Multiobjective Optimization Algorithm Benchmarking and Design Under Parameter Uncertainty
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This research aims to improve our understanding of multiobjective optimization, by comparing the performance of five multiobjective optimization algorithms, and by proposing a new formulation to consider input uncertainty in multiobjective optimization problems. Four deterministic multiobjective optimization algorithms and one probabilistic algorithm were compared: the Weighted Sum, the Adaptive Weighted Sum, the Normal Constraint, the Normal Boundary Intersection methods, and the Nondominated Sorting Genetic Algorithm-II (NSGA-II). The algorithms were compared using six test problems, which included a wide range of optimization problem types (bounded vs. unbounded, constrained vs. unconstrained). Performance metrics used for quantitative comparison were the total run (CPU) time, number of function evaluations, variance in solution distribution, and numbers of dominated and non-optimal solutions. Graphical representations of the resulting Pareto fronts were also presented. No single method outperformed the others for all performance metrics, and the two different classes of algorithms were effective for different types of problems. NSGA-II did not effectively solve problems involving unbounded design variables or equality constraints. On the other hand, the deterministic algorithms could not solve a problem with a non-continuous objective function. In the second phase of this research, design under uncertainty was considered in multiobjective optimization. The effects of input uncertainty on a Pareto front were quantitatively investigated by developing a multiobjective robust optimization framework. Two possible effects on a Pareto front were identified: a shift away from the Utopia point, and a shrinking of the Pareto curve. A set of Pareto fronts were obtained in which the optimum solutions have different levels of insensitivity or robustness. Four test problems were used to examine the Pareto front change. Increasing the insensitivity requirement of the objective function with regard to input variations moved the Pareto front away from the Utopia point or reduced the length of the Pareto front. These changes were quantified, and the effects of changing robustness requirements were discussed. The approach would provide designers with not only the choice of optimal solutions on a Pareto front in traditional multiobjective optimization, but also an additional choice of a suitable Pareto front according to the acceptable level of performance variation.