A surrogate assisted parallel multiobjective evolutionary algorithm for robust engineering design

Multiple parameter optimization methodology by integrating a game theory principle into priority-based decision making

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.

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 .

Robust design optimization (RDO) has been eminent, ascertaining optimal configuration of engineering systems in presence of uncertainties. However, computational aspect of conventional RDO can often get computationally intensive as neighborhood assessments of every solution are required to compute the performance variance and ensure feasibility. Surrogate assisted optimization is one of the efficient approaches in order to mitigate this issue of computational expense. However, the performance of a surrogate model plays a key factor in determining the optima in multi-modal and highly non-linear landscapes, in presence of uncertainties. In other words, the approximation accuracy of the model is principal in yielding the actual optima and thus, avoiding any misguide to the decision maker on the basis of false or, local optimum points. Therefore, an extensive survey has been carried out by employing most of the well-known surrogate models in the framework of RDO. It is worth mentioning that the numerical study has revealed consistent performance of a model out of all the surrogates utilized. Finally, the best performing model has been utilized in solving a large-scale practical RDO problem. All the results have been compared with that of Monte Carlo simulation results.

Robust design optimization of the vibrating rotor-shaft system subjected to selected dynamic constraints

Surrogate duality for robust optimization

The paper is devoted to model uncertainties (or model form uncertainties) induced by modeling errors in computational sciences and engineering (such as in computational structural dynamics, fluid-structure interaction, and vibroacoustics, etc.) for which a parametric high-fidelity computational model (HFM) is used for addressing optimization problems (such as a robust design optimization problem), which are solved by introducing a parametric reduced-order model (ROM) constructed using an adapted reduced-order basis (ROB) derived from the parametric HFM. Two main methodologies are available to take into account such modeling errors. The first one is the usual output-predictive error method that has been introduced for many years. This approach can induce some difficulties because the parametric HFM and ROM do not learn from data. The second one is the nonparametric probabilistic approach of model uncertainties introduced in the framework of structural dynamics fifteen years ago. This approach is adapted, but is mainly limited to linear operators of the parametric HFM. The present paper deals with this challenging problem and proposes a novel nonparametric probabilistic approach of the modeling errors for any parametric nonlinear HFM for which a parametric nonlinear ROM can be constructed from the HFM. The methodology proposed consists in substituting the deterministic ROB with a stochastic ROB for which the probability measure in constructed on a subset of a compact Stiefel manifold. The stochastic model depends on a small number of hyperparameters for which the identification is performed by solving a statistical inverse problem. An application is presented in nonlinear computational structural dynamics.

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

Mars entry trajectory planning using robust optimization and uncertainty quantification

This article demonstrates the main capabilities of IOSO (Indirect Optimization based on Self-organization) technology algorithms, tools, and software, which can be used for the optimization of complex systems and objects. IOSO algorithms have higher efficiency, provide a wider range of capabilities, and are practically insensitive with respect to the types of objective function and constraints. They could be smooth, non-differentiable, and stochastic, with multiple optima, with the portions of the design space where objective function and constraints could not be evaluated at all, with the objective function and constraints dependent on mixed variables, etc. The capabilities of IOSO software are demonstrated using examples of solving complex multi-objective (up to 8 simultaneous objectives) problems, which are solved in deterministic and robust design optimization statements. The results of this study show the Pareto set probability statement, which decreases technical risks when developing modern objects and systems with the highest level of efficiency.

Swarm intelligence hybridized with genetic search in multi-objective design optimization under constrained-Pareto dominance

Demand in a water distribution network (WDN) is an aleatory variable, owing to the unpredictable behaviors of water users. Therefore, it is one of the main reasons for uncertainty in the design of this infrastructure. The increasing number of water demand data sets offers opportunities to improve the traditional deterministic design approaches of WDNs by combining statistical and optimization methods. Robust optimization (RO) takes demand uncertainty into account by studying solutions that perform well under any possible demand scenario, that is, any possible realizations of this variable in the lifetime of a WDN. The right choice of scenario is therefore essential to ensure the reliability of the designed network. This paper presents a statistical methodology for generating scenarios to be used to solve a robust design optimization problem. It involves three steps: (a) descriptive analytics of historical data to derive the marginal distributions of peak hour demand in each node of the WDN, (b) generation of a very large number of snapshots by stratified sampling from the correlated marginal distributions of nodal peak demand, (c) generation of the peak demand scenarios by reducing the number of snapshots. Two heuristic techniques are proposed to reduce the number of snapshots, and for each of them, two different numbers of scenarios are derived. Two multi‐objective RO models are solved: the first model includes cost minimization and a mean‐variance Generalized Resilience and Failure index maximization objectives, and the second one additionally considers the minimization of the maximum undelivered demand, formalized using a regret function.

Interval‐based multi‐objective robust optimization aims to achieve high‐performance solutions insensitive to uncertainty, has garnered significant attention. However, efficiently analyzing the maximum fluctuations of solutions in both objective and constraint functions to assess their robustness remains challenging. To address this issue, this article proposes an adaptive coevolution method, which can evaluate the maximum fluctuations of multiple candidate solutions with respect to a function in a single run. This method is integrated with the multi‐objective evolutionary algorithm (MOEA) to develop a framework termed adaptive coevolution‐based multi‐objective robust optimization (AC‐MORO) for solving multi‐objective robust optimization problems. To evaluate the performance of AC‐MORO, it is compared with MODE‐RO on a set of benchmark problems; meanwhile, a performance metric is proposed to test the accuracy of the adaptive coevolution method in analyzing the robustness of solutions. The impact of various parameter settings on the efficiency of the proposed method is also investigated. Subsequently, a variant of the adaptive coevolution method is explored to further enhance the performance of AC‐MORO. Finally, AC‐MORO is applied to address robust optimization problems in practical engineering.

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.

In this paper, we define a distributionally robust k-sum optimization problem as the problem of finding a solution that minimizes the worst-case expected sum of up to the k largest costs of the elements in the solution. The costs are random with a joint probability distribution that is not completely specified but rather assumed to be known to lie in a set of probability distributions. For k=1, this reduces to a distributionally robust bottleneck optimization problem while for k=n, this reduces to distributionally robust minimum sum optimization problem. Our main result is that for a Fréchet class of discrete marginal distributions with finite support, the distributionally robust k-sum combinatorial optimization problem is solvable in polynomial time if the deterministic minimum sum problem is solvable in polynomial time. This extends the result of Punnen and Aneja [On k-sum optimization, Oper. Res. Lett. 18(5)(1996), pp. 233–236] from the deterministic to the robust case. We show that this choice of the set of distributions helps preserve the submodularity of the k-sum objective function which is an useful structural property for optimization problems.