Monotonicity and Active Set Strategies in Probabilistic Design Optimization

Abstract

Probabilistic design optimization addresses the presence of uncertainty in design problems. Extensive studies on ReliabilityBased Design Optimization (RBDO), i.e., problems with random variables and probabilistic constraints, have focused on improving computational efficiency of estimating values for the probabilistic functions. In the presence of many probabilistic inequality constraints, computational costs can be reduced if probabilistic values are computed only for constraints that are known to be active or likely active. This article presents an extension of monotonicity analysis concepts from deterministic problems to probabilistic ones, based on the fact that several probability metrics are monotonic transformations. These concepts can be used to construct active set strategies that reduce the computational cost associated with handling inequality constraints, similarly to the deterministic case. Such a strategy is presented as part of a sequential linear programming algorithm along with a numerical example.

A method for reliability-based optimization with multiple non-normal stochastic parameters: a simplified airshed management study

“…2, we begin with the sequential linear programming (SLP) algorithm for RBDO developed in Chan et al (2006Chan et al ( , 2007. Since the objective function is deterministic, the real challenge of the algorithm lies in the method of calculating the constraints.…”

“…As discussed in Fletcher and Leyffer (2002), similar structure and convergence arguments could be used in creating a filter-based SQP algorithm. In Chan et al (2006), we showed that active set strategies can be integrated with the filter-SQP algorithm to improve the efficiency of the algorithm by only calculating constraints that are likely to be active in the next iteration. For problems with a large number of constraints or computational expensive constraints (such as the probability constraints in RBDO), the active set based algorithm significantly improves the computational efficiency.…”

“…Here we propose using extensions of the RBDO methodology developed in Chan et al (2006Chan et al ( , 2007 to study environmental policy design problems. This requires using realistic probability distributions that have been developed to analyze pollutant emissions and transport.…”

“…The main contribution of this paper is that the filterbased SQP algorithm developed for RBDO problems in Chan et al (2007) and the active set strategies proposed in Chan et al (2006) are extended to handle a primary challenge in environmental decision-making, namely, handling non-normal and discrete random parameters. A further issue is the joint reliability of constraint satisfaction.…”

“…Section 2 solves Eq. 2 under discrete random parameters, non-normal random parameters, and joint constraint reliability building from the approach detailed in Chan et al (2006). The application of the extended RBDO approach to the simplified airshed management problem of Fig.…”