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

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 ◽  
Single Objective

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.

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

2020 ◽  
Author(s):  
Tingting Xia ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Pareto Solutions ◽  
Worst Case ◽  
Multi Objective ◽  
Pareto Points ◽  
Engineering Design Problems ◽  
Single Objective

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.

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

2012 ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Oscar Brito Augusto
Keyword(s):  
Robust Optimization ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Multi Objective Optimization ◽  
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.

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

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

2005 ◽  
Vol 128 (4) ◽  
pp. 874-883 ◽  
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

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 Optimization ◽  
Multi Objective ◽  
Single 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.

The Relationship between Metaheuristics Stopping Criteria and Performances

2014 ◽  
Vol 5 (3) ◽  
pp. 44-70 ◽  
Author(s):  
Mohamed-Mahmoud Ould Sidi ◽  
Bénédicte Quilot-Turion ◽  
Abdeslam Kadrani ◽  
Michel Génard ◽  
Françoise Lescourret
Keyword(s):  
Optimization Problem ◽  
Optimization Problems ◽  
Statistical Tests ◽  
Particle Swarm ◽  
Computational Time ◽  
Plant Management ◽  
Stopping Criterion ◽  
Nsga Ii ◽  
Multi Objective ◽  
Crowding Distance

A major difficulty in the use of metaheuristics (i.e. evolutionary and particle swarm algorithms) to deal with multi-objective optimization problems is the choice of a convenient point at which to stop computation. Indeed, it is difficult to find the best compromise between the stopping criterion and the algorithm performance. This paper addresses this issue using the Non-dominated Sorting Genetic Algorithm (NSGA-II) and the Multi-Objective Particle Swarm Optimization with Crowding Distance (MOPSO-CD) for the model-based design of sustainable peach fruits. The optimization problem of interest contains three objectives: maximize fruit fresh mass, maximize fruit sugar content, and minimize the crack density on the fruit skin. This last objective targets a reduction in the use of fungicides and can thus enhance preservation of the environment and human health. Two versions of the NSGA-II and two of the MOPSO-CD were applied to tackle this difficult optimization problem: the original versions with a maximum number of generations used as stopping criterion and modified versions using the stopping criterion proposed by the authors of (Roudenko & Schoenauer, 2004). This second stopping criterion is based on the stabilization of the maximal crowding distance and aims to stop computation when many generations are performed without further improvement in the quality of the solutions. A benchmark consisting of four plant management scenarios has been used to compare the performances of the original versions (OV) and the modified versions (MV) of the NSGA-II and the MOPSO-CD. Twelve independent simulations were performed for each version and for each scenario, and a high number of generations were generated for the OV (e.g., 1500 for the NSGA-II and 200 for the MOPSO-CD). This paper compares the performances of the two versions of both algorithms using four standard metrics and statistical tests. For both algorithms, the MV obtained solutions similar in quality to those of the OV but after significantly fewer generations. The resulting reduction in computational time for the optimization step will provide opportunities for further studies on the sustainability of peach ideotypes.

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

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 ◽  
Distributed Constraint Optimization Problem

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 ◽  
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 ◽  
Single Objective

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.

Design strategies for multi-objective optimization of aerodynamic surfaces

2017 ◽  
Vol 34 (5) ◽  
pp. 1724-1753 ◽  
Author(s):  
Anand Amrit ◽  
Leifur Leifsson ◽  
Slawomir Koziel
Keyword(s):  
Pareto Front ◽  
Design Space ◽  
Optimization Problems ◽  
Computational Time ◽  
High Fidelity ◽  
Design Strategies ◽  
Aerodynamic Optimization ◽  
Content Type ◽  
Multi Objective ◽  
Aerodynamic Surfaces

Purpose This paper aims to investigates several design strategies to solve multi-objective aerodynamic optimization problems using high-fidelity simulations. The purpose is to find strategies which reduce the overall optimization time while still maintaining accuracy at the high-fidelity level. Design/methodology/approach Design strategies are proposed that use an algorithmic framework composed of search space reduction, fast surrogate models constructed using a combination of physics-based surrogates and kriging and global refinement of the Pareto front with co-kriging. The strategies either search the full or reduced design space with a low-fidelity model or a physics-based surrogate. Findings Numerical investigations of airfoil shapes in two-dimensional transonic flow are used to characterize and compare the strategies. The results show that searching a reduced design space produces the same Pareto front as when searching the full space. Moreover, as the reduced space is two orders of magnitude smaller (volume-wise), the number of required samples to setup the surrogates can be reduced by an order of magnitude. Consequently, the computational time is reduced from over three days to less than half a day. Originality/value The proposed design strategies are novel and holistic. The strategies render multi-objective design of aerodynamic surfaces using high-fidelity simulation data in moderately sized search spaces computationally tractable.

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

2005 ◽  
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.

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