An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design problems. To find the robust Pareto fronts, multi-objective robust optimization methods with inner-outer optimization structures generally have high computational complexity, which is always an important issue to address. Based on the general experience, robust Pareto solutions usually lie somewhere near the nominal Pareto points. Starting from the obtained nominal Pareto points, the search process for robust solutions could be more efficient. In this paper, we propose a method that sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points. MOOPs are solved by the SARPF in two optimization stages. The deterministic optimization problem and the robustness metric optimization problem are solved in the first stage, and nominal Pareto solutions and the robust-most solutions can be found respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and the robust-most points. The proposed SARPF method can save a significant amount of computation time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values which can save computational efforts further. The global solvers, NSGA-II for the multi-objective case and genetic algorithm (GA) for the single-objective case, are used in corresponding optimization processes. Two examples with comparison to a previous method are presented for the applicability and efficiency demonstration.

An Efficient Multi-Objective Robust Optimization Method by Sequentially Searching From Nominal Pareto Solutions

Journal of Computing and Information Science in Engineering ◽

10.1115/1.4049996 ◽

2021 ◽

Vol 21 (4) ◽

Author(s):

Tingting Xia ◽

Mian Li

Keyword(s):

Robust Optimization ◽

Pareto Front ◽

Optimization Problem ◽

Optimization Problems ◽

Computational Time ◽

Pareto Solutions ◽

Worst Case ◽

Multi Objective ◽

Pareto Points ◽

Abstract Multi-objective optimization problems (MOOPs) with uncertainties are common in engineering design. To find robust Pareto fronts, multi-objective robust optimization (MORO) methods with inner–outer optimization structures usually have high computational complexity, which is a critical issue. Generally, in design problems, robust Pareto solutions lie somewhere closer to nominal Pareto points compared with randomly initialized points. The searching process for robust solutions could be more efficient if starting from nominal Pareto points. We propose a new method sequentially approaching to the robust Pareto front (SARPF) from the nominal Pareto points where MOOPs with uncertainties are solved in two stages. The deterministic optimization problem and robustness metric optimization are solved in the first stage, where nominal Pareto solutions and the robust-most solutions are identified, respectively. In the second stage, a new single-objective robust optimization problem is formulated to find the robust Pareto solutions starting from the nominal Pareto points in the region between the nominal Pareto front and robust-most points. The proposed SARPF method can reduce a significant amount of computational time since the optimization process can be performed in parallel at each stage. Vertex estimation is also applied to approximate the worst-case uncertain parameter values, which can reduce computational efforts further. The global solvers, NSGA-II for multi-objective cases and genetic algorithm (GA) for single-objective cases, are used in corresponding optimization processes. Three examples with the comparison with results from the previous method are presented to demonstrate the applicability and efficiency of the proposed method.

Volume 3: Advanced Composite Materials and Processing; Robotics; Information Management and PLM; Design Engineering ◽

Toward the Use of Pareto Performance Solutions and Pareto Robustness Solutions for Multi-Objective Robust Optimization Problems

10.1115/esda2012-82099 ◽

2012 ◽

Cited By ~ 1

Author(s):

Weijun Wang ◽

Stéphane Caro ◽

Fouad Bennis ◽

Oscar Brito Augusto

Keyword(s):

Robust Optimization ◽

Pareto Front ◽

Optimization Problem ◽

Optimization Problems ◽

Robust Solutions ◽

Multi Objective ◽

Robust Optimization Problem ◽

Design Solutions ◽

Further Development

For Multi-Objective Robust Optimization Problem (MOROP), it is important to obtain design solutions that are both optimal and robust. To find these solutions, usually, the designer need to set a threshold of the variation of Performance Functions (PFs) before optimization, or add the effects of uncertainties on the original PFs to generate a new Pareto robust front. In this paper, we divide a MOROP into two Multi-Objective Optimization Problems (MOOPs). One is the original MOOP, another one is that we take the Robustness Functions (RFs), robust counterparts of the original PFs, as optimization objectives. After solving these two MOOPs separately, two sets of solutions come out, namely the Pareto Performance Solutions (PP) and the Pareto Robustness Solutions (PR). Make a further development on these two sets, we can get two types of solutions, namely the Pareto Robustness Solutions among the Pareto Performance Solutions (PR(PP)), and the Pareto Performance Solutions among the Pareto Robustness Solutions (PP(PR)). Further more, the intersection of PR(PP) and PP(PR) can represent the intersection of PR and PP well. Then the designer can choose good solutions by comparing the results of PR(PP) and PP(PR). Thanks to this method, we can find out the optimal and robust solutions without setting the threshold of the variation of PFs nor losing the initial Pareto front. Finally, an illustrative example highlights the contributions of the paper.

Volume 1: Advances in Aerospace Technology; Energy Water Nexus; Globalization of Engineering; Posters ◽

A Bi-Level Multi-Objective Robust Design Optimization Method With Multiple Robustness Index Capabilities

10.1115/imece2011-64336 ◽

2011 ◽

Author(s):

Todd Letcher ◽

M.-H. Herman Shen

Keyword(s):

Robust Optimization ◽

Optimization Problem ◽

Optimization Problems ◽

Optimization Method ◽

Robust Design Optimization ◽

Nsga Ii ◽

Worst Case ◽

Multi Objective ◽

Robustness Index ◽

Multiple Robustness

A multi-objective robust optimization framework that incorporates a robustness index for each objective has been developed in a bi-level approach. The top level of the framework consists of the standard optimization problem formulation with the addition of a robustness constraint. The bottom level uses the Worst Case Sensitivity Region (WCSR) concept previously developed to solve single objective robust optimization problems. In this framework, a separate robustness index for each objective allows the designer to choose the importance of each objective. The method is demonstrated on a commonly studied two-bar truss structural optimization problem. The results of the problem demonstrate the effectiveness and usefulness of the multiple robustness index capabilities added to this framework. A multi-objective genetic algorithm, NSGA-II, is used in both levels of the framework.

A New Deterministic Approach Using Sensitivity Region Measures for Multi-Objective Robust and Feasibility Robust Design Optimization

Journal of Mechanical Design ◽

10.1115/1.2202884 ◽

2005 ◽

Vol 128 (4) ◽

pp. 874-883 ◽

Cited By ~ 67

Author(s):

Mian Li ◽

Shapour Azarm ◽

Art Boyars

Keyword(s):

Pareto Front ◽

Optimization Problems ◽

Continuous Functions ◽

Robust Design Optimization ◽

Worst Case ◽

New Approach ◽

Multi Objective ◽

Tolerance Region ◽

Parameter Variations ◽

Gradient Based

We present a deterministic non-gradient based approach that uses robustness measures in multi-objective optimization problems where uncontrollable parameter variations cause variation in the objective and constraint values. The approach is applicable for cases that have discontinuous objective and constraint functions with respect to uncontrollable parameters, and can be used for objective or feasibility robust optimization, or both together. In our approach, the known parameter tolerance region maps into sensitivity regions in the objective and constraint spaces. The robustness measures are indices calculated, using an optimizer, from the sizes of the acceptable objective and constraint variation regions and from worst-case estimates of the sensitivity regions’ sizes, resulting in an outer-inner structure. Two examples provide comparisons of the new approach with a similar published approach that is applicable only with continuous functions. Both approaches work well with continuous functions. For discontinuous functions the new approach gives solutions near the nominal Pareto front; the earlier approach does not.

An Interactive Neural Network for Constrained Multi-Objective Optimization with Application to the Design of Digital Filters

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.433-440.2808 ◽

2012 ◽

Vol 433-440 ◽

pp. 2808-2816

Author(s):

Jian Jin Zheng ◽

You Shen Xia

Keyword(s):

Neural Network ◽

Neural Networks ◽

Computational Complexity ◽

Digital Filters ◽

Optimization Problem ◽

Optimization Problems ◽

Optimal Solution ◽

Multi Objective ◽

This paper presents a new interactive neural network for solving constrained multi-objective optimization problems. The constrained multi-objective optimization problem is reformulated into two constrained single objective optimization problems and two neural networks are designed to obtain the optimal weight and the optimal solution of the two optimization problems respectively. The proposed algorithm has a low computational complexity and is easy to be implemented. Moreover, the proposed algorithm is well applied to the design of digital filters. Computed results illustrate the good performance of the proposed algorithm.

Distributed Constraint Optimization Problem

A Two-Phase Complete Algorithm for Multi-Objective Distributed Constraint Optimization

Journal of Advanced Computational Intelligence and Intelligent Informatics ◽

10.20965/jaciii.2014.p0573 ◽

2014 ◽

Vol 18 (4) ◽

pp. 573-580

Author(s):

Alexandre Medi ◽

◽

Tenda Okimoto ◽

Katsumi Inoue ◽

◽

...

Keyword(s):

Branch And Bound ◽

Pareto Front ◽

Optimization Problem ◽

Optimization Problems ◽

Branch And Bound Algorithm ◽

Constraint Optimization ◽

Distributed Constraint Optimization ◽

Multi Objective ◽

Multi Agent ◽

A Distributed Constraint Optimization Problem (DCOP) is a fundamental problem that can formalize various applications related to multi-agent cooperation. Many application problems in multi-agent systems can be formalized as DCOPs. However, many real world optimization problems involve multiple criteria that should be considered separately and optimized simultaneously. A Multi-Objective Distributed Constraint Optimization Problem (MO-DCOP) is an extension of a mono-objective DCOP. Compared to DCOPs, there exists few works on MO-DCOPs. In this paper, we develop a novel complete algorithm for solving an MO-DCOP. This algorithm utilizes a widely used method called Pareto Local Search (PLS) to generate an approximation of the Pareto front. Then, the obtained information is used to guide the search thresholds in a Branch and Bound algorithm. In the evaluations, we evaluate the runtime of our algorithm and show empirically that using a Pareto front approximation obtained by a PLS algorithm allows to significantly speed-up the search in a Branch and Bound algorithm.

Biobjective Optimization of Well Placement: Algorithm, Validation, and Field Testing

SPE Journal ◽

10.2118/203960-pa ◽

2021 ◽

pp. 1-28

Author(s):

Faruk Alpak ◽

Vivek Jain ◽

Yixuan Wang ◽

Guohua Gao

Keyword(s):

Multiobjective Optimization ◽

Objective Function ◽

Pareto Front ◽

Optimization Problem ◽

Optimization Problems ◽

Field Testing ◽

Efficient Implementation ◽

Field Development ◽

Biobjective Optimization ◽

Summary We describe the development and validation of a novel algorithm for field-development optimization problems and document field-testing results. Our algorithm is founded on recent developments in bound-constrained multiobjective optimization of nonsmooth functions for problems in which the structure of the objective functions either cannot be exploited or are nonexistent. Such situations typically arise when the functions are computed as the result of numerical modeling, such as reservoir-flow simulation within the context of field-development planning and reservoir management. We propose an efficient implementation of a novel parallel algorithm, namely BiMADS++, for the biobjective optimization problem. Biobjective optimization is a special case of multiobjective optimization with the property that Pareto points may be ordered, which is extensively exploited by the BiMADS++ algorithm. The optimization algorithm generates an approximation of the Pareto front by solving a series of single-objective formulations of the biobjective optimization problem. These single-objective problems are solved using a new and more efficient implementation of the mesh adaptive direct search (MADS) algorithm, developed for nonsmooth optimization problems that arise within reservoir-simulation-based optimization workflows. The MADS algorithm is extensively benchmarked against alternative single-objective optimization techniques before the BiMADS++ implementation. Both the MADS optimization engine and the master BiMADS++ algorithm are implemented from the ground up by resorting to a distributed parallel computing paradigm using message passing interface (MPI) for efficiency in industrial-scaleproblems. BiMADS++ is validated and field tested on well-location optimization (WLO) problems. We first validate and benchmark the accuracy and computational performance of the MADS implementation against a number of alternative parallel optimizers [e.g., particle-swarm optimization (PSO), genetic algorithm (GA), and simultaneous perturbation and multivariate interpolation (SPMI)] within the context of single-objective optimization. We also validate the BiMADS++ implementation using a challenging analytical problem that gives rise to a discontinuous Pareto front. We then present BiMADS++ WLO applications on two simple, intuitive, and yet realistic problems, and a model for a real problem with known Pareto front. Finally, we discuss the results of the field-testing work on three real-field deepwater models. The BiMADS++ implementation enables the user to identify various compromise solutions of the WLO problem with a single optimization run without resorting to ad hoc adjustments of penalty weights in the objective function. Elimination of this “trial-and-error” procedure and distributed parallel implementation renders BiMADS++ easy to use and significantly more efficient in terms of computational speed needed to determine alternative compromise solutions of a given WLO problem at hand. In a field-testing example, BiMADS++ delivered a workflow speedup of greater than fourfold with a single biobjective optimization run over the weighted-sumsobjective-function approach, which requires multiple single-objective-function optimization runs.

A New Deterministic Approach Using Sensitivity Region Measures for Multi-Objective Robust and Feasibilty Robust Design Optimization

Volume 2: 31st Design Automation Conference, Parts A and B ◽

10.1115/detc2005-85095 ◽

2005 ◽

Cited By ~ 1

Author(s):

Mian Li ◽

Shapour Azarm ◽

Art Boyars

Keyword(s):

Robust Optimization ◽

Pareto Front ◽

Continuous Functions ◽

Robust Design Optimization ◽

Deterministic Approach ◽

Worst Case ◽

New Approach ◽

Multi Objective ◽

Tolerance Region ◽

Gradient Based

We present a deterministic, non-gradient based approach that uses robustness measures for robust optimization in multi-objective problems where uncontrollable parameters variations cause variation in the objective and constraint values. The approach is applicable for cases with discontinuous objective and constraint functions, and can be used for objective or feasibility robust optimization, or both together. In our approach, the parameter tolerance region maps into sensitivity regions in the objective and constraint spaces. The robustness measures are indices calculated, using an optimizer, from the sizes of the acceptable objective and constraint variation regions and from worst-case estimates of the sensitivity regions’ sizes, resulting in an outer-inner structure. Two examples provide comparisons of the new approach with a similar published approach that is applicable only with continuous functions. Both approaches work well with continuous functions. For discontinuous functions the new approach gives solutions near the nominal Pareto front; the earlier approach does not.

SETNDS: A SET-Based Non-Dominated Sorting Algorithm for Multi-Objective Optimization Problems

Applied Sciences ◽

10.3390/app10196858 ◽

2020 ◽

Vol 10 (19) ◽

pp. 6858

Author(s):

Lingling Xue ◽

Peng Zeng ◽

Haibin Yu

Keyword(s):

Time Complexity ◽

Optimization Problems ◽

Sorting Algorithm ◽

Pareto Solutions ◽

Worst Case ◽

Multi Objective ◽

Sorting Algorithms ◽

Dominant Set ◽

The One

Non-dominated sorting, used to find pareto solutions or assign solutions to different fronts, is a key but time-consuming process in multi-objective evolutionary algorithms (MOEAs). The best-case and worst-case time complexity of non-dominated sorting algorithms currently known are O(MNlogN) and O(MN2); M and N represent the number of objectives and the population size, respectively. In this paper, a more efficient SET-based non-dominated sorting algorithm, shorted to SETNDS, is proposed. The proposed algorithm can greatly reduce the number of comparisons on the promise of ensuring a shorter running time. In SETNDS, the rank of a solution to be sorted is determined by only comparing with the one with the highest rank degree in its dominant set. This algorithm is compared with six generally existing non-dominated sorting algorithms—fast non-dominated sorting, the arena’s principle sort, the deductive sort, the corner sort, the efficient non-dominated sort and the best order sort on several kinds of datasets. The compared results show that the proposed algorithm is feasible and effective and its computational efficiency outperforms other existing algorithms.

MCDM towards knowledge incorporation in ANN models for phase transformation in continuous cooling of steel

Multidiscipline Modeling in Materials and Structures ◽

10.1108/mmms-01-2018-0002 ◽

2019 ◽

Vol 15 (1) ◽

pp. 170-186 ◽

Cited By ~ 2

Author(s):

Subhamita Chakraborty ◽

Prasun Das ◽

Naveen Kumar Kaveti ◽

Partha Protim Chattopadhyay ◽

Shubhabrata Datta

Keyword(s):

Optimization Problem ◽

Pareto Solutions ◽

Single Model ◽

Content Type ◽

Multi Objective ◽

Ann Models ◽

Knowledge Incorporation ◽

Single Objective ◽

System Knowledge

Purpose The purpose of this paper is to incorporate prior knowledge in the artificial neural network (ANN) model for the prediction of continuous cooling transformation (CCT) diagram of steel, so that the model predictions become valid from materials engineering point of view. Design/methodology/approach Genetic algorithm (GA) is used in different ways for incorporating system knowledge during training the ANN. In case of training, the ANN in multi-objective optimization mode, with prediction error minimization as one objective and the system knowledge incorporation as the other, the generated Pareto solutions are different ANN models with better performance in at least one objective. To choose a single model for the prediction of steel transformation, different multi-criteria decision-making (MCDM) concepts are employed. To avoid the problem of choosing a single model from the non-dominated Pareto solutions, the training scheme also converted into a single objective optimization problem. Findings The prediction results of the models trained in multi and single objective optimization schemes are compared. It is seen that though conversion of the problem to a single objective optimization problem reduces the complexity, the models trained using multi-objective optimization are found to be better for predicting metallurgically justifiable result. Originality/value ANN is being used extensively in the complex materials systems like steel. Several works have been done to develop ANN models for the prediction of CCT diagram. But the present work proposes some methods to overcome the inherent problem of data-driven model, and make the prediction viable from the system knowledge.