Efficient Robust Optimization Research Articles

ERGO-II: An Improved Bayesian Optimization Technique for Robust Design With Multiple Objectives, Failed Evaluations, and Stochastic Parameters

Abstract In this work, the efficient robust global optimization (ERGO) method is revisited with the aim of enhancing and expanding its existing capabilities. The original objective of ERGO was to address the computational challenges associated with optimization-under-uncertainty through the use of Bayesian optimization (BO). ERGO tackles robust optimization problems which are characterized by sensitivity in the objective function due to stochasticity in the design space. It does this by concurrently minimizing the mean and variance of the objective in a multi-objective setting. To handle the computational complexity arising from the uncertainty propagation, ERGO exploits the analytical expression of the surrogate model underlying BO. In this study, ERGO is extended to accommodate multiple objectives, incorporate an improved predictive error estimation approach, investigate the treatment of failed function evaluations, and explore the handling of stochastic parameters next to stochastic design variables. To evaluate the effectiveness of these improvements, the enhanced ERGO scheme is compared with the original method using an analytical test problem with varying dimensionality. Additionally, the novel optimization technique is applied to an aerodynamic design problem to validate its performance.

Efficient Robust Production Optimization with Reduced Sampling

Summary Model-based robust production optimization is performed by propagating the uncertainty in the reservoir model description through the nonlinear fluid flow dynamics to predict the stochastic production response and the objective function of interest. Propagation of uncertainty is often accomplished through Monte Carlo simulation by sampling from the distribution of uncertain input parameters and computing the corresponding value of the objective function after performing reservoir simulation. The computational burden of performing hundreds of reservoir simulation runs to characterize the distribution of the objective function (or its point statistics) over the sampled input parameters makes the process prohibitive for large-scale deployment. To reduce the computational load of robust production optimization under geologic uncertainty, we propose an efficient stochastic production optimization algorithm based on a reduced sampling strategy. In this approach, at each iteration of the optimization procedure, instead of evaluating the objective function and its gradient over a large number of fixed realizations, we approximate them by using a small subset of model realizations that are selected randomly. This approximation leads to solving a series of “small” subproblems that require significantly fewer reservoir simulation runs, thereby reducing the computational complexity of the robust optimization workflow. Because the samples are selected randomly, over several iterations of the optimization process, a large number of samples are included. For a fixed set of realization, the proposed approach results in each sample being used multiple times during the optimization process. The proposed method is applied to several numerical examples, including a field-scale reservoir model, to show its computational efficiency over traditional robust production optimization under geologic uncertainty.

ERGO: A New Robust Design Optimization Technique Combining Multi-Objective Bayesian Optimization With Analytical Uncertainty Quantification

Abstract In this work, robust design optimization (RDO) is treated, motivated by the increasing desire to account for variability in the design phase. The problem is formulated in a multi-objective setting with the objective of simultaneously minimizing the mean of the objective and its variance due to variability of design variables and/or parameters. This allows the designer to choose its robustness level without the need to repeat the optimization as typically encountered when formulated as a single objective. To account for the computational cost that is often encountered in RDO problems, the problem is fitted in a Bayesian optimization framework. The use of surrogate modeling techniques to efficiently solve problems under uncertainty has effectively found its way in the optimization community leading to surrogate-assisted optimization-under-uncertainty schemes. The Gaussian processes, the surrogates on which Bayesian optimization builds, are often considered cheap-to-sample black-boxes and are sampled to obtain the desired quantities of interest. However, since the analytical formulation of these surrogates is known, an analytical treatment of the problem is available. To obtain the quantities of interest without sampling an analytical uncertainty, propagation through the surrogate is presented. The multi-objective Bayesian optimization framework and the analytical uncertainty quantification are linked together through the formulation of the robust expected improvement, obtaining the novel efficient robust global optimization scheme. The method is tested on a series of test cases to examine its behavior for varying difficulties and validated on an aerodynamic test function which proves the effectiveness of the novel scheme.

Point coverage with heterogeneous sensor networks: A robust optimization approach under target location uncertainty

In this study, we consider the point coverage problem and aim to locate heterogeneous sensor networks under target uncertainty. Target location uncertainty phenomenon is commonly seen in some fields such as military and security, where only probabilistic information regarding target location is usually available via intelligence, historical data, and expert opinions. In these circumstances, possible scenarios are usually generated and several courses of action are developed to address these scenarios. Taking into account this real-life phenomenon as well as the realistic constraints associated with it, we firstly develop an integer nonlinear program for locating a heterogeneous sensor network with hub-spoke topology for a given target scenario. In this topology, wireless sensors constitute the lower level network, while the hubs constitute the upper level by collecting data from the sensors and fusioning the transmitted detection data. Secondly, we propose a linear approximation for the nonlinear model which provides computational efficiency for solving large size problem instances. Then, utilizing the p-robustness concept, we develop a robust counterpart of the deterministic formulation which accounts for multiple scenarios simultaneously and generates a compromise solution that ensures a predefined threshold for regret percentages of individual scenarios. For the first time in the point coverage literature, we define the heterogeneous sensor network location problem within target location uncertainty concept and propose an efficient robust optimization approach for solving it. Finally, we present an illustrative case study to show the applicability of the proposed robust solution approach.

Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics

Efficient Robust Design Optimization (RDO) strategies coupling a parsimonious uncertainty quantification (UQ) method with a surrogate-based multi-objective genetic algorithm (SMOGA) are investigated for a test problem in computational fluid dynamics (CFD), namely the inverse robust design of an expansion nozzle. The low-order statistics (mean and variance) of the stochastic cost function are computed through either a gradient-enhanced kriging (GEK) surrogate or through the less expensive, lower fidelity, first-order method of moments (MoM). Both the continuous (non-intrusive) and discrete (intrusive) adjoint methods are evaluated for computing the gradients required for GEK and MoM. In all cases, the results are assessed against a reference kriging UQ surrogate not using gradient information. Subsequently, the GEK and MoM UQ solvers are fused together to build a multi-fidelity surrogate with adaptive infill enrichment for the SMOGA optimizer. The resulting hybrid multi-fidelity SMOGA RDO strategy ensures a good tradeoff between cost and accuracy, thus representing an efficient approach for complex RDO problems.

Robust design optimization and stochastic performance analysis of a grid-connected photovoltaic system with battery storage and hydrogen storage

Balancing of intermittent energy such as solar energy can be achieved by batteries and hydrogen-based storage. However, combining these systems received limited attention in a grid-connected framework and its design optimization is often performed assuming fixed parameters. Hence, such optimization induces designs highly sensitive to real-world uncertainties, resulting in a drastic mismatch between simulated and actual performances. To fill the research gap on design optimization of grid-connected, hydrogen-based renewable energy systems, we performed a computationally efficient robust design optimization under different scenarios and compared the stochastic performance based on the corresponding cumulative density functions. This paper provides the optimized stochastic designs and the advantage of each design based on the financial flexibility of the system owner. The results illustrate that the economically preferred solution is a photovoltaic array when the self-sufficiency ratio is irrelevant (≤30%). When a higher self-sufficiency ratio threshold is of interest, i.e. up to 59%, photovoltaic-battery designs and photovoltaic-battery-hydrogen designs provide the cost-competitive alternatives which are least-sensitive to real-world uncertainty. Conclusively, including storage systems improves the probability of attaining an affordable levelized cost of electricity over the system lifetime. Future work will focus on the integration of the heat demand.

Robust Design Optimization of a Five-Phase PM Hub Motor for Fault-Tolerant Operation Based on Taguchi Method

This article investigates the efficient robust design optimization of a five-phase permanent magnet (PM) hub motor for electric vehicles. Besides the requirement of high-performance, like high torque density, low torque ripple and efficiency, fault-tolerant operation capability are also considered in the design optimization. To ensure that the motor performance is not sensitive to the variations of manufacturing tolerances, robust design optimization is employed to the investigated motor. To improve the fault tolerant capability of the motor, the motor performances under fault operation are also considered in the optimization. A Fuzzy-based sequential Taguchi robust optimization method is proposed to improve the comprehensive performance and save computing time. The proposed method is efficient because it holds the advantages of Taguchi method, fuzzy theory, and sequential optimization strategy. The motor performance is improved significantly by using the proposed method. Experimental results verify the accuracy of the model used in this study.

Quantile-based robust optimization of a supersonic nozzle for organic rankine cycle turbines

Organic Rankine Cycle (ORC) turbines usually operate in thermodynamic regions characterized by high-pressure ratios and strong non-ideal gas effects, complicating the aerodynamic design significantly. Systematic optimization methods accounting for multiple uncertainties due to variable operating conditions, referred to as Robust Optimization may benefit to ORC turbines aerodynamic design. This study presents an original and fast robust shape optimization approach to overcome the limitation of a deterministic optimization that neglects operating conditions variability, applied to a well-known supersonic turbine nozzle for ORC applications. The flow around the blade is assumed inviscid and adiabatic and it is reconstructed using the open-source SU2 code. The non-ideal gasdynamics is modeled through the Peng-Robinson-Stryjek-Vera equation of state. We propose here a mono-objective formulation which consists in minimizing the α-quantile of the targeted Quantity of Interest (QoI) under a probabilistic constraint, at a low computational cost. This problem is solved by using an efficient robust optimization approach, coupling a state-of-the-art quantile estimation and a classical Bayesian optimization method. First, the advantages of a quantile-based formulation are illustrated with respect to a conventional mean-based robust optimization. Secondly, we demonstrate the effectiveness of applying this robust optimization framework with a low-fidelity inviscid solver by comparing the resulting optimal design with the ones obtained with a deterministic optimization using a fully turbulent solver.

Towards Efficient Robust Optimization using Data based Optimal Segmentation of Uncertain Space

Performing multi-objective optimization under uncertainty is a common requirement in industries and academia. Robust optimization (RO) is considered as an efficient and tractable approach provided one has access to behavioral data for the uncertain parameters. However, solutions of RO may be far from the real solution and less reliable due to inability to map the uncertain space accurately, especially when the data appears discontinuous and scattered in the uncertain domain. Amalgamating machine learning algorithms with RO, this paper proposes a data-driven methodology, where a novel fuzzy clustering mechanism is implemented along-with boundary construction, to transcript the uncertain space such that the specific regions of uncertainty are identified. Subsequently, using intelligent Sobol sampling, samples are generated in the mapped uncertain regions. Results of two test cases are presented along with a comprehensive comparison study. Considered case-studies include highly nonlinear model for continuous casting process from steelmaking industries, where a multi-objective optimization problem under uncertainty is solved to balance the conflict between productivity and energy consumption. The Pareto-optimal solutions of the resulting RO problem are obtained through Non-Dominated Sorting Genetic Algorithm – II, and ~23–29% improvement is observed in the uncertain objective function. Further, the spread and diversity metrics are enhanced by ~10–95% as compared to those obtained using other standard uncertainty sets.

An efficient robust optimization model for the unit commitment and dispatch of multi-energy systems and microgrids

Multi-energy systems and microgrids can play an important role in increasing the efficiency of distributed energy systems and favoring an increasing penetration from renewable sources, by serving as control hubs for the optimal management of Distributed Energy Resources. Predictive operation planning via Mixed Integer Linear Programming is an effective way of tackling the optimal management of these systems. However, the uncertainty of demand and renewable production forecasts can hinder the optimality of the scheduling solution and even lead to outages. This paper proposes a new Affinely Adjustable Robust Formulation of the day-ahead scheduling problem for a generic multi-energy system/microgrid subject to multiple uncertainty factors. Piece-wise linear decision rules are considered in the robust formulation, and their potential use for real-time control is assessed. Novel features include an ad hoc characterization of the polyhedral uncertainty space aimed at reducing solution conservativeness, aggregation of uncertain factors and partial-past recourse which allows speeding up the computational time. The advantages and limitations of the Affinely Adjustable Robust Formulation are thoroughly discussed and quantified through artificial and real-world test cases. The comparison with a conventional deterministic approach shows that, despite the limitations of the affine decision rules, the adjustable robust formulation can ensure full system reliability while attaining at the same time better performances.

Surrogate-Assisted Robust Optimization of Large-Scale Networks Based on Graph Embedding

Robust optimization of complex networks has attracted much attention in recent years. Although existing methods have been successful in achieving promising results, the computational cost for robust optimization tasks is extremely high, which prevents them from being further applied to large-scale networks. Thus, computationally efficient robust optimization methods are in high demand. This article proposes a low-cost method for estimating the robustness of networks with the help of graph embedding techniques and surrogate models. An evolutionary algorithm is then developed to find large-scale robust networks by combining the surrogate-assisted low-cost robustness estimator with the time-consuming real robustness measure by means of a model management strategy. The experimental results on different kinds of synthetic and real networks demonstrate the highly competitive search ability of the proposed algorithm. In addition, the algorithm is able to save up to 80% of the computation time for enhancing the robustness of large-scale networks compared with the state-of-the-art methods.

A reduced random sampling strategy for fast robust well placement optimization

Model-based decision-making in oilfield development often involves hundreds of computationally demanding reservoir simulation runs. In particular, well placement optimization under uncertainty in the geologic representation of the reservoir model is an overly time-consuming procedure as the performance of any proposed well configuration needs to be evaluated over multiple realizations, using computationally expensive flow simulations. To reduce computation, we propose an efficient robust optimization procedure in which at each iteration of the optimization procedure, instead of evaluating the well configuration over all available realizations, we approximate the expected performance using a small subset of randomly selected model realizations. Since the samples are selected randomly, all the realizations are expected to eventually be included in the performance evaluation after a certain number of iterations. However, using only a few random realizations to compute the expected cost function introduces noise in the estimated objective function, necessitating the use of a stochastic optimizer. In this paper, we use the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm, which is known to be robust against noise in the objective function. We first evaluate the performance of different forms of the SPSA algorithm (including discrete, continuous, and adaptive) using several numerical experiments, followed by a discussion of the properties of the proposed reduced random sampling approach and comparison with global optimization techniques. The method is applied to several numerical experiments, including case studies involving vertical, horizontal, and lateral wells, to evaluate its performance. The results from these experiments indicate that the reduced random sampling approach can provide significant computational gain with minimal impact on the attained optimization performance.

An efficient robust cost optimization procedure for rice husk ash concrete mix

As rice husk ash (RHA) is not produced in controlled manufacturing process like cement, its properties vary significantly even within the same lot. In fact, properties of Rice Husk Ash Based Concrete (RHABC) are largely dictated by uncertainty leading to huge deviations from their expected values. This paper proposes a Robust Cost Optimization (RCO) procedure for RHABC, which minimizes such unwanted deviation due to uncertainty and provides guarantee of achieving desired strength and workability with least possible cost. The RCO simultaneously minimizes cost of RHABC production and its deviation considering feasibility of attaining desired strength and workability in presence of uncertainty. RHA related properties have been modeled as uncertain-but-bounded type as associated probability density function is not available. Metamodeling technique is adopted in this work for generating explicit expressions of constraint functions required for formulation of RCO. In doing so, the Moving Least Squares Method is explored in place of conventional Least Square Method (LSM) to ensure accuracy of the RCO. The efficiency by the proposed MLSM based RCO is validated by experimental studies. The error by the LSM and accuracy by the MLSM predictions are clearly envisaged from the test results. The experimental results show good agreement with the proposedMLSMbased RCO predicted mix properties. The present RCO procedure yields RHABC mixes which is almost insensitive to uncertainty (i.e., robust solution) with nominal deviation from experimental mean values. At the same time, desired reliability of satisfying the constraints is achieved with marginal increment in cost.

Robust Design Optimization of a High-Temperature Superconducting Linear Synchronous Motor Based on Taguchi Method

This paper investigates the efficient robust design and optimization of a high-temperature superconducting (HTS) linear synchronous motor by using the Taguchi parameter design approach. The manufacturing tolerances of the HTS magnets, primary iron core and the air gap are considered in the robust design to ensure that the optimal design is less sensitive to these uncertainties. To overcome the disadvantages of the conventional Taguchi parameter design approach, a sequential Taguchi robust optimization method is presented for improvement of the motor performance and manufacturing quality. The proposed method is efficient because it holds the advantages of both Taguchi method and sequential optimization strategy. It can significantly increase the average thrust and decrease the thrust ripple of the investigated HTS linear synchronous motor.

Robust optimization for intensity-modulated proton therapy with soft spot sensitivity regularization.

Proton dose distribution is sensitive to uncertainties in range estimation and patient positioning. Currently, the proton robustness is managed by worst-case scenario optimization methods, which are computationally inefficient. To overcome these challenges, we develop a novel intensity-modulated proton therapy (IMPT) optimization method that integrates dose fidelity with a sensitivity term that describes dose perturbation as the result of range and positioning uncertainties. In the integrated optimization framework, the optimization cost function is formulated to include two terms: a dose fidelity term and a robustness term penalizing the inner product of the scanning spot sensitivity and intensity. The sensitivity of an IMPT scanning spot to perturbations is defined as the dose distribution variation induced by range and positioning errors. To evaluate the sensitivity, the spatial gradient of the dose distribution of a specific spot is first calculated. The spot sensitivity is then determined by the total absolute value of the directional gradients of all affected voxels. The fast iterative shrinkage-thresholding algorithm is used to solve the optimization problem. This method was tested on three skull base tumor (SBT) patients and three bilateral head-and-neck (H&N) patients. The proposed sensitivity-regularized method (SenR) was implemented on both clinic target volume (CTV) and planning target volume (PTV). They were compared with conventional PTV-based optimization method (Conv) and CTV-based voxel-wise worst-case scenario optimization approach (WC). Under the nominal condition without uncertainties, the three methods achieved similar CTV dose coverage, while the CTV-based SenR approach better spared organs at risks (OARs) compared with the WC approach, with an average reduction of [Dmean, Dmax] of [4.72, 3.38] GyRBE for the SBT cases and [2.54, 3.33]GyRBE for the H&N cases. The OAR sparing of the PTV-based SenR method was comparable with the WC method. The WC method, and SenR approaches all improved the plan robustness from the conventional PTV-based method. On average, under range uncertainties, the lowest [D95%, V95%, V100%] of CTV were increased from [93.75%, 88.47%, 47.37%] in the Conv method, to [99.28%, 99.51%, 86.64%] in the WC method, [97.71%, 97.85%, 81.65%] in the SenR-CTV method and [98.77%, 99.30%, 85.12%] in the SenR-PTV method, respectively. Under setup uncertainties, the average lowest [D95%, V95%, V100%] of CTV were increased from [95.35%, 94.92%, 65.12%] in the Conv method, to [99.43%, 99.63%, 87.12%] in the WC method, [96.97%, 97.13%, 77.86%] in the SenR-CTV method, and [98.21%, 98.34%, 83.88%] in the SenR-PTV method, respectively. The runtime of the SenR optimization is eight times shorter than that of the voxel-wise worst-case method. We developed a novel computationally efficient robust optimization method for IMPT. The robustness is calculated as the spot sensitivity to both range and shift perturbations. The dose fidelity term is then regularized by the sensitivity term for the flexibility and trade-off between the dosimetry and the robustness. In the stress test, SenR is more resilient to unexpected uncertainties. These advantages in combination with its fast computation time make it a viable candidate for clinical IMPT planning.

Robust Optimization for Micromachine Design Problems Involving Multimodal Distributions

The conventional robust optimization methods usually focus on problems with unimodal random variables. In real applications, input random variables may follow multimodal distributions with multiple peaks in their probability density. When multimodal random variables are involved, the conventional methods, such as the mean-variance-based methods, will be not accurate. This paper presents an efficient robust optimization method, which provides a potential computational tool for engineering problems involving multimodal random variables. A robustness metric is formulated by introducing the concept of accepting/rejecting the limit to calculate the failure probability of the performance response, which can directly capture the multimodal characteristics of the performance. A second-order higher moment method is presented to efficiently conduct the probability calculation in the inner loop of design optimization. The proposed decoupling strategy drives the probability calculation and the design optimization sequentially and alternately. This method is applied to the three micromachine design problems, including a sweat-rate sensor, a piezoelectric sensor, and an image sensing module. The numerical results show that the method has excellent engineering practicality due to the comprehensive performance in terms of efficiency, accuracy, and convergence.

Toward a Comprehensive and Efficient Robust Optimization Framework for (Bio)chemical Processes

Model-based design principles have received considerable attention in biotechnology and the chemical industry over the last two decades. However, parameter uncertainties of first-principle models are critical in model-based design and have led to the development of robustification concepts. Various strategies have been introduced to solve the robust optimization problem. Most approaches suffer from either unreasonable computational expense or low approximation accuracy. Moreover, they are not rigorous and do not consider robust optimization problems where parameter correlation and equality constraints exist. In this work, we propose a highly efficient framework for solving robust optimization problems with the so-called point estimation method (PEM). The PEM has a fair trade-off between computational expense and approximation accuracy and can be easily extended to problems of parameter correlations. From a statistical point of view, moment-based methods are used to approximate robust inequality and equality constraints for a robust process design. We also apply a global sensitivity analysis to further simplify robust optimization problems with a large number of uncertain parameters. We demonstrate the performance of the proposed framework with two case studies: (1) designing a heating/cooling profile for the essential part of a continuous production process; and (2) optimizing the feeding profile for a fed-batch reactor of the penicillin fermentation process. According to the derived results, the proposed framework of robust process design addresses uncertainties adequately and scales well with the number of uncertain parameters. Thus, the described robustification concept should be an ideal candidate for more complex (bio)chemical problems in model-based design.

Robust Turbine Blade Optimization in the Face of Real Geometric Variations

Because of manufacturing variations, no real turbine blade exactly conforms to its nominal geometry. Even minimal deviations are known to affect aerodynamic performance, blade temperatures, and blade lifespan negatively. Rather than conventional deterministic design with its costly adherence to strict control of tolerance limits, robust design optimization aims to incorporate inevitable variations into the design process itself, so that both performance mean and scatter can be optimized simultaneously. Such a workflow is presented and applied in this paper to aerodynamically optimize an industrial turbine rotor blade against realistic manufacturing variations. A set of digitized three-dimensional laser scans from two turbofan engines forms the core of this study. On the basis of these deviations, the approach uses high-fidelity geometric models, nonintrusive uncertainty quantification, and efficient robust optimization with constraints to effectively locate Pareto-optimal designs. One selected robust blade is validated and shown to be desensitized to the observed manufacturing variability. The underlying measurement data are crucial to obtain realistic results and, as a consequence, are vital to design real robust turbine blades.

Efficient robust design optimization of rail bridge hollow pier considering uncertain but bounded type parameters in metamodeling framework

A robust design optimization (RDO) of rail bridge hollow pier (RBHP) under uncertainty is explored in the present paper. Uncertainty information about such structural system is often observed to be insufficient to define involved uncertain parameters probabilistically. Rather, information about bounds of the variability of these uncertain parameters is only available. In such case, parameters are modeled as uncertain but bounded (UBB) type. The conventional reliability-based design optimization cannot be applied with UBB type uncertainty. Keeping this in view, an RDO, which is a relatively newer approach, is explored for RBHP under UBB type uncertainty in the present study. The RDO minimizes cost and deviation of cost simultaneously ensuring constraint feasibility under uncertainty. Literature addressing RDO on RBHP with UBB parameters considering seismic effects is observed to be scarce. As the simulation-based approach would require a tremendous computational time, a moving least squares method (MLSM)-based metamodeling approach has been adopted here to alleviate the computational burden. The RDO results show that guarantee in safe performance and substantially reduced deviation of structural performance can be achieved by the RDO. The behavior of the structure is quite satisfactorily captured by the MLSM predictions than the conventional least squares method-based metamodeling. The computational time is observed to be drastically reduced in the present case in comparison to that by the direct Monte Carlo Simulation.

Uncertainty quantification-based robust aerodynamic optimization of laminar flow nacelle

The aerodynamic performance of laminar flow nacelle is highly sensitive to uncertain working conditions, especially the surface roughness. An efficient robust aerodynamic optimization method on the basis of non-deterministic computational fluid dynamic (CFD) simulation and Efficient Global Optimization (EGO)algorithm was employed. A non-intrusive polynomial chaos method is used in conjunction with an existing well-verified CFD module to quantify the uncertainty propagation in the flow field. This paper investigates the roughness modeling behavior with the [Formula: see text]-Ret shear stress transport model including modeling flow transition and surface roughness effects. The roughness effects are modeled to simulate sand grain roughness. A Class-Shape Transformation-based parametrical description of the nacelle contour as part of an automatic design evaluation process is presented. A Design-of-Experiments (DoE) was performed and surrogate model by Kriging method was built. The new design nacelle process demonstrates that significant improvements of both mean and variance of the efficiency are achieved and the proposed method can be applied to laminar flow nacelle design successfully.