The Smart Normal Constraint Method for Directly Generating a Smart Pareto Set

Abstract

In design situations where a single solution must be selected, it is often desirable to present the designer with a smart Pareto set of solutions-a minimal set of nondominated solutions that sufficiently represents the tradeoff characteristics of the design space.These sets are generally created by finding many well-distributed solutions and then either filtering out the excess ones or searching more closely in those regions that appear to have significant tradeoff. Such methods suffer from the inherent inefficiency of creating numerous solutions that will never be presented to the designer. This paper introduces the Smart Normal Constraint (SNC) method-a Pareto set generation method capable of directly generating a smart Pareto set. Direct generation is achieved by iteratively updating an approximation of the design space geometry and searching only in those regions capable of yielding new smart Pareto solutions. This process is made possible through the use of a new, computationally benign calculation for identifying regions of high tradeoff in a design space. Examples are provided that show the SNC method performing significantly more efficiently than the predominant existing method for generating smart Pareto sets.

Multiobjective and multi-physics topology optimization using an updated smart normal constraint bi-directional evolutionary structural optimization method

“…The authors showed that this improved on the NNC method's ability to generate a more uniformly distributed Pareto frontier. This idea was recently taken further by Hancock and Mattson (2013), developing the smart normal constraint (SNC) method. Additional linear constraints, known as smart constraints (Haddock et al (2008)), are used to determine which approximate point is most likely to produce a smart Pareto solution.…”

“…The authors show that the SNC method alleviates the need for a Pareto filter to generate smart Pareto sets. This improves the computational expense of the algorithm, since the Pareto filter first generates many solutions, then reduces the set by removing solutions that are considered insignificantly different from other Pareto points (Hancock and Mattson (2013)). Such an approach is computationally inefficient as a large number of designs, which are later removed from consideration, are generated.…”

“…Such an approach is computationally inefficient as a large number of designs, which are later removed from consideration, are generated. Hancock and Mattson (2013) apply the SNC method to three numerical examples from the literature: the TNK problem (Tanaka et al (1995)), a gear box design (Huang et al (2006)) and the WATER problem (Ray et al (2001)). While these examples do demonstrate the effectiveness and efficiency of the SNC method in generating smart Pareto sets, they all have objective functions that depend only on a small number of design variables (2, 7 and 3, respectively).…”

“…It is also unable to generate an evenly distributed Pareto front or guarantee a smart Pareto set will be obtained. Therefore, the multiobjective algorithm adopted for this work is a modified SNC method, which has been shown to capture the entire design domain ), produce evenly distributed Pareto solutions (Martinez et al (2007)) and efficiently obtain a smart Pareto set (Hancock and Mattson (2013)).…”

A cognitive approach for the multi-objective optimization of RC structural problems

“…In this context, the use of Pareto set does not require a previous articulation of preferences. Many methods have been created with the intent of abbreviating the set of Pareto solutions [13]. The ''knee method'' [14] is a posteriori method that identifies the points for which an improvement in one objective results in a significant worsening of at least one other objective.…”