Probabilistic Dominance in Robust Multi-Objective Optimization

Abstract

Real-world problems typically require the simultaneous optimization of several, often conflicting objectives. Many of these multi-objective optimization problems are characterized by wide ranges of uncertainties in their decision variables or objective functions, which further increases the complexity of optimization. To cope with such uncertainties, robust optimization is widely studied aiming to distinguish candidate solutions with uncertain objectives specified by confidence intervals, probability distributions or sampled data. However, existing techniques mostly either fail to consider the actual distributions or assume uncertainty as instances of uniform or Gaussian distributions. This paper introduces an empirical approach that enables an efficient comparison of candidate solutions with uncertain objectives that can follow arbitrary distributions. Given two candidate solutions under comparison, this operator calculates the probability that one solution dominates the other in terms of each uncertain objective. It can substitute for the standard comparison operator of existing optimization techniques such as evolutionary algorithms to enable discovering robust solutions to problems with multiple uncertain objectives. This paper also proposes to incorporate various uncertainties in well-known multi-objective problems to provide a benchmark for evaluating uncertainty-aware optimization techniques. The proposed comparison operator and benchmark suite are integrated into an existing optimization tool that features a selection of multi-objective optimization problems and algorithms. Experiments show that in comparison with existing techniques, the proposed approach achieves higher optimization quality at lower overheads.

Non‐parametric measure approximations for constrained multi‐objective optimisation under uncertainty

“…Note that this formulation naturally tackles the existence of dependency between the random variables $$$ \mathbf{A} $$$ and $$$ \mathbf{B} $$$, which greatly differs from the assumptions made in References 18,37.…”

“…Gaussian variable), the dominance probability cannot reach exactly 1. It has been proposed in Reference37 to relax this threshold. The $$$ \epsilon $$$‐relaxed probabilistic constrained Pareto dominance finally reads: $$$$ {\displaystyle \begin{array}{ll}\hfill \mathrm{Domination}:\kern0.3em & \mathbf{A}\underset{\epsilon \kern0.3em }{\succ_c}\mathbf{B}\Longleftrightarrow {\mathbb{P}}_{\mathbf{A},\mathbf{B}}\left[\mathbf{A}{\succ}_c\mathbf{B}\right]\ge 1-\epsilon, \\ {}\hfill \mathrm{Strict}\ \mathrm{domination}:\kern0.3em & \mathbf{A}\underset{\epsilon \kern0.3em }{\succ {\succ}_c}\mathbf{B}\Longleftrightarrow {\mathbb{P}}_{\mathbf{A},\mathbf{B}}\left[\mathbf{A}\succ {\succ}_c\mathbf{B}\right]\ge 1-\epsilon, \\ {}\hfill \mathrm{Indifference}:\kern0.3em & \mathbf{A}\underset{\epsilon \kern0.3em }{\sim_c}\mathbf{B}\Longleftrightarrow \mathbf{A}\underset{\epsilon \kern0.3em }{\nsucc_c}\mathbf{B}\kern0.3em \mathrm{and}\kern0.3em \mathbf{B}\underset{\epsilon \kern0.3em }{\nsucc_c}\mathbf{A}.\end{array}} $$$$ …”

“…These approximated measures are treated as a random vector and are compared and ranked using the concept of probabilistic Pareto dominance. 18,36,37 The approach proposed in this article for drawing joint realisation is to be contrasted with other works concerning the construction of Sparse GP. [38][39][40] This article aims to employ sparse approximations specifically for estimating model error estimation of statistical measures from an OUU perspective.…”

“…Interval and worst-case approaches are tackled in [22,23,24], within the classical NSGA-II and MOEA frameworks. Recent developments [25,26,27,28] extends these computations to non-parametric noise distributions through the construction of histogram-based approximate densities, which permits fast computations of the probability of dominance. A straightforward extension of the Pareto dominance rule to interval uncertainty in the presence of constraints is proposed in [29] under the name of Bounding-Boxes.…”

Uncertainty-Aware Compositional System-Level Reliability Analysis

“…However, extending this approach to consider diversely distributed uncertain objectives requires solving difficult integrals demanding a huge computational effort. To this end, approximate simulation-based approaches, e. g., in [30], provide trade-offs between execution time and accuracy of calculating this probability. In [33], it is proposed to compare uncertain objectives with respect to their lower and upper bounds.…”