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The challenge of equality constraints in robust design optimization: examination and new approach

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Abstract

In recent years, robust design optimization (RDO) has emerged as a significant area of research. The focus of RDO is to obtain a design that minimizes the effects of uncertainty on product reliability and performance. The effectiveness of the resulting solution in RDO highly depends on how the objective function and the constraints are formulated to account for uncertainties. Inequality constraint and objective function formulations under uncertainty have been studied extensively in the literature. However, the approaches for formulating equality constraints in the RDO literature are in a state of disharmony. Moreover, we observe that these approaches are generally applicable only to certain special cases of equality constraints. There is a need for a systematic approach for handling equality constraints in RDO, which is the motivation for this research. In this paper, we examine critical issues pertinent to formulating equality constraints in RDO. Equality constraints in RDO can be classified as belonging to two classes: (1) those that cannot be satisfied, because of the uncertainty inherently present in the RDO problem, and (2) those that must be satisfied, regardless of the uncertainty present in the problem. In this paper, we propose a linearization- based approach to classify equality constraints into the above two classes, and propose respective formulation methods. The theoretical developments presented in this paper are illustrated with the help of two numerical examples.

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Abbreviations

A :

Matrix of linearized equality constraints (25)

A d :

Block columns of matrix A related to the dependent variables (19)

A ̅d :

Block columns of matrix A related to the independent variables (19)

C :

Cost function (10)

Cov(-,-):

Covariance between two variables (8)

g :

Vector of behavioral inequality constraints (2)

h :

Vector of behavioral equality constraints (3)

J :

Objective function (1)

n d :

Number of dependent variables (20)

n ̅d :

Number of independent variables (21)

n e :

Number of equality constraints (3)

n q :

Number of behavioral inequality constraints (2)

n ̅s :

Number of Type S equality constraints (35)

n x :

Number of design variables (4)

N :

Total number of Monte Carlo simulation cycles (51)

N s :

Number of Monte Carlo simulations for which the equality constraint lies within a tolerance (51)

P cs :

Probability of equality constraint satisfaction within a tolerance (51)

Var(X):

Variance of the random variable X (8)

x :

Vector of design variables (5)

x max :

Vector of upper bounds of x (4)

x min :

Vector of lower bounds of x (4)

X :

Vector of random design variables (5)

X d :

Vector of dependent design variables (14)

X ̅d :

Vector of independent design variables (14)

α k :

Shift in the k-th inequality constraint (5)

δ μ̅s :

Parameter for the mean of Type ̅S constraint (32)

δ σ̅s :

Parameter for the standard deviation of Type ̅S constraint (32)

△ μ̅s :

Approximate moment matching function for the mean values (32)

△ σ̅s :

Approximate moment matching function for the standard deviation values (32)

μ ( ) :

Mean of ( ) (5)

σ ( ) :

Standard deviation of ( ) (5)

σ min :

Minimum acceptable standard deviation for X (13)

σ max :

Maximum acceptable standard deviation for X (13)

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Authors and Affiliations

  1. Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Sirisha Rangavajhala, Anoop Mullur & Achille Messac

Authors
  1. Sirisha Rangavajhala
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  2. Anoop Mullur
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  3. Achille Messac
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Correspondence to Achille Messac.

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Rangavajhala, S., Mullur, A. & Messac, A. The challenge of equality constraints in robust design optimization: examination and new approach. Struct Multidisc Optim 34, 381–401 (2007). https://doi.org/10.1007/s00158-007-0104-8

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  • DOI: https://doi.org/10.1007/s00158-007-0104-8

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