Multi-objective Robust Optimization Problem Research Articles

Solving multi-objective robust optimization problems via Stakelberg-based game model

Real-world multi-objective engineering problems frequently involve uncertainties stemming from environmental factors, production inaccuracies, and other sources. A critical aspect of addressing these problems, termed Multi-Objective Robust Optimization (MORO) problems, is the development of solutions that are both optimal and resilient to uncertainties. This paper proposes addressing these uncertainties through the application of Stackelberg game models, a novel approach involving the interaction of two players. The Leader searches for optimal and robust solutions and the Follower generates uncertainties based on the Leader’s chosen solutions. The Follower seeks to tackle the most challenging uncertainties associated with the Leader’s candidate solutions. Additionally, this paper introduces a novel metric to assess the robustness of a given set of solutions concerning specified uncertainties.Based on the proposed approach, a co-evolutionary algorithm is developed. A numerical study is then conducted to evaluate the algorithm by comparing its performance with those obtained by four benchmark algorithms on nine benchmark MORO problems. The numerical study also aims to assess its sensitivity to run parameter variations. The experimental results demonstrate the proposed approach’s effectiveness in identifying a non-dominated robust set of solutions.

Robust design optimization of a multi-body system with aleatory and epistemic uncertainty

There are several issues in robust design optimization of the multi-body system with aleatory and epistemic uncertainty, such as the difficulty in the multi-source uncertainty characterization and low efficiency in dynamic modeling, analysis and the corresponding solving strategy for its robust design optimization. To address these problems, this paper firstly constructs a unified analysis framework for aleatory and epistemic uncertainty for the multi-body system based on probability theory and evidence theory. Taking a multi-body system of a shipborne artillery model as an example, the transfer matrix method is next employed to rapidly calculate the dynamic response of the muzzle. The results are compared with those by the finite element method to verify the accuracy and efficiency of the transfer matrix method. A decision model of the robust design optimization with both aleatory and epistemic uncertainty is proposed. Based on the polynomial chaos expansion-Kriging model, subset simulation optimization method and a weighted single-objective solving strategy, the multi-objective robust design optimization problem involving aleatory and epistemic uncertainty is then transformed into a single-objective optimization one. Finally, results of both the multi-objective and weighted single-objective robust design optimization methods for the shipborne artillery model are compared to demonstrate the performance of the proposed method.

Optimality Conditions of the Approximate Efficiency for Nonsmooth Robust Multiobjective Fractional Semi-Infinite Optimization Problems

This paper is devoted to the investigation of optimality conditions and saddle point theorems for robust approximate quasi-weak efficient solutions for a nonsmooth uncertain multiobjective fractional semi-infinite optimization problem (NUMFP). Firstly, a necessary optimality condition is established by using the properties of the Gerstewitz’s function. Furthermore, a kind of approximate pseudo/quasi-convex function is defined for the problem (NUMFP), and under its assumption, a sufficient optimality condition is obtained. Finally, we introduce the notion of a robust approximate quasi-weak saddle point to the problem (NUMFP) and prove corresponding saddle point theorems.

Sufficient Optimality Conditions for a Robust Multiobjective Problem

In this note, we establish sufficient optimality conditions in terms of upper convexificators for a robust multiobjective optimization problem (UCP) involving nonsmooth/nonconvex real-valued functions. For this purpose, two kinds of generalized convex functions are defined in terms of upper convexificators.

Necessary Optimality Conditions for Robust Nonsmooth Multiobjective Optimization Problems

Abstract This paper deals with a robust multiobjective optimization problem involving nonsmooth/nonconvex real-valued functions. Under an appropriate constraint qualification, we establish necessary optimality conditions for weakly robust efficient solutions of the considered problem. These optimality conditions are presented in terms of Karush-Kuhn-Tucker multipliers and convexificators of the related functions. Examples illustrating our findings are also given.

On approximate optimality conditions for robust multi-objective convex optimization problems

In this paper, we are interested in the study of approximate optimality conditions for weakly ϵ-efficient solutions to robust multi-objective optimization problems ((RMOP) for short) in view of its associated minimax optimization problem (MMOP). To this end, we first establish the relationship between a weakly ϵ-efficient solution to the problem (RMOP) and an α-solution to the problem (MMOP), where and . Then, we explore the representation of the so-called β-normal set (where is a given parameter) to a closed convex set at some reference point by two methods. At last, by employing the α-subdifferential of the max-function and the obtained representation of the β-normal set, we establish an approximate necessary optimality condition for the problem (RMOP). Moreover, we also give an example to illustrate our results.

Robust optimal attitude determination using interval analysis

The attitude determination of the satellite leads to solving the Wahba problem. The Wahba problem utilizes a set of at least two independent sensor measurements and reference vectors for attitude determination. These input vectors are not accurate due to sensor noises, misalignment, and low-order approximations. But these uncertainties do not consider in the classic Wahba problem directly. This paper presents a robust multi-objective optimization method for solving the static attitude determination with bounded uncertainty. The bounded uncertainty due to modeling error, measurement noise, and biases have modeled using interval analysis. Then, the Wahba problem is reformulated to a robust multi-objective optimization problem that consists of two performance indexes. The effectiveness of the proposed approach is evaluating by solving several examples.

Bayesian optimization for robust design of steel frames with joint and individual probabilistic constraints

This work proposes a Bayesian optimization (BO) method for solving multi-objective robust design optimization (RDO) problems of steel frames under aleatory uncertainty in external loads and material properties. Joint and individual probabilistic constrained RDO problems are formulated to consider two different ways the frame reaches its collapse state. Each problem involves three conflicting objective functions, namely, the total mass of the frame, the mean and variance of the maximum inter-story drift. Since the uncertain objective and probabilistic constraint functions of both problems are implicit within a finite element analysis program and the computation of the probabilistic constraints is an NP-hard problem, BO is used to guide the optimization process toward better solutions after it completes an iteration and offers a set of near Pareto-optimal solutions when it terminates. Specifically, Bayesian regression models called Gaussian processes (GPs) serve as surrogates for the structural responses. Two acquisition functions are then developed for the two RDO problems and a maximization problem of these functions is formulated as a mixed-integer nonlinear programming (MINLP) problem. A new random search coupled with simulated annealing is devised to solve the MINLP problem, thereby locating the most promising point in the input variable space at which the current solutions maximize their chance to be improved and the GP models are refined before the BO starts a new iteration. A test problem and two design examples show that exact or good Pareto-optimal solutions to the RDO problems can be found by the proposed method with 20 iterations.

Minimax programming as a tool for studying robust multi-objective optimization problems

Minimax programming as a tool for studying robust multi-objective optimization problems

Robust Multiobjective Nonlinear Constrained Optimization with Ensemble Stochastic Gradient Sequential Quadratic Programming-Filter Algorithm

Summary As the crucial step in closed-loop reservoir management, robust life-cycle production optimization is defined as maximizing/minimizing the expected value of a predefined objective (cost) function over geological uncertainties (i.e., uncertainties in the reservoir permeability, porosity, endpoint relative permeability, etc.). However, with robust optimization, there is no control over downside risk defined as the minimum net present value (NPV) among the individual NPVs of the different reservoir models. Yet, field operators generally wish to keep this minimum NPV reasonably large to try to ensure that the reservoir is commercially viable. In addition, the field operator may desire to maximize the NPV of production over a much shorter time period than the life of the reservoir under the limitation of surface facilities (e.g., field liquid and water production rates). Thus, it is important to consider multiobjective robust production optimization with nonlinear constraints and when geological uncertainties are incorporated. The three objectives considered in this paper are; to maximize the average life-cycle NPV, to maximize the average short-term NPV, and to maximize the minimum NPV of the set of realizations. Generally, these objectives are in conflict; for example, the well controls that give a global maximum for robust life-cycle production optimization do not usually correspond to the controls that maximize the short-term average NPV of production. Moreover, handling the nonlinear state constraints (e.g., field liquid production rates and field water production rates for the bottom-hole pressure controlled producers in the robust production optimization) is also a challenge because those nonlinear constraints should be satisfied at each control steps for each geological realization. To provide potential solutions to the multiobjective robust optimization problem with state constraints, we developed a modified lexicographic method with a minimizing-maximum scheme to attempt to obtain a set of Pareto optimal solutions and to satisfy all nonlinear constraints. We apply the sequential quadratic programming filter with modified stochastic gradients to solve a sequence of optimization problems, where each solution is designed to generate a single point on the Pareto front. In the modified lexicographic method, the objective is always considered to be the primary objective, and the other objectives are considered by specifying bounds on them to convert them to state constraints. The temporal damping and truncation schemes are applied to improve the quality of the stochastic gradient on nonlinear constraints, and the minimizing–maximum procedure is applied to enforce constraints on the normal state constraints. The main advantage that the modified lexicographic method has over the standard lexicographic method is that it allows for the generation of potential Pareto optimal points, which are uniformly spaced in the values of the second and/or third objective that one wishes to improve by multiobjective optimization.

Identification of an Epidemiological Model to Simulate the COVID-19 Epidemic Using Robust Multiobjective Optimization and Stochastic Fractal Search.

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.

Multi-Objective Robust Optimization of a Dual-Flux-Modulator Magnetic Geared Machine With Hybrid Uncertainties

In order to improve the robustness and torque performance of a dual-flux-modulator magnetic-geared machine (DFM-MGM) considering simultaneously both random and interval uncertainties of design parameters, a multi-objective robust optimization (MORO) method with multiple Monte Carlo simulations (MCSs) is proposed. In this method, the multiple MCSs are adopted to evaluate the effects of parametric hybrid uncertainties on the robustness of optimization results. To build a MORO model of the DFM-MGM, the three-dimensional finite element model is established firstly and then validated by the experiment. Through a parametric study, it is found that five dimensional parameters of the permanent magnets (PMs) and stator have more significant effects on the stall torque (ST) and ST per PM volume (STPPV). Finally, a multi-objective particle swarm optimization algorithm with surrogate models, sigma criteria design and multiple MCSs method is implemented to solve the MORO problem. Both the average standard deviations and standard deviation differences of the ST and STPPV are used to deal with hybrid uncertainties during MORO. The optimized DFM-MGM by MORO has a STPPV 6.3% higher than that of the initial design under the same ST constraint. Moreover, the average standard deviations and standard deviation differences obtained by MORO are much smaller than those achieved by the deterministic optimization, indicating that the robustness of optimal results can also be significantly improved by the MORO.

On Approximate Efficiency for Nonsmooth Robust Vector Optimization Problems

In this article, we use the robust optimization approach (also called the worst-case approach) for finding e-efficient solutions of the robust multiobjective optimization problem defined as a robust (worst-case) counterpart for the considered nonsmooth multiobjective programming problem with the uncertainty in both the objective and constraint functions. Namely, we establish both necessary and sufficient optimality conditions for a feasible solution to be an e-efficient solution (an approximate efficient solution) of the considered robust multiobjective optimization problem. We also use a scalarizing method in proving these optimality conditions.

On a class of nonsmooth fractional robust multi-objective optimization problems. Part I: Optimality conditions

On a class of nonsmooth fractional robust multi-objective optimization problems. Part I: Optimality conditions

Evolutionary Multiobjective Optimization With Robustness Enhancement

Uncertainty is an important feature abstracted from real-world applications. Multiobjective optimization problems (MOPs) with uncertainty can always be characterized as robust MOPs (RMOPs). Over recent years, multiobjective optimization evolutionary algorithms (EAs) have demonstrated the success in solving MOPs. However, most of them do not consider disturbance in the design. In order to handling the uncertainty in the optimization problem, we first give a thorough analysis of three important issues on robust optimization. Then, a novel EA called multiobjective optimization EA with robustness enhancement is developed, where the seamless integration of robustness and optimality is achieved by a proposed novel archive updating mechanism applied on the evolutionary process as well as the new robust optimal front building strategy designed to construct the final robust optimal front. Furthermore, the new designed archive updating mechanism makes the robust optimization process free of the enormous computational workload induced from sampling. The experimental results on a set of benchmark functions show the superiority of the proposed design in terms of both solutions' quality under the disturbance and computational efficiency in solving RMOPs.

Robust optimization of aerodynamic loadings for airfoil inverse designs

For the modern design of more realistic aerodynamic shapes, disturbances caused by uncertain operating conditions must be conveniently considered, since they can affect significantly the performance of the designed systems. In this situation, the concept of robust design in conjunction with optimization tools is strongly recommended, since the interest is to maximize the performance and its robustness, simultaneously. Clearly, the great number of exact evaluations normally required to compute the robustness makes the robust optimization in aerodynamics computationally prohibitive, particularly when direct methods are chosen. To overcome this drawback, inverse methods appear as an interesting option, provided a robust aerodynamic loading can be furnished previously at low computational costs. However, few works have been dedicated to this subject in the open literature, which motivates the present study. The focus is to apply the robustness concept to optimize the velocity (or pressure) distributions for airfoil inverse designs, using a boundary layer method to predict the aerodynamic coefficients prior to the knowledge of the final airfoil shape. Here, the velocity distribution is parameterized using B-spline polygons with a set of control points, where the design variables are the ordinates of these points in the parameterization. The resulting robust multiobjective optimization problem involves the performance of the airfoil as a first objective function and its robustness introduced as additional objective to be optimized simultaneously. To illustrate the usefulness of the proposed robust design method, an example of drag minimization for an isolated airfoil is addressed and the aerodynamic coefficients for the optimal airfoils are compared with the corresponding obtained by experiments from the open literature.

Robust Multiobjective Optimization via Evolutionary Algorithms

Uncertainty inadvertently exists in most real-world applications. In the optimization process, uncertainty poses a very important issue and it directly affects the optimization performance. Nowadays, evolutionary algorithms (EAs) have been successfully applied to various multiobjective optimization problems (MOPs). However, current researches on EAs rarely consider uncertainty in the optimization process and existing algorithms often fail to handle the uncertainty, which have limited EAs’ applications in real-world problems. When MOPs come with uncertainty, they are referred to as robust MOPs (RMOPs). In this paper, we aim at solving RMOPs using EA-based optimization search. We propose a novel robust multiobjective optimization EA (RMOEA) with two distinct, yet complement, parts: 1) multiobjective optimization finding global Pareto optimal front ignoring disturbance at first and 2) robust optimization searching for the robust optimal front afterward. Furthermore, a comprehensive performance evaluation method is proposed to quantify the performance of RMOEA in solving RMOPs. Experimental results on a group of benchmark functions demonstrate the superiority of the proposed design in terms of both solutions’ quality under the disturbance and computational efficiency in solving RMOPs.

Ensemble prediction-based dynamic robust multi-objective optimization methods

Abstract Many real-world multi-objective optimization problems are subject to environmental changes over time, resulting in changing Pareto-optima. Wide studies on solving dynamic multi-objective optimization problems have so far concentrated on tracking moving Pareto-optima as soon as possible. In practice, however, a new solution every time the environmental change may be different from the previous optima, causing the expensive switching-cost. To this end, dynamic robust multi-objective optimization method is developed to find robust Pareto-optima over time whose performance is acceptable for the current and subsequently changed environments. With the purpose of measuring the robustness of a candidate, its fitness values in the subsequent environments are estimated by ensemble prediction methods constructed by moving average(MA), autoregressive(AR), and single exponential smoothing(SES). MA-, SES- and AR-based sub-prediction models are synthesized by the weight sum. The weights can be the pre-set constant or the binary/real number adjusted in terms of the prediction error. To examine the performance of the developed algorithm, the proposed prediction strategies are compared with three single prediction methods for 11 dynamic benchmark functions. The experimental results indicate that ensemble prediction methods have the better robustness than the single prediction models and can effectively tackle dynamic robust multi-objective optimization problems.

On approximate solutions for nonsmooth robust multiobjective optimization problems

ABSTRACTWe introduce a new concept of generalized convexity of ‘degree n’ for a multiobjective optimization problem and is compared it to the previous notions of generalized convex functions. Some examples to justify the importance of the term ‘degree n’ are provided. Namely, the conclusions of our results may fail if this term is dropped. By applying our new definition to nonsmooth robust multiobjective optimization problems, we establish the nonsmooth robust optimality conditions and robust duality theory for robust ϵ-quasi-(weakly) efficient solutions. A robust ϵ-Mond-Weir type duality of degree n for an uncertain multi-objective optimization problem under our generalized convexity assumption is presented. Furthermore, we introduce an ϵ-approximate scalar saddle-point and an ϵ-approximate weak vector saddle-point of degree n for the robust multi-objective optimization problem. The relationships between these two concepts with robust ϵ-approximate condition and robust ϵ-weakly efficient solutions are also given.

Robust Multiobjective Optimization With Robust Consensus

Consider a multiobjective robust optimization problem, where a set of weighted decision makers provides their preferences a priori . The preferences are provided either in the objective space or in the decision variable space using fuzzy numbers. To solve this problem, an indicator to measure consensus, an indicator to measure the robustness of the solutions to their degree of consensus, and a reformulation of the multiobjective robust optimization problem, are required. It is necessary for the reformulated problem to generate robust solutions that also enjoy high degree of consensus. In this paper, we have addressed these three issues. For this purpose, we have proposed two approaches to define consensus . Then, we have extended these approaches to define robust consensus , an indicator to measure the robustness of a given solution to its degree of consensus. Though these approaches can be used to define a countless number of measures, we have proposed 12 definitions of consensus, and hence, robust consensus. Furthermore, we have proposed two ways for the reformulation. Experimental results illustrate that the behavior of the proposed definitions and of the reformulations are consistent with our expectations.