Efficient Robust Optimization Research Articles

Robust topology optimization for continuum structures with random loads

PurposeThis paper aims to tackle the challenge topic of continuum structural layout in the presence of random loads and to develop an efficient robust method.Design/methodology/approachAn innovative robust topology optimization approach for continuum structures with random applied loads is reported. Simultaneous minimization of the expectation and the variance of the structural compliance is performed. Uncertain load vectors are dealt with by using additional uncertain pseudo random load vectors. The sensitivity information of the robust objective function is obtained approximately by using the Taylor expansion technique. The design problem is solved using bi-directional evolutionary structural optimization method with the derived sensitivity numbers.FindingsThe numerical examples show the significant topological changes of the robust solutions compared with the equivalent deterministic solutions.Originality/valueA simple yet efficient robust topology optimization approach for continuum structures with random applied loads is developed. The computational time scales linearly with the number of applied loads with uncertainty, which is very efficient when compared with Monte Carlo-based optimization method.

An efficient robust optimization method with random and interval uncertainties

This paper develops a new mixed uncertainty robust optimization (MURO) method with both random and interval uncertainties. Existing strategies in literature always treat the system performance as a sequence of probability distribution with the interval factors varying within their domains. Moreover, the robust design objective and constraints are modeled in the form of combination of interval mean and interval deviations of performances, which cannot offer a quantitative robustness measurement of a design. The new MURO method is based on the sensitivity region concept and a hybrid robustness index is developed to represent the possibility that the uncertain vector locates within the worst-case sensitivity region (WCSR). This proposed index can offer a more quantitative and intuitive way to evaluate the robustness of a design. With the hybrid indices, the traditional robust optimization problem can be converted to an ordinary optimization with the robustness index constraints. Two numerical examples and two engineering examples with different combinations of interval and random factors are illustrated to demonstrate the applicability and efficiency of the proposed algorithm. The comparison results show that the new method can reduce the conservatism of previous method significantly with fewer computational efforts.

Ensemble clustering for efficient robust optimization of naturally fractured reservoirs

In the development of naturally fractured reservoirs (NFRs), the existence of natural fractures induces severe fingering and breakthrough. To manage the flooding process and improve the ultimate recovery, we propose a numerical workflow to generate optimal production schedules for smart wells, in which the inflow control valve (ICV) settings can be controlled individually. To properly consider the uncertainty introduced by randomly distributed natural fractures, the robust optimization would require a large ensemble size and it would be computationally demanding. In this work, a hierarchical clustering method is proposed to select representative models for the robust optimization in order to avoid redundant simulation runs and improve the efficiency of the robust optimization. By reducing the full ensemble of models into a small subset ensemble, the efficiency of the robust optimization algorithm is significantly improved. The robust optimization is performed using the StoSAG scheme to find the optimal well controls that maximize the net-present-value (NPV) of the NFR’s development. Due to the discrete property of a natural fracture field, traditional feature extraction methods such as model-parameter-based clustering may not be directly applicable. Therefore, two different kinds of clustering-based optimization methods, a state-based (e.g., s w profiles) clustering and a response-based (e.g., production rates) clustering, are proposed and compared. The computational results show that the robust clustering optimization could increase the computational efficiency significantly without sacrificing much expected NPV of the robust optimization. Moreover, the performance of different clustering algorithms varies widely in correspondence to different selections of clustering features. By properly extracting model features, the clustered subset could adequately represent the uncertainty of the full ensemble.

A Lagrangian relaxation approach to fuzzy robust multi-objective facility location network design problem

This study considers a multi-objective combined budget constrained facility location/network design problem (FL/NDP) in which the system uncertainty is considered. The most obvious practical examples of the problem are territorial designing and locating of academies, airline networks, and medical service centers. In order to assure the network reliability versus uncertainty, an efficient robust optimization approach is applied to model the proposed problem. The formulation is minimizing the total expected costs, including, transshipment costs, facility location (FL) costs, fixed cost of road/link utilization as well as minimizing the total penalties of uncovered demand nodes. Then, in order to consider of several system uncertainty, the proposed model is changed to a fuzzy robust model by suitable approaches. An efficient Sub-gradient based Lagrangian relaxation algorithm is applied. In addition, a practical example is studied. At the following, a series of experiments, including several test problems, is designed and solved to evaluate of the performance of the algorithm. The obtained results emphasize that considering of practical factors (e.g., several uncertainties, system disruptions, and customer satisfaction) in modelling of the problem can lead to significant improvement of the system yield and subsequently more efficient utilization of the established network.

Flexible production resources and capacity utilization rates: A robust optimization perspective

This paper presents a novel application of robust optimization (RO) in the context of optimal choices of product-dedicated and flexible capacities in a spatial setting. Specifically, we examine the impact of acquiring flexible production technologies on a firm's overall capacity utilization rates under demand uncertainty. Uncorrelated, negatively, and positively correlated demands are considered. We implement an efficient RO algorithm, which can solve large samples of robust min-max problems in a realistic amount of time. This paper examines the impact of three critical factors that lead to different capacity utilization and resource flexibility outcomes: the degree of solution robustness selected by the decision-maker to accommodate uncertain demand realizations, the flexible capacity costs relative to dedicated capacity costs, and the level of correlation between product demands. One of our results show that according to robust solutions, the total capacity may not be fully utilized because of the distinction between largest (in unit terms) and costliest demand realizations. The second result shows that increasing the proportion of flexible capacity can increase capacity utilization; however, our results report less than full utilization even when using flexible capacity only. We show that the optimal amount of flexible capacity, and its impact on overall capacity utilization rate substantially depends on demand correlation.

Robust optimization of supersonic ORC nozzle guide vanes

An efficient Robust Optimization (RO) strategy is developed for the design of 2D supersonic Organic Rankine Cycle turbine expanders. The dense gas effects are not-negligible for this application and they are taken into account describing the thermodynamics by means of the Peng-Robinson-Stryjek-Vera equation of state. The design methodology combines an Uncertainty Quantification (UQ) loop based on a Bayesian kriging model of the system response to the uncertain parameters, used to approximate statistics (mean and variance) of the uncertain system output, a CFD solver, and a multi-objective non-dominated sorting algorithm (NSGA), also based on a Kriging surrogate of the multi-objective fitness function, along with an adaptive infill strategy for surrogate enrichment at each generation of the NSGA. The objective functions are the average and variance of the isentropic efficiency. The blade shape is parametrized by means of a Free Form Deformation (FFD) approach. The robust optimal blades are compared to the baseline design (based on the Method of Characteristics) and to a blade obtained by means of a deterministic CFD-based optimization.

Expected improvement based infill sampling for global robust optimization of constrained problems

A novel adaptive sampling scheme for efficient global robust optimization of constrained problems is proposed. The method addresses expensive to simulate black-box constrained problems affected by uncertainties for which only the bounds are known, while the probability distribution is not available. An iterative strategy for global robust optimization that adaptively samples the Kriging metamodel of the computationally expensive problem is proposed. The presented approach is tested on several benchmark problems and the average performance based on 100 runs is evaluated. The applicability of the method to engineering problems is also illustrated by applying robust optimization on an integrated photonic device affected by manufacturing uncertainties. The numerical results show consistent convergence to the global robust optimum using a limited number of expensive simulations.

System Robust Optimization of Ring Resonator-Based Optical Filters

Fabrication variations can have a detrimental effect on the performance of optical filters based on ring resonators. However, by using robust optimization these effects can be minimized and device yield can be significantly improved. This paper presents an efficient robust optimization technique for designing manufacturable optical filters based on serial ring resonators. The serial ring resonator is treated as a system which has computationally expensive (directional coupler section) and cheap components (ring section). Cheap mathematical models are constructed of the directional coupler sections in the resonators. The approximate system response based on the cheap model is then robustly optimized. The robust bandpass filter performance is compared against designs that do not take uncertainties into account. The optimality of the robust solutions is confirmed by simulating it on the expensive physical model as a post-processing step. Results indicate that the employed approach can provide an efficient means for robust optimization of ring resonator-based optical filters.

Robust multi-objective dynamic optimization of chemical processes using the Sigma Point method

Dynamic optimization solutions largely rely on the accuracy of the underlying mathematical models. However, these models only represent an approximation of the real dynamic process and their predictions are dependent on a set of parameter values. These parameter values can be hard to estimate exactly (e.g., thermal conductivity) or vary over time (e.g., due to fouling) potentially leading to hazardous situations when applying a model based optimal solution. Robust dynamic optimization deals with the uncertainty related to these parameters in order to quantify their effect and deliver safer (i.e., more robust) operating conditions. This paper discusses a computationally efficient robust dynamic optimization approach based on the Sigma Point method and shows how it outperforms a linearization-based method for a nonlinear dynamic chemical vapor deposition reactor case-study with multiple uncertainties. Moreover, by accounting for uncertainty a trade-off between process safety and performance of the reactor is introduced. This aspect is cast in a multi-objective dynamic optimization framework. In particular, it is illustrated how an increasing robustness (i.e., process safety) induces a worsening of other investigated objective functions and results in robustified Pareto sets.

Robust and reliable medical services network design under uncertain environment and system disruptions

This paper addresses an efficient mixed integer linear programming model for a robust and reliable medical service (MS) center location network design problem (RR/MSL/NDP), which simultaneously takes uncertain parameters, system disruptions, and investment budget constraint into account. The proposed model is formulated based on an efficient robust optimization approach to protect the network against uncertainty. Furthermore, a mixed integer linear programming model with augmented PR-robust constraints is proposed to control the system reliability under unforeseen situations. A practical case study is presented in detail to illustrate the application of the proposed model.

Solving mixed-integer robust optimization problems with interval uncertainty using Benders decomposition

Uncertainty and integer variables often exist together in economics and engineering design problems. The goal of robust optimization problems is to find an optimal solution that has acceptable sensitivity with respect to uncertain factors. Including integer variables with or without uncertainty can lead to formulations that are computationally expensive to solve. Previous approaches for robust optimization problems under interval uncertainty involve nested optimization or are not applicable to mixed-integer problems where the objective or constraint functions are neither quadratic, nor linear. The overall objective in this paper is to present an efficient robust optimization method that does not contain nested optimization and is applicable to mixed-integer problems with quasiconvex constraints (⩽ type) and convex objective funtion. The proposed method is applied to a variety of numerical examples to test its applicability and numerical evidence is provided for convergence in general as well as some theoretical results for problems with linear constraints.

A simplex-based numerical framework for simple and efficient robust design optimization

The Simplex Stochastic Collocation (SSC) method is an efficient algorithm for uncertainty quantification (UQ) in computational problems with random inputs. In this work, we show how its formulation based on simplex tessellation, high degree polynomial interpolation and adaptive refinements can be employed in problems involving optimization under uncertainty. The optimization approach used is the Nelder-Mead algorithm (NM), also known as Downhill Simplex Method. The resulting SSC/NM method, called Simplex2, is based on (i) a coupled stopping criterion and (ii) the use of an high-degree polynomial interpolation in the optimization space for accelerating some NM operators. Numerical results show that this method is very efficient for mono-objective optimization and minimizes the global number of deterministic evaluations to determine a robust design. This method is applied to some analytical test cases and a realistic problem of robust optimization of a multi-component airfoil.

Uncertainty‐based robust aerodynamic optimization of rotor blades

SUMMARYThe aerodynamic performance of a compressor is highly sensitive to uncertain working conditions. This paper presents an efficient robust aerodynamic optimization method on the basis of nondeterministic computational fluid dynamic (CFD) simulation and multi‐objective genetic algorithm (MOGA). A nonintrusive polynomial chaos method is used in conjunction with an existing well‐verified CFD module to quantify the uncertainty propagation in the flow field. This method is validated by comparing with a Monte Carlo method through full 3D CFD simulations on an axial compressor (National Aeronautics and Space Administration rotor 37). On the basis of the validation, the nondeterministic CFD is coupled with a surrogate‐based MOGA to search for the Pareto front. A practical engineering application is implemented to the robust aerodynamic optimization of rotor 37 under random outlet static pressure. Two curve angles and two sweep angles at tip and hub are used as design variables. Convergence analysis shows that the surrogate‐based MOGA can obtain the Pareto front properly. Significant improvements of both mean and variance of the efficiency are achieved by the robust optimization. The comparison of the robust optimization results with that of the initial design, and a deterministic optimization demonstrate that the proposed method can be applied to turbomachinery successfully. Copyright © 2012 John Wiley & Sons, Ltd.

Efficient robust optimization based on polynomial chaos and tabu search algorithm

The study of robust design methodologies and techniques has become a new topical area in design optimizations in nearly all engineering and applied science disciplines in the last 10 years due to inevitable and unavoidable imprecision or uncertainty which is existed in real word design problems. To develop a fast optimizer for robust designs, a methodology based on polynomial chaos and tabu search algorithm is proposed. In the methodology, the polynomial chaos is employed as a stochastic response surface model of the objective function to efficiently evaluate the robust performance parameter while a mechanism to assign expected fitness only to promising solutions is introduced in tabu search algorithm to minimize the requirement for determining robust metrics of intermediate solutions. The proposed methodology is applied to the robust design of a practical inverse problem with satisfactory results.

Efficient reanalysis techniques for robust topology optimization

The article focuses on the reduction of the computational effort involved in robust topology optimization procedures. The performance of structures designed by means of topology optimization may be seriously degraded due to fabrication errors. Robust formulations of the optimization problem were shown to yield optimized designs that are tolerant with respect to such manufacturing uncertainties. The main drawback of such procedures is the added computational cost associated with the need to evaluate a set of designs by performing multiple finite element analyses. In this article, we propose efficient robust topology optimization procedures based on reanalysis techniques. The approach is demonstrated on two compliant mechanism design problems where robust design is achieved by employing either a worst case formulation or a stochastic formulation. It is shown that the time spent on finite element analysis within robust topology optimization can be reduced significantly, without affecting the outcome of the optimization process.

A modified Benders decomposition method for efficient robust optimization under interval uncertainty

The goal of robust optimization problems is to find an optimal solution that is minimally sensitive to uncertain factors. Uncertain factors can include inputs to the problem such as parameters, decision variables, or both. Given any combination of possible uncertain factors, a solution is said to be robust if it is feasible and the variation in its objective function value is acceptable within a given user-specified range. Previous approaches for general nonlinear robust optimization problems under interval uncertainty involve nested optimization and are not computationally tractable. The overall objective in this paper is to develop an efficient robust optimization method that is scalable and does not contain nested optimization. The proposed method is applied to a variety of numerical and engineering examples to test its applicability. Current results show that the approach is able to numerically obtain a locally optimal robust solution to problems with quasi-convex constraints (? type) and an approximate locally optimal robust solution to general nonlinear optimization problems.

Robust optimization based on a polynomial expansion of chaos constructed with integration point rules

Robust optimization is a probabilistic approach to engineering design under uncertainty. The main idea is to select designs insensitive to changes in given parameters. Robust optimization using numerical simulations for black-box problems has received increasing interest. However, when the simulation programmes are computationally expensive, robust optimization is difficult to implement due to the intensive computational demand of uncertainty propagation. Based on polynomial chaos expansion (PCE), an efficient robust optimization method is presented. The PCE is constructed with points of monomial cubature rules (MCRs) to approximate the original model. As the number of points of MCRs is small and all of the points are sampled, the robust optimization procedure is computationally efficient and stable. Two engineering design problems are employed to demonstrate the availability of the proposed method.

Efficient robust optimization for robust control with constraints

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of $${\mathcal{O}}(N^{6})$$ per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires $${\mathcal{O}}(N^3)$$ time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.

A response surface based hierarchical approach to multidisciplinary robust optimization design

A multidisciplinary robust optimization design framework, concurrent subsystem robust design optimization, is proposed to obtain robust optimum solution in the large-scaled and coupled system. In this framework, response surfaces in the form of artificial neural networks provide information pertaining to system performance characteristics, and individual subsystems engage in performing robust optimization design in parallel while communicating with the system level. This optimization approach incorporates uncertainty analysis and generates a global robust optimum solution in an iterative fashion. Two applications are considered, and the results demonstrate that the approach yields a reasonable robust optimum solution, and it is a potential and efficient multidisciplinary robust optimization approach .

A robust optimization model for stochastic logistic problems

The main difficulty of a logistic management problem is in the face of uncertainty about the future. Since many logistic models encounter uncertainty and noisy data in which variables or parameters have the probability of occurrence, a highly promising technique of solving stochastic optimization problems is the robust programming proposed by Mulvey et al. (Operations Research 43(2) (1995a) 264–281) and Mulvey and Ruszczynski (Operations Research 43 (3) (1995b) 477–490). However, heavy computational burden has prevented wider applications in practice. In this study, we reformulate a stochastic management problem as a highly efficient robust optimization model capable of generating solutions that are progressively less sensitive to the data in the scenario set. The method proposed herein to transform a robust model into a linear program only requires adding n+ m variables (where n and m are the number of scenarios and total control constraints, respectively). Whereas, the current robust programming methods proposed by Mulvey et al., Mulvey and Ruszczynski and Bai et al. (Management Science 43 (7)(1997) 895–907) require adding 2 n+2 m. Two logistic examples, logistic management problems involving a wine company and an airline company, demonstrate the computational efficiency of the proposed model.