Robust Optimal Designs Research Articles

Model Robust Optimal Designs for Kronecker Model for Mixture Experiments

In comparison to Scheffè’s canonical polynomial models (S-models), the Kronecker models (K-models) for mixture experiments are symmetric, compact in notation, and based on the Kronecker algebra of vectors and matrices. Further, there is a corresponding transition from S-models to K-models in the form of model re-parameterization. In the literature, it has been recommended to use second-degree K-models in practice compared to the widely used second-degree S-models especially when the moment matrix is of an ill-conditioning type. The motivation of the present article is to discriminate between K-models and S-models in terms of the model-robust D- and A-optimality criteria. These optimality criteria are discussed when there is uncertainty in selecting an appropriate model out of two rival models for a mixture experiment. International Journal of Statistical Sciences, Vol.24(1), March, 2024, pp 31-48

Physically constrained eigenspace perturbation for turbulence model uncertainty estimation

Aerospace design is increasingly incorporating design under uncertainty-based approaches to lead to more robust and reliable optimal designs. These approaches require dependable estimates of uncertainty in simulations for their success. The key contributor of predictive uncertainty in computational fluid dynamics (CFD) simulations of turbulent flows are the structural limitations of Reynolds-averaged Navier–Stokes models, termed model-form uncertainty. Currently, the common procedure to estimate turbulence model-form uncertainty is the eigenspace perturbation framework (EPF), involving perturbations to the modeled Reynolds stress tensor within physical limits. The EPF has been applied with success in design and analysis tasks in numerous prior works from the industry and academia. Owing to its rapid success and adoption in several commercial and open-source CFD solvers, in-depth verification and validation of the EPF is critical. In this work, we show that under certain conditions, the perturbations in the EPF can lead to Reynolds stress dynamics that are not physically realizable. This analysis enables us to propose a set of necessary physics-based constraints, leading to a realizable EPF. We apply this constrained procedure to the illustrative test case of a converging-diverging channel, and we demonstrate that these constraints limit physically implausible dynamics of the Reynolds stress tensor, while enhancing the accuracy and stability of the uncertainty estimation procedure.

Generating Robust Optimal Mixture Designs Due to Missing Observation Using a Multi-Objective Genetic Algorithm

Missing observation is a common problem in scientific and industrial experiments, particularly in a small-scale experiment. They often present significant challenges when experiment repetition is infeasible. In this research, we propose a multi-objective genetic algorithm as a practical alternative for generating optimal mixture designs that remain robust in the face of missing observation. Our algorithm prioritizes designs that exhibit superior D-efficiency while maintaining a high minimum D-efficiency due to missing observations. The focus on D-efficiency stems from its ability to minimize the impact of missing observations on parameter estimates, ensure reliability across the experimental space, and maximize the utility of available data. We study problems with three mixture components where the experimental region is an irregularly shaped polyhedral within the simplex. Our designs have proven to be D-optimal designs, demonstrating exceptional performance in terms of D-efficiency and robustness to missing observations. We provide a well-distributed set of optimal designs derived from the Pareto front, enabling experimenters to select the most suitable design based on their priorities using the desirability function.

Robust Optimum Life-Testing Plans under Progressive Type-I Interval Censoring Schemes with Cost Constraint

This paper considers optimal design problems for the Weibull distribution, which can be used to model symmetrical or asymmetrical data, in the presence of progressive interval censoring in life-testing experiments. Two robust approaches, Bayesian and minimax, are proposed to deal with the dependence of the D-optimality and c-optimality on the unknown model parameters. Meanwhile, the compound design method is applied to ensure a compromise between the precision of estimation of the model parameters and the precision of estimation of the quantiles. Furthermore, to make the design become more practical, the cost constraints are taken into account in constructing the optimal designs. Two algorithms are provided for finding the robust optimal solutions. A simulated example and a real life example are given to illustrate the proposed methods. The sensitivity analysis is also studied. These new design methods can help the engineers to obtain robust optimal designs for the censored life-testing experiments.

Recent Advances in Robust Design for Accelerated Failure Time Models with Type I Censoring

Many fields including clinical and manufacturing areas usually perform life-testing experiments and accelerated failure time models (AFT) play an essential role in these investigations. In these models the covariate causes an accelerant effect on the course of the event through the term named acceleration factor (AF). Despite the influence of this factor on the model, recent studies state that the form of AF is weakly or partially known in most real applications. In these cases, the classical optimal design theory may produce low efficient designs since they are highly model dependent. This work explores planning and techniques that can provide the best robust designs for AFT models with type I censoring when the form of the AF is misspecified, which is an issue little explored in the literature. Main idea is focused on considering the AF to vary over a neighbourhood of perturbation functions and assuming the mean square error matrix as the basis for measuring the design quality. A key result of this research was obtaining the asymptotic MSE matrix for type I censoring under the assumption of known variance regardless the selected failure time distribution. In order to illustrate the applicability of previous result to a study case, analytical characterizations and numerical approaches were developed to construct optimal robust designs under different contaminating scenarios for a failure time following a log-logistic distribution.

Minimax A‐, c‐, and I‐optimal regression designs for models with heteroscedastic errors

It is well known that it is difficult to construct minimax optimal designs. Furthermore, since in practice we never know the true error variance, it is important to allow small deviations and construct robust optimal designs. We investigate a class of minimax optimal regression designs for models with heteroscedastic errors that are robust against possible misspecification of the error variance. Commonly used A‐, c‐, and I‐optimality criteria are included in this class of minimax optimal designs. Several theoretical results are obtained, including a necessary condition and a reflection symmetry for these minimax optimal designs. In this article, we focus mainly on linear models and assume that an approximate error variance function is available. However, we also briefly discuss how the methodology works for nonlinear models. We then propose an effective algorithm to solve challenging nonconvex optimization problems to find minimax designs on discrete design spaces. Examples are given to illustrate minimax optimal designs and their properties.

Optimal allocation to treatments in a sequential multiple assignment randomized trial.

One of the main questions in the design of a trial is how many subjects should be assigned to each treatment condition. Previous research has shown that equal randomization is not necessarily the best choice. We study the optimal allocation for a novel trial design, the sequential multiple assignment randomized trial, where subjects receive a sequence of treatments across various stages. A subject's randomization probabilities to treatments in the next stage depend on whether he or she responded to treatment in the current stage. We consider a prototypical sequential multiple assignment randomized trial design with two stages. Within such a design, many pairwise comparisons of treatment sequences can be made, and a multiple-objective optimal design strategy is proposed to consider all such comparisons simultaneously. The optimal design is sought under either a fixed total sample size or a fixed budget. A Shiny App is made available to find the optimal allocations and to evaluate the efficiency of competing designs. As the optimal design depends on the response rates to first-stage treatments, maximin optimal design methodology is used to find robust optimal designs. The proposed methodology is illustrated using a sequential multiple assignment randomized trial example on weight loss management.

Multi-criteria decision making under uncertainties in composite materials selection and design

During a composite application’s initial design stages, the main objective is to have the optimal performance of the final structure. There is a vast demand for lightweight structures with minimum cost and enhanced safety features in all heavy-duty and performance-based industries such as aerospace, automobile, and sports. In order to make prudent decisions and establish the reliability of industrial decision-makers, it is paramount to consider the impacts of uncertainties on the strength and cost of the structure. For that reason, every source of uncertainty should be included when designing an optimal engineering device.This work focuses on applying multidisciplinary optimization tools for the optimal design of fiber-reinforced composites under uncertainties arising from different scales. For demonstration, we consider a composite leafspring for optimization under uncertainties. Material microstructure accounts for microscale uncertainties while composite layers stacking sequence and structural loading account for meso and macroscale uncertainties, respectively. Using a Sparse Polynomial Chaos Expansion (SPCE) method, a data-driven model that establishes a relationship between input parameters and system objectives is constructed by analyzing data. Results are provided with respect to both variations and probability distributions. The stiffness and the cost of the leafspring are the design objectives. Finally, the robust optimal designs are discussed using the Pareto front.

Hierarchical Yield-Aware Synthesis Methodology Covering Device-, Circuit-, and System-Level for Radiofrequency ICs

This paper presents an innovative yield-aware synthesis strategy based on a hierarchical bottom-up methodology that uses a multiobjective evolutionary optimization algorithm to design a complete radiofrequency integrated circuit from the passive component level up to the system level. Within it, performances’ calculation aims for the highest possible accuracy. A surrogate model calculates the performances for the inductive devices, with accuracy comparable to full electromagnetic simulation; and, an electrical simulator calculates circuit- and system-level performances. Yield is calculated using Monte-Carlo (MC) analysis with the foundry-provided models without any model approximation. The computation of the circuit yield throughout the hierarchy is estimated employing parallelism and reducing the number of simulations by performing MC analysis only to a reduced number of candidate solutions, alleviating the computational requirements during the optimization. The yield of the elements not accurately evaluated is assigned using their degree of similitude to the simulated solutions. The result is a novel synthesis methodology that reduces the total optimization time compared to a complete MC yield-aware optimization. Ultimately, the methodology proposed in this work is compared against other methodologies that do not consider yield throughout the system’s complete hierarchy, demonstrating that it is necessary to consider it over the entire hierarchy to achieve robust optimal designs.

Bayesian optimal designs for cox regression model with random and nonrandom intercept based on type I censored data

In this article, we aim to develop A and D-optimal designs for censored data via Bayesian strategy. In this regard, based on some different sorts of prior distributions for the unknown parameters, the appropriate designs are calculated. In the sequel, we define the random intercept model for the Cox regression model assuming the random intercept distributed normally. It is observed that the likelihood function of the observations cannot be acquired in a closed form. Thus, we obtain the quasi-information matrix based on the quasi-likelihood function. We then, use this matrix to obtain the A and D-optimal designs. It is seen that the Fisher information matrix depends on the unknown parameters as a result the locally optimal designs will be inefficient when the true guess about the parameters is failed. To overcome this problem, we propose the Bayesian strategy to obtain more robust optimal designs. Some numerical techniques such as Monte Carlo integration and numerical optimization methods are implemented to calculate the designs. Further, the generalized equivalence theorem is adapted to confirm the optimality of the proposed designs. All of the numerical results are done with R software. The results of this article will help to remove the weakness of the locally optimal designs that have been addressed in Schmidt and Schwabe when the parameters are misspecfied.

Energy maximising robust control for spectral and pseudospectral methods with application to wave energy systems

Spectral and Pseudospectral methods have been widely considered in diverse optimal control applications, usually where energy optimisation is required. Although such methods are a good way to ensure a good balance between performance and computational effort, in most of the literature, nominal mathematical models are considered without taking into account possible dynamic deviations from the nominal case. The main aim of this study is to propose a novel framework where spectral and pseudospectral problems include some structured uncertainty, achieving robust optimal control designs guaranteeing the ‘best worst-case performance’. In this paper, the objective function used for optimisation is inspired by wave energy converters. Two solution methodologies are developed. Firstly, an analytical solution, for circular and convex polytopic uncertainty boundaries, is proposed. Then, a numerical formulation is introduced to consider uncertainty sets of arbitrary shape, adding the ability to consider physical system constraints. Finally, an application example shows the benefit of this new control formulation.

Copula-based robust optimal block designs.

Blocking is often used to reduce known variability in designed experiments by collecting together homogeneous experimental units. A common modeling assumption for such experiments is that responses from units within a block are dependent. Accounting for such dependencies in both the design of the experiment and the modeling of the resulting data when the response is not normally distributed can be challenging, particularly in terms of the computation required to find an optimal design. The application of copulas and marginal modeling provides a computationally efficient approach for estimating population‐average treatment effects. Motivated by an experiment from materials testing, we develop and demonstrate designs with blocks of size two using copula models. Such designs are also important in applications ranging from microarray experiments to experiments on human eyes or limbs with naturally occurring blocks of size two. We present a methodology for design selection, make comparisons to existing approaches in the literature, and assess the robustness of the designs to modeling assumptions.

Design of order-of-addition experiments

Summary In an order-of-addition experiment, each treatment is a permutation of $m$ components. It is often unaffordable to test all the $m!$ possible treatments, and thus the design problem arises. We consider a flexible model that incorporates the order of each pair of components and can also account for the distance between the two components in every such pair. Under this model, the optimality of the uniform design measure is established, via the approximate theory, for a broad range of criteria. Coupled with an eigenanalysis, this result serves as a benchmark that paves the way for assessing the efficiency and robustness of any exact design. The closed-form construction of a class of robust optimal fractional designs that can also facilitate model selection is explored and illustrated.

Integrating robustness indicators into multi-objective optimization to find robust optimal low-energy building designs

Uncertainties can have a large influence on building performance and cause deviations between predicted performance and performance during operation. It is therefore important to quantify this influence and identify robust designs that have potential to deliver the desired performance under uncertainties. Generally, robust building designs are identified by assessing the performance of multiple design configurations under various uncertainties. When exploring a large design space, this approach becomes computationally expensive and infeasible in practice. Therefore, we propose a simulation framework based on multi-objective optimization and sampling strategies to find robust optimal designs at low computational costs. The genetic algorithm parameters of optimization are fine tuned to further enhance the computational efficiency. Furthermore, a modified fitness function is implemented to use minimax regret robustness method in the optimization loop. The implemented simulation framework can save up to 94–99% of computational time compared to full factorial approach, while identifying the same robust designs.

Optimal robust parameter designs with qualitative and quantitative factors

SummaryRobust parameter design is an effective methodology for reducing variance and improving the quality of a product and a process. Recent work has mainly concentrated on two‐level robust parameter designs. We consider general robust parameter designs with factors having two or more or mixed levels these levels being either qualitative or quantitative. We propose a methodology and develop a generalised minimum aberration optimality criterion for selecting optimal robust parameter designs. A catalogue of 18‐run optimal designs is constructed and tabulated.

An Efficient Direct Search Methodology for Robust Optimization of Electromagnetic Devices

A direct search formulation to treat the robust performances as additional constrained functions is proposed for robust optimization of electromagnetic devices under uncertainties. To solve the resulting robust optimization problem, a robust-oriented quantum-inspired evolutionary algorithm (QEA) is proposed. In the proposed robust-oriented QEA, a mechanism to activate the robust performance checking procedure only to promising solution is introduced, and an automatic updating mechanism for the increment angle is designed to guarantee a good balance between exploration and exploitation searches. Also, the stochastic approximation method is employed to minimize the computational burdens in determining the gradient information. The numerical results on two case studies are reported in order to validate the feasibility and the merit of the proposed direct search methodology in solving both benchmark problems and practical engineering robust optimal designs.

Robust Optimization With Parameter and Model Uncertainties Using Gaussian Processes

Uncertainty is unavoidable in engineering design, which may result in variations in the objective functions and/or constraints. The former may degrade the designed performance while the latter can even change the feasibility of the obtained optimal solutions. Taking uncertainty into consideration, robust optimization (RO) algorithms aim to find optimal solutions that are also insensitive to uncertainty. Uncertainty may include variation in parameters and/or design variables, inaccuracy in simulation models used in design problems, and other possible errors. Most existing RO algorithms only consider uncertainty in parameters, but overlook that in simulation models by assuming that the simulation model used can always provide identical outputs to those of the real physical systems. In this paper, we propose a new RO framework using Gaussian processes, considering not only parameter uncertainty but also uncertainty in simulation models. The consideration of model uncertainty in RO could reduce the risk for the obtained robust optimal designs becoming infeasible even if the parameter uncertainty has been considered. Two test examples with different degrees of complexity are utilized to demonstrate the applicability and effectiveness of our proposed algorithm.

Robust Transceiver Optimization for Power-Splitting Based Downlink MISO SWIPT Systems

This letter considers a downlink multi-input single-out (MISO) system where each user performs simultaneous wireless information and power transfer (SWIPT) based on a power splitting receiver architecture. Assuming imperfect channel state information (CSI) at the base station, we develop two robust joint beamforming and power splitting (BFPS) designs that minimize the transmission power under both the signal-to-interference-plus-noise ratio (SINR) and energy harvesting (EH) constraints per user. In the first design, we consider the worst-case (WC) SINR and EH constraints, and show that the WC-BFPS problem can be relaxed as a semidefinite program (SDP) through a linear matrix inequality representation for (infinitely many) robust quadratic matrix inequality constraints. In the second design, we consider the chance constraints (CCs) for SINR and EH, and resort to both semidefinite relaxation and Bernstein-type inequality restriction to transform the CC-BFPS problem into another convex SDP. Based on these convex reformulations, the (near-)optimal robust BFPS designs can be efficiently solved. Numerical results are provided to demonstrate the merit of the proposed robust designs.

Robust beamforming designs for multiuser MISO downlink with per-antenna power constraints

This paper studies the robust beamforming designs for a multiuser multiple-input single-output (MISO) downlink system. Different from the conventional sum-power constraint across all transmit antennas, we consider individual power constraints per antenna at the base station. Assuming that the channel uncertainty is bounded by a spherical region, we develop the optimal robust designs to maximize the minimum worst-case signal-to-interference-plus-noise ratio (SINR) among all users. Specifically, we show that the optimal max-min SINR beamformers can be obtained by solving a sequence of “dual” min-max power problems. Relying on the S-procedure and the linear matrix inequality representation for the cone of Lorentz-positive maps (LPMs), respectively, two designs, which are referred to as Robust-SP and Robust-LPM, are proposed to efficiently solve such min-max power problems. Building on either Robust-SP or Robust LPM, a bisection search algorithm is then developed to find the robust max-min SINR beamformers with guaranteed global optimality and geometrically fast convergence speed. Using the max-min SINR solutions as a cornerstone, we further put forth the optimal robust design for the worst-case weighted sum-rate (WSR) maximization. By formulating the worst-case WSR maximization problem into a monotonic program (MP), we develop a polyblock outer approximation algorithm to obtain the globally optimal solution. Numerical results are presented to demonstrate the merits of the proposed robust beamforming designs.

Uncertainty analysis and probabilistic design optimization of hybrid rocket motors for manned lunar landing

To obtain a conceptual design for a hybrid rocket motor (HRM) to be used as the Ascent Propulsion System in the Apollo lunar module, the deterministic design optimization (DDO) method is applied to the HRM design. Based on the results of an uncertainty analysis of HRMs, an uncertainty-based design optimization (UDO) method is also adopted to improve the design reliability. The HRM design process, which is a multidisciplinary system, is analyzed, and a mathematical model for the system design is established to compute the motor performance from the input parameters, including the input variables and model parameters. The input parameter uncertainties are quantified, and a sensitivity analysis of the model parameter uncertainties is conducted to identify the most important model parameter uncertainties for HRMs. The DDO and probabilistic UDO methods are applied to conceptual designs for an HRM to be used as a substitute for the liquid rocket motor (LRM) of the Ascent Propulsion System. The conceptual design results show that HRMs have several advantages as an alternative to the LRM of the Ascent Propulsion System, including nontoxic propellant combination, small motor volume, and comparable functions, such as restarting and throating. Comparisons of the DDO and UDO results indicate that the UDO method achieves more robust and reliable optimal designs than the DDO method. The probabilistic UDO method can be used to develop better conceptual designs for HRMs.