Searching Feasible Design Space by Solving Quantified Constraint Satisfaction Problems

Abstract

In complex systems design, multidisciplinary constraints are imposed by stakeholders. Engineers need to search feasible design space for a given problem before searching for the optimum design solution. Searching feasible design space can be modeled as a constraint satisfaction problem (CSP). By introducing logical quantifiers, CSP is extended to quantified constraint satisfaction problem (QCSP) so that more semantics and design intent can be captured. This paper presents a new approach to formulate searching design problems as QCSPs in a continuous design space based on generalized interval, and to numerically solve them for feasible solution sets, where the lower and upper bounds of design variables are specified. The approach includes two major components. One is a semantic analysis which evaluates the logic relationship of variables in generalized interval constraints based on Kaucher arithmetic, and the other is a branch-and-prune algorithm that takes advantage of the logic interpretation. The new approach is generic and can be applied to the case when variables occur multiple times, which is not available in other QCSP solving methods. A hybrid stratified Monte Carlo method that combines interval arithmetic with Monte Carlo sampling is also developed to verify the correctness of the QCSP solution sets obtained by the branch-and-prune algorithm.

A Constraint Satisfaction Algorithm for the Generalized Inverse Phase Stability Problem

“…Subsequently, this work was extended to more general set-based representations [34]. Hu et al proposed a method that uses generalized interval to solve for the feasible set [35,36]. The NUMER-ICA [37] modeling language in particular guarantees correctness, completeness, and certainty.…”

Exploration of the High Entropy Alloy Space as a Constraint Satisfaction Problem

“…Most techniques developed to solve CCSPs are based on interval arithmetic, branch and bound, or the root inclusion test. However, often these techniques require an analytical expression to determine if a subregion of the search space contains a feasible solution [40]. Since the phase-stability space is nonanalytical-phase boundaries represent abrupt transitions between the presence and absence of specific phases-these methods cannot be used for the solution to the IPSP.…”

Sensitivity Analysis in Quantified Interval Constraint Satisfaction Problems

“…More information of how to formulate QCSPs from a given design problem and how to solving QCSPs can be found in Ref. [30].…”

“…QCSP is a generalization of CSP, and CSP is a special case of QCSP where all variables are associated with 9. In the QCSP formulation of feasible design problems, design intent of controllability, materials properties, process sequences, and others can be captured by assigning appropriate quantifiers to variables so that logic interpretations of quantified constraints become available [30]. The interval-based SA approach can also be applied to the compromise programming (CP) formulation [31] in set-based design [32,33], as well as its extensions (e.g., Refs.…”