous ethanol is required for mobility applications. We also consider the impact of noisy measurements on the optimality operating region. The proposed approach can be extended to the optimization of merged discrete and continuous systems. Bayesian optimization techniques represent black-box optimization strategies that can be employed for the continuous online optimization of intricate separation systems. These techniques aid in mitigating sustainability issues by reducing mass and energy requirements.

Sampling design methods for making improved lake management decisions

AbstractThe ecological status of lakes is important for understanding an ecosystem's biodiversity as well as for service water quality and policies related to land use and agricultural run‐off. If the status is weak, then decisions about management alternatives need to be made. We assess the value of information of lake monitoring in Finland, where lakes are abundant. With reasonable ecological values and restoration costs, the value of information analysis can be compared with the survey's costs. Data are worth gathering if the expected value from the data exceeds the costs. From existing data, we specify a hierarchical Bayesian spatial logistic regression model for the ecological status of lakes. We then rely on functional approximations and Laplace approximations to get closed‐form expressions for the value of information of a sampling design. The case study contains thousands of lakes. The combinatorially difficult design problem is to wisely pick the right subset of lakes for data gathering. To solve this optimization problem, we study the performance of various heuristics: greedy forward algorithms, exchange algorithms and Bayesian optimization approaches. The value of information increases quickly when adding lakes to a small design but then flattens out. Good designs are usually composed of lakes that are difficult to manage, while also balancing a variety of covariates and geographic coverage. The designs achieved by forward selection are reasonably good, but we can outperform them with the more nuanced search algorithms. Statistical designs clearly outperform other designs selected according to simpler criteria.

A Bayesian Optimization Approach for Tuning a Grouping Genetic Algorithm for Solving Practically Oriented Pickup and Delivery Problems

Background: The Multi Depot Pickup and Delivery Problem with Time Windows and Heterogeneous Vehicle Fleets (MDPDPTWHV) is a strongly practically oriented routing problem with many real-world constraints. Due to its complexity, solution approaches with sufficiently good quality ideally contain several operators with certain probabilities.Thus, automatically selecting the best parameter configurations enhances the overall solution quality. Methods: To solve the MDPDPTWHV, we present a Grouping Genetic Algorithm (GGA) framework with several operators and population management variants. A Bayesian Optimization (BO) approach is introduced to optimize the GGA’s parameter configuration. The parameter tuning is evaluated on five data sets which differ in several structural characteristics and contain 1200 problem instances. The outcomes of the parameter-tuned GGA are compared to both the initial GGA parameter configuration and a state-of-the-art Adaptive Large Neighborhood Search (ALNS). Results: The presented GGA framework achieves a better solution quality than the ALNS, even for the initial parameter configuration used. The mean value of the relative error is less than 0.9% and its standard deviation is less than 1.31% for every problem class. For the ALNS, these values are up to three times higher and the GGA is up to 38% faster than the ALNS. Conclusions: It is shown that the BO, as a parameter tuning approach, is a good choice in improving the performance of the considered meta-heuristic over all instances in each data set. In addition, the best parameter configuration per problem class with the same characteristics is able to improve both the frequency of finding the best solution, as well as the relative error to this solution, significantly.

Robust control of wind turbines to reduce wind power fluctuation

AbstractThe wind power generation system of a 5 MW horizontal axis wind turbine has a high wind energy conversion efficiency. The proportion of installed capacity in practical production applications is increasing year on year, so that the stability of its operation becomes a central factor in determining the productivity of the wind farm in question. This paper takes a 5 MW wind turbine as the research object and proposes a parameter‐adaptive robust model predictive control method to achieve self‐optimization of controller parameters through a Bayesian optimization approach. A robust model predictive control strategy, aiming to reduce the power fluctuation while maximizing the power output, is developed in this paper to enhance the dynamic economic performance under uncertain wind speed variation. A Bayesian algorithm is used in this paper to optimize the parameters of the controller. Moreover, wind speeds are simulated using TurbSim for different turbulence intensities of 5%, 10%, and 15% turbulence. Finally, the robust model predictive control toolbox in MATLAB is designed and simulated. The results show that the operational instability of the wind energy system is overcome. Meanwhile, the robustness of the wind energy system operation is improved compared to the traditional model predictive control approach.

A Multi-Fidelity Bayesian Optimization Approach for Constrained Multi-Objective Optimization Problems

Abstract In this paper, a multi-fidelity Bayesian optimization approach is presented to tackle computationally expensive constrained multiobjective optimization problems (MOPs). The proposed approach consists of a three-stage optimization framework designed to search for promising candidate points. In the first stage, an acquisition function is proposed to identify a feasible solution if none is available in the current set of sampling points. Subsequently, a new multi-fidelity weighted expected hypervolume improvement function is developed to find better solutions. In the third stage, a constrained weighted lower confidence bound acquisition function is presented to enhance the constraint predictions and refine the solutions near the constraint boundary. Additionally, a filter strategy is suggested to determine whether constraint updating is necessary, aiming to save computational resources and improve optimization efficiency. Moreover, to expedite the optimization process, a parallel optimization approach is further developed based on the suggested three-stage optimization framework. To achieve this, a multi-fidelity influence function is introduced, allowing the proposed approach to determine a desired number of candidate points within a single iteration. Lastly, the proposed approach is demonstrated through six numerical benchmark examples, which verifies its significant advantages in addressing expensive constrained MOPs. Besides, the proposed approach is applied to the multiobjective optimization of a metamaterial vibration isolator, resulting in the attainment of satisfactory solutions.

Data-driven adaptive building thermal controller tuning with constraints: A primal–dual contextual Bayesian optimization approach

We study the problem of tuning the parameters of a room temperature controller to minimize its energy consumption, subject to the constraint that the daily cumulative thermal discomfort of the occupants is below a given threshold. We formulate it as an online constrained black-box optimization problem where, on each day, we observe some relevant environmental context and adaptively select the controller parameters. In this paper, we propose to use a data-driven Primal-Dual Contextual Bayesian Optimization (PDCBO) approach to solve this problem.In a simulation case study on a single room, we apply our algorithm to tune the parameters of a Proportional Integral (PI) heating controller and the pre-heating time. Our results show that PDCBO can save up to 4.7% energy consumption compared to other state-of-the-art Bayesian optimization-based methods while keeping the daily thermal discomfort below the given tolerable threshold on average. Additionally, PDCBO can automatically track time-varying tolerable thresholds while existing methods fail to do so. We then study an alternative constrained tuning problem where we aim to minimize the thermal discomfort with a given energy budget. With this formulation, PDCBO reduces the average discomfort by up to 63% compared to state-of-the-art safe optimization methods while keeping the average daily energy consumption below the required threshold.