Advanced Robust Optimization Approach for Design Optimization With Interval Uncertainty Using Sequential Quadratic Programming

2013 ◽  
Author(s):  
Jianhua Zhou ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Linear Equations ◽  
Variable Parameter ◽  
Optimal Solution ◽  
Interval Uncertainty ◽  
Computational Time ◽  
Optimization Approach ◽  
Engineering Optimization ◽  
Loop Optimization ◽  
New Approach

Uncertainty is inevitable in real world. It has to be taken into consideration, especially in engineering optimization; otherwise the obtained optimal solution may become infeasible. Robust optimization (RO) approaches have been proposed to deal with this issue. Most existing RO algorithms use double-looped structures in which a large amount of computational efforts have been spent in the inner loop optimization to determine the robustness of candidate solutions. In this paper, an advanced approach is presented where no optimization run is required to be performed for robustness evaluations in the inner loop. Instead, a concept of Utopian point is proposed and the corresponding maximum variable/parameter variation will be obtained by just solving a set of linear equations. The obtained robust optimal solution from the new approach may be conservative, but the deviation from the true robust optimal solution is very small given the significant improvement in the computational efficiency. Six numerical and engineering examples are tested to show the applicability and efficiency of the proposed approach, whose solutions and computational time are compared with those from a similar but double-looped approach, SQP-RO, proposed previously.

Advanced Robust Optimization With Interval Uncertainty Using a Single-Looped Structure and Sequential Quadratic Programming

2013 ◽  
Vol 136 (2) ◽  
Author(s):  
Jianhua Zhou ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Variable Parameter ◽  
Optimal Solution ◽  
Parameter Variation ◽  
Interval Uncertainty ◽  
Quadratic Program ◽  
Engineering Optimization ◽  
Loop Optimization ◽  
New Approach ◽  
Matrix Operations

Uncertainty is inevitable and has to be taken into consideration in engineering optimization; otherwise, the obtained optimal solution may become infeasible or its performance can degrade significantly. Robust optimization (RO) approaches have been proposed to deal with this issue. Most existing RO algorithms use double-looped structures in which a large amount of computational efforts have been spent in the inner loop optimization to determine the robustness of candidate solutions. In this paper, an advanced approach is presented where no optimization run is required for robustness evaluation in the inner loop. Instead, a concept of Utopian point is proposed and the corresponding maximum variable/parameter variation will be obtained just by performing matrix operations. The obtained robust optimal solution from the new approach may be conservative, but the deviation from the true robust optimal solution is small enough and acceptable given the significant improvement in the computational efficiency. Six numerical and engineering examples are tested to show the applicability and efficiency of the proposed approach, whose solutions and computational efforts are compared to those from a previously proposed double-looped approach, sequential quadratic program-robust optimization (SQP-RO).

Multiobjective Collaborative Robust Optimization With Interval Uncertainty and Interdisciplinary Uncertainty Propagation

2008 ◽  
Vol 130 (8) ◽  
Author(s):  
M. Li ◽  
S. Azarm
Keyword(s):  
Robust Optimization ◽  
Probability Distributions ◽  
Uncertainty Propagation ◽  
Multiple Objectives ◽  
Interval Uncertainty ◽  
Optimization Approach ◽  
Solution Approach ◽  
Time In Literature ◽  
Collaborative Robust Optimization ◽  
First Time

We present a new solution approach for multidisciplinary design optimization (MDO) problems that, for the first time in literature, has all of the following characteristics: Each discipline has multiple objectives and constraints with mixed continuous-discrete variables; uncertainty exists in parameters and as a result, uncertainty propagation exists within and across disciplines; probability distributions of uncertain parameters are not available but their interval of uncertainty is known; and disciplines can be fully (two-way) coupled. The proposed multiobjective collaborative robust optimization (McRO) approach uses a multiobjective genetic algorithm as an optimizer. McRO obtains solutions that are as best as possible in a multiobjective and multidisciplinary sense. Moreover, for McRO solutions, the variation of objective and/or constraint functions can be kept within an acceptable range. McRO includes a technique for interdisciplinary uncertainty propagation. The approach can be used for robust optimization of MDO problems with multiple objectives, or constraints, or both together at system and subsystem levels. Results from an application of McRO to a numerical and an engineering example are presented. It is concluded that McRO can solve fully coupled MDO problems with interval uncertainty and obtain solutions that are comparable to a single-disciplinary robust optimization approach.

A Multi-Objective Robust Optimization Approach Under Interval Uncertainty Based on Kriging and Support Vector Machine

2018 ◽  
Author(s):  
Tingli Xie ◽  
Ping Jiang ◽  
Qi Zhou ◽  
Leshi Shu ◽  
Yang Yang
Keyword(s):  
Support Vector Machine ◽  
Robust Optimization ◽  
Kriging Model ◽  
Interval Uncertainty ◽  
Support Vector ◽  
Optimization Approach ◽  
Multi Objective ◽  
Large Numbers ◽  
Design Alternatives ◽  
Engineering Design Problems

Interval uncertainty can cause uncontrollable variations in the objective and constraint values, which could seriously deteriorate the performance or even change the feasibility of the optimal solutions. Robust optimization is to obtain solutions that are optimal and minimally sensitive to uncertainty. Because large numbers of complex engineering design problems depend on time-consuming simulations, the robust optimization approaches might become computationally intractable. To address this issue, a multi-objective robust optimization approach based on Kriging and support vector machine (MORO-KS) is proposed in this paper. Firstly, the feasible domain of main problem in MORO-KS is iteratively restricted by constraint cuts formed in the subproblem. Secondly, each objective function is approximated by a Kriging model to predict the response value. Thirdly, a Support Vector Machine (SVM) model is constructed to replace all constraint functions classifying design alternatives into two categories: feasible and infeasible. A numerical example and the design optimization of a microaerial vehicle fuselage are adopted to test the proposed MORO-KS approach. Compared with the results obtained from the MORO approach based on Constraint Cuts (MORO-CC), the effectiveness and efficiency of the proposed MORO-KS approach are illustrated.

scholarly journals Non-Convex Feasibility Robust Optimization Via Scenario Generation and Local Refinement

2019 ◽  
Vol 142 (5) ◽  
Author(s):  
Eliot Rudnick-Cohen ◽  
Jeffrey W. Herrmann ◽  
Shapour Azarm
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Computational Cost ◽  
Optimal Solution ◽  
Optimization Techniques ◽  
Scenario Generation ◽  
Test Problems ◽  
Optimization Approach ◽  
Uncertain Parameters ◽  
Worst Case

Abstract Feasibility robust optimization techniques solve optimization problems with uncertain parameters that appear only in their constraint functions. Solving such problems requires finding an optimal solution that is feasible for all realizations of the uncertain parameters. This paper presents a new feasibility robust optimization approach involving uncertain parameters defined on continuous domains. The proposed approach is based on an integration of two techniques: (i) a sampling-based scenario generation scheme and (ii) a local robust optimization approach. An analysis of the computational cost of this integrated approach is performed to provide worst-case bounds on its computational cost. The proposed approach is applied to several non-convex engineering test problems and compared against two existing robust optimization approaches. The results show that the proposed approach can efficiently find a robust optimal solution across the test problems, even when existing methods for non-convex robust optimization are unable to find a robust optimal solution. A scalable test problem is solved by the approach, demonstrating that its computational cost scales with problem size as predicted by an analysis of the worst-case computational cost bounds.

scholarly journals Unconstrained Quadratic Programming Problem with Uncertain Parameters

2021 ◽  
pp. 1-7
Author(s):  
Sie Long Kek ◽  
Fong Peng Lim ◽  
Harley Ooi
Keyword(s):  
Quadratic Programming ◽  
Programming Problem ◽  
Linear Equations ◽  
Optimal Solution ◽  
Theoretical Work ◽  
Quadratic Programming Problem ◽  
Model Parameters ◽  
Optimization Approach ◽  
Uncertain Parameters ◽  
Unconstrained Quadratic Programming

In this paper, an unconstrained quadratic programming problem with uncertain parameters is discussed. For this purpose, the basic idea of optimizing the unconstrained quadratic programming problem is introduced. The solution method of solving linear equations could be applied to obtain the optimal solution for this kind of problem. Later, the theoretical work on the optimization of the unconstrained quadratic programming problem is presented. By this, the model parameters, which are unknown values, are considered. In this uncertain situation, it is assumed that these parameters are normally distributed; then, the simulation on these uncertain parameters are performed, so the quadratic programming problem without constraints could be solved iteratively by using the gradient-based optimization approach. For illustration, an example of this problem is studied. The computation procedure is expressed, and the result obtained shows the optimal solution in the uncertain environment. In conclusion, the unconstrained quadratic programming problem, which has uncertain parameters, could be solved successfully.

A New Hybrid Algorithm for Multi-Objective Robust Optimization With Interval Uncertainty

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Shuo Cheng ◽  
Jianhua Zhou ◽  
Mian Li
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Hybrid Approach ◽  
Interval Uncertainty ◽  
Engineering Optimization ◽  
Objective Functions ◽  
Robust Solutions ◽  
Multi Objective ◽  
And Performance

Uncertainty is a very critical but inevitable issue in design optimization. Compared to single-objective optimization problems, the situation becomes more difficult for multi-objective engineering optimization problems under uncertainty. Multi-objective robust optimization (MORO) approaches have been developed to find Pareto robust solutions. While the literature reports on many techniques in MORO, few papers focus on using multi-objective differential evolution (MODE) for robust optimization (RO) and performance improvement of its solutions. In this article, MODE is first modified and developed for RO problems with interval uncertainty, formulating a new MODE-RO algorithm. To improve the solutions’ quality of MODE-RO, a new hybrid (MODE-sequential quadratic programming (SQP)-RO) algorithm is proposed further, where SQP is incorporated into the procedure to enhance the local search. The proposed hybrid approach takes the advantage of MODE for its capability of handling not-well behaved robust constraint functions and SQP for its fast local convergence. Two numerical and one engineering examples, with two or three objective functions, are tested to demonstrate the applicability and performance of the proposed algorithms. The results show that MODE-RO is effective in solving MORO problems while, on the average, MODE-SQP-RO improves the quality of robust solutions obtained by MODE-RO with comparable numbers of function evaluations.

A multi-objective robust optimization approach for engineering design under interval uncertainty

2018 ◽  
Vol 35 (2) ◽  
pp. 580-603 ◽  
Author(s):  
Qi Zhou ◽  
Xinyu Shao ◽  
Ping Jiang ◽  
Tingli Xie ◽  
Jiexiang Hu ◽  
...  
Keyword(s):  
Robust Optimization ◽  
Engineering Design ◽  
Optimization Problems ◽  
Interval Uncertainty ◽  
Optimization Approach ◽  
Content Type ◽  
Computational Burden ◽  
Multi Objective ◽  
Design Alternatives ◽  
Kriging Metamodel

Purpose Engineering system design and optimization problems are usually multi-objective and constrained and have uncertainties in the inputs. These uncertainties might significantly degrade the overall performance of engineering systems and change the feasibility of the obtained solutions. This paper aims to propose a multi-objective robust optimization approach based on Kriging metamodel (K-MORO) to obtain the robust Pareto set under the interval uncertainty. Design/methodology/approach In K-MORO, the nested optimization structure is reduced into a single loop optimization structure to ease the computational burden. Considering the interpolation uncertainty from the Kriging metamodel may affect the robustness of the Pareto optima, an objective switching and sequential updating strategy is introduced in K-MORO to determine (1) whether the robust analysis or the Kriging metamodel should be used to evaluate the robustness of design alternatives, and (2) which design alternatives are selected to improve the prediction accuracy of the Kriging metamodel during the robust optimization process. Findings Five numerical and engineering cases are used to demonstrate the applicability of the proposed approach. The results illustrate that K-MORO is able to obtain robust Pareto frontier, while significantly reducing computational cost. Practical implications The proposed approach exhibits great capability for practical engineering design optimization problems that are multi-objective and constrained and have uncertainties. Originality/value A K-MORO approach is proposed, which can obtain the robust Pareto set under the interval uncertainty and ease the computational burden of the robust optimization process.

An on-line Kriging metamodel assisted robust optimization approach under interval uncertainty

2017 ◽  
Vol 34 (2) ◽  
pp. 420-446 ◽  
Author(s):  
Qi Zhou ◽  
Ping Jiang ◽  
Xinyu Shao ◽  
Hui Zhou ◽  
Jiexiang Hu
Keyword(s):  
Robust Optimization ◽  
Optimization Problems ◽  
Computational Cost ◽  
Interval Uncertainty ◽  
Optimization Approach ◽  
Content Type ◽  
Computational Burden ◽  
Design Alternative ◽  
Kriging Metamodel ◽  
On Line

Purpose Uncertainty is inevitable in real-world engineering optimization. With an outer-inner optimization structure, most previous robust optimization (RO) approaches under interval uncertainty can become computationally intractable because the inner level must perform robust evaluation for each design alternative delivered from the outer level. This paper aims to propose an on-line Kriging metamodel-assisted variable adjustment robust optimization (OLK-VARO) to ease the computational burden of previous VARO approach. Design/methodology/approach In OLK-VARO, Kriging metamodels are constructed for replacing robust evaluations of the design alternative delivered from the outer level, reducing the nested optimization structure of previous VARO approach into a single loop optimization structure. An on-line updating mechanism is introduced in OLK-VARO to exploit the obtained data from previous iterations. Findings One nonlinear numerical example and two engineering cases have been used to demonstrate the applicability and efficiency of the proposed OLK-VARO approach. Results illustrate that OLK-VARO is able to obtain comparable robust optimums as to that obtained by previous VARO, while at the same time significantly reducing computational cost. Practical implications The proposed approach exhibits great capability for practical engineering design optimization problems under interval uncertainty. Originality/value The main contribution of this paper lies in the following: an OLK-VARO approach under interval uncertainty is proposed, which can significantly ease the computational burden of previous VARO approach.

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