Sequential efficiency optimality conditions for Robust multiobjective fractional optimization problems

This paper presents a new approach aims to solve robust multidisciplinary design optimization MDO problem called Improved Multi-objective Robust Collaborative Optimization . This method combines the Multi-objective Robust Collaborative Optimization method, the Worst Possible Point constraint cuts and the Genetic algorithm NSGA-II type as an optimizer to solve the robust optimization problem of complex structure named Y-stiffened panel under interval uncertainty . The proposed approach hierarchically decomposes the optimization problem into a structure level considered as an upper level in the Y-stiffened panel and a second level considered as a lower level of the studied panel. A robust multi-objective optimization problem intended to optimize the eigenfrequency, the global mass and the displacement at a fixed point of the Y-stiffened panel at the first level and each structure’s robust optimization problem allows optimizing its eigenfrequency and mass limited by their local constraint functions at the second one. Tor demonstrate our method, an engineering example of Y-stiffened panel is treated. A good performance of proposed method is proved by a comparison between obtained results and Non-Distributed Multi-objective Robust Optimization .

In the present paper, we explore issues in handling equality constraints in multiobjective robust design optimization (RDO) problems. Satisfying an equality constraint exactly under uncertainty can be a challenging task. The challenge of handling equality constraints is compounded in multiobjective RDO problems. Modeling the tradeoffs between the mean of the performance and the variation of the performance for each design objective in a multiobjective RDO problem is a complex task by itself. Equality constraints add to this complexity because of the additional tradeoffs that are introduced between constraint satisfaction under uncertainty and multiobjective performance. Satisfying equality constraints at their mean values, an approach typically followed, could lead to undesirable results. In this paper, we study the implications of equality constraint satisfaction in a multiobjective RDO problem, and provide a new problem formulation approach to resolve the above discussed tradeoffs. We illustrate that the proposed formulation can be used as an effective multiobjective design space exploration tool, with emphasis on equality constraint satisfaction under uncertainty. We present numerical examples to illustrate our theoretical developments.

Many engineering design optimization problems are multi-objective, constrained, and contain uncertain parameters. For these problems, it is desirable to obtain solutions that are robustly optimum. A robust optimal solution is one which remains feasible and optimal despite perturbations in the uncertainty parameters. This article presents a scenario-generation-based method for solving multi-objective robust optimization (MORO) problems. The approach follows a sequential, single-level optimization problem workflow, with the uncertainty handled by a fixed sampling method. The proposed approach is applied to an unmanned surface vessel (USV) case study problem. The results obtained and subsequent performance analysis show that the proposed approach generally performs well.

This paper proposes a novel nonmonotone proximal gradient method for solving nondifferentiable composite multiobjective optimization problems. The method integrates a nonmonotone line search technique into the proximal gradient framework, offering enhanced flexibility and efficiency for handling nonsmooth or nonconvex objective functions. We analyze the convergence properties of the proposed algorithm, demonstrate that all accumulation points of the generated sequence satisfy Pareto stationarity when the problem is nonconvex. Furthermore, under convexity and mild conditions, we establish that the sequence converges to a weakly Pareto optimal solution of the problem. Numerical experiments on robust multiobjective optimization problems and Lasso multiobjective optimization problems are provided to illustrate the computational efficiency and the practical utility of the proposed method.

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.

Ensemble prediction-based dynamic robust multi-objective optimization methods

Engineering design optimization problems are commonly multi-objective, involve computationally expensive simulations, and have uncertainty. In these problems, consideration of uncertainty can produce feasibly robust optimized design solutions, i.e., optimized and feasible under uncertainty. In general, methods for solving multi-objective robust optimization problems can be computationally expensive. To ease the computational burden and make these methods more accessible, a surrogate-based bi-objective Bayesian feasibility robust optimization method from the literature has been extended in this paper to higher numbers of objectives. A key component of the proposed method is a new acquisition function. The acquisition function proposed in this paper uses a Monte Carlo approach to approximate the expected centroidal improvement of the solution set. The proposed acquisition function is compared against exact (closed form) acquisition function-based approaches and other methods from the literature. The experimental results from this study suggest the multi-objective robust optimization framework is insensitive to the selection of acquisition function.

Surrogate duality for robust optimization

Traditionally, the identification of parameters in the formulation and solution of inverse problems considers that models, variables, and mathematical parameters are free of uncertainties. This aspect simplifies the estimation process, but does not consider the influence of relatively small changes in the design variables in terms of the objective function. In this work, the SIDR (Susceptible, Infected, Dead, and Recovered) model is used to simulate the dynamic behavior of the novel coronavirus disease (COVID-19), and its parameters are estimated by formulating a robust inverse problem, that is, considering the sensitivity of design variables. For this purpose, a robust multiobjective optimization problem is formulated, considering the minimization of uncertainties associated with the estimation process and the maximization of the robustness parameter. To solve this problem, the Multiobjective Stochastic Fractal Search algorithm is associated with the Effective Mean concept for the evaluation of robustness. The results obtained considering real data of the epidemic in China demonstrate that the evaluation of the sensitivity of the design variables can provide more reliable results.

A number of multi-objective evolutionary algorithms have been proposed in recent years and many of them have been used to solve engineering design optimization problems. However, designs need to be robust for real-life implementation, i.e. performance should not degrade substantially under expected variations in the variable values or operating conditions. Solutions of constrained robust design optimization problems should not be too close to the constraint boundaries so that they remain feasible under expected variations. A robust design optimization problem is far more computationally expensive than a design optimization problem as neighbourhood assessments of every solution are required to compute the performance variance and to ensure neighbourhood feasibility. A framework for robust design optimization using a surrogate model for neighbourhood assessments is introduced in this article. The robust design optimization problem is modelled as a multi-objective optimization problem with the aim of simultaneously maximizing performance and minimizing performance variance. A modified constraint-handling scheme is implemented to deal with neighbourhood feasibility. A radial basis function (RBF) network is used as a surrogate model and the accuracy of this model is maintained via periodic retraining. In addition to using surrogates to reduce computational time, the algorithm has been implemented on multiple processors using a master–slave topology. The preliminary results of two constrained robust design optimization problems indicate that substantial savings in the actual number of function evaluations are possible while maintaining an acceptable level of solution quality.

This paper proposes a framework for solving high-dimensional robust multi-objective optimization problems. A decision variable classification-based framework is developed to search for robust Pareto-optimal solutions. The decision variables are classified as highly and weakly robustness-related variables based on their contributions to the robustness of candidate solutions. In the case study, an order scheduling problem in the apparel industry is investigated via the proposed framework. The experimental results reveal that the performance of robust evolutionary optimization can be greatly improved via analyzing the properties of decision variables and then decomposing the high-dimensional robust multi-objective optimization problem.

Mars entry trajectory planning using robust optimization and uncertainty quantification

An Active Robust Optimisation Problem (AROP) aims at finding robust adaptable solutions, i.e. solutions that actively gain robustness to environmental changes through adaptation. Existing AROP studies have considered only a single performance objective. This study extends the Active Robust Optimisation methodology to deal with problems with more than one objective. Once multiple objectives are considered, the optimal performance for every uncertain parameter setting is a set of configurations, offering different trade-offs between the objectives. To evaluate and compare solutions to this type of problems, we suggest a robustness indicator that uses a scalarising function combining the main aims of multi-objective optimisation: proximity, diversity and pertinence. The Active Robust Multi-objective Optimisation Problem is formulated in this study, and an evolutionary algorithm that uses the hypervolume measure as a scalarasing function is suggested in order to solve it. Proof-of-concept results are demonstrated using a simplified gearbox optimisation problem for an uncertain load demand.

In Chapter 4, several polynomially solvable cases of robust discrete optimization problems are discussed in detail. However, by results presented in Chapter 3 we also know that most robust discrete optimization problems belong to the NP-hard class. In this chapter, we present our approach for solving these difficult robust discrete optimization problems. We are in this chapter restricting our attention to robust discrete optimization problems with equivalent single scenario problems that can be efficiently solved with a polynomial or pseudo-polynomial procedure. The solution procedures are based on branch-and-bound with both upper and lower bounds generated by surrogate relaxation. To be exact, the upper bound (for a maximization problem) is obtained from surrogate relaxation and the lower bound is obtained as a by-product via a heuristic based on the surrogate relaxation result. In the case when input data satisfies bounded percentage deviation condition (to be defined in Section 5.2), the heuristic is shown to provide a constant approximation. Computational results in Section 5.3 demonstrate the effectiveness of the bounds and the solution procedure.

:A robust multi-objective wing design optimization procedure using CFD is presented in this work. Instead of directly applying CFD to wing design optimization, a Kriging aerodynamic model that approximates CFD result is proposed to save computational cost. By introducing 6σ robust approach as a sub-objective function, multi-objective wing design problems can be solved with aerodynamic robustness. Non-Uniform Rational B-Spline curves (NURBS) are introduced to depict the wing geometry in design optimization process. The most important parameters of the wing geometry can then be screened by design of experiment and treated as design variables due to the local variation property of NURBS. Under this formulation, a multi-objective genetic algorithm based on non-dominated sorting is employed to handle multiple flight conditions in wing design. The optimum wing geometry is selected according to trade-off among design objects on the non-dominated front. To validate the proposed approach, two robust design optimization cases are studied for ONERA-M6-Wing. It turns out that the drag coefficient is insensitive to Mach number between Ma0.8–Ma0.9 after robust optimization. The result also verifies the effectiveness of our method in boosting performance and robustness of multiple flight conditions: the lift curve slope (Ma0.3) of optimum wing and the 6σ objective function of transonic drag (Ma0.8–Ma0.9) increase by 11.9% and 25.4%, respectively. Observations in the optimization cases are concluded as follows: 1) For wing aerodynamic design problems, the multi-objective robust optimization can both improve the performance at different flight conditions and provide robustness. 2) The Kriging aerodynamic model derived from CFD result can satisfy the precision requirement of wing design. 3) Even though the optimization result is subject to the weights of the 6σ function in sub-object, the influence is not comparable to the trade-off among design objects. 4) The optimum solution obtained by proposed approach is superior to gradient based optimization method, while the computational cost is acceptable.