Multi-Objective Robust Optimization Using Differential Evolution and Sequential Quadratic Programming

2013 ◽  
Author(s):  
Shuo Cheng ◽  
Mian Li
Keyword(s):  
Quadratic Programming ◽  
Robust Optimization ◽  
Differential Evolution ◽  
Performance Improvement ◽  
Sequential Quadratic Programming ◽  
Optimization Problems ◽  
Multi Objective ◽  
Engineering Problems ◽  
And Performance

Multi-Objective Robust Optimization (MORO) can find Pareto solutions to multi-objective engineering problems while keeping the variation of the solutions being within an acceptable range when parameters vary. While the literature reports on many techniques in MORO, few papers focus on the implementation of Multi-Objective Differential Evolution (MODE) for robust optimization and the performance improvement of solutions. In this paper, MODE is first modified and implemented for robust optimization, formulating a new MODE-RO algorithm. To improve the solutions’ quality of MODE-RO, a new hybrid MODE-SQP-RO algorithm is further proposed, where Sequential Quadratic Programming (SQP) is incorporated to enhance the local search. In the hybrid algorithm, two criteria, indicating the convergence speed of MODE-RO and the switch between MODE and SQP are proposed respectively. One numerical and one engineering examples are tested to demonstrate the applicability and performance of the proposed algorithms. The results show that MODE-RO is effective in solving Multi-Objective Robust Optimization problems; while on the average, MODE-SQP-RO significantly improves the quality of robust solutions with comparable numbers of function evaluations.

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.

Sequential Quadratic Programming for Robust Optimization With Interval Uncertainty

2012 ◽  
Vol 134 (10) ◽  
Author(s):  
Jianhua Zhou ◽  
Shuo Cheng ◽  
Mian Li
Keyword(s):  
Quadratic Programming ◽  
Robust Optimization ◽  
Sequential Quadratic Programming ◽  
Optimization Problems ◽  
Critical Role ◽  
Interval Uncertainty ◽  
Design Solution ◽  
Novel Approach ◽  
Design Variables ◽  
Feasibility Robustness

Uncertainty plays a critical role in engineering design as even a small amount of uncertainty could make an optimal design solution infeasible. The goal of robust optimization is to find a solution that is both optimal and insensitive to uncertainty that may exist in parameters and design variables. In this paper, a novel approach, sequential quadratic programming for robust optimization (SQP-RO), is proposed to solve single-objective continuous nonlinear optimization problems with interval uncertainty in parameters and design variables. This new SQP-RO is developed based on a classic SQP procedure with additional calculations for constraints on objective robustness, feasibility robustness, or both. The obtained solution is locally optimal and robust. Eight numerical and engineering examples with different levels of complexity are utilized to demonstrate the applicability and efficiency of the proposed SQP-RO with the comparison to its deterministic SQP counterpart and RO approaches using genetic algorithms. The objective and/or feasibility robustness are verified via Monte Carlo simulations.

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.

scholarly journals Multi-Objective Robust Optimization Using a Postoptimality Sensitivity Analysis Technique: Application to a Wind Turbine Design

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
Weijun Wang ◽  
Stéphane Caro ◽  
Fouad Bennis ◽  
Ricardo Soto ◽  
Broderick Crawford
Keyword(s):  
Sensitivity Analysis ◽  
Robust Optimization ◽  
Wind Turbine ◽  
Optimization Design ◽  
Optimization Problems ◽  
Design Environment ◽  
Multi Objective ◽  
Pareto Ranking ◽  
Analysis Technique ◽  
Robust Optimization Design

Toward a multi-objective optimization robust problem, the variations in design variables (DVs) and design environment parameters (DEPs) include the small variations and the large variations. The former have small effect on the performance functions and/or the constraints, and the latter refer to the ones that have large effect on the performance functions and/or the constraints. The robustness of performance functions is discussed in this paper. A postoptimality sensitivity analysis technique for multi-objective robust optimization problems (MOROPs) is discussed, and two robustness indices (RIs) are introduced. The first one considers the robustness of the performance functions to small variations in the DVs and the DEPs. The second RI characterizes the robustness of the performance functions to large variations in the DEPs. It is based on the ability of a solution to maintain a good Pareto ranking for different DEPs due to large variations. The robustness of the solutions is treated as vectors in the robustness function space (RF-Space), which is defined by the two proposed RIs. As a result, the designer can compare the robustness of all Pareto optimal solutions and make a decision. Finally, two illustrative examples are given to highlight the contributions of this paper. The first example is about a numerical problem, whereas the second problem deals with the multi-objective robust optimization design of a floating wind turbine.

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