Handling Equality Constraints in Robust Design Optimization

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Handling Equality Constraints in Robust Design
Optimization Christopher A. Mattson Achille Messac



Corresponding Author

Prof. Achille Messac
Mechanical, Aerospace, and Nuclear Engineering Department
110 8 th Street, JEC-2049 Troy, NY, 12180-3590
USA
Email: messac@rpi.edu
Tel: (518) 276-8145
Fax: (518) 276-6025
www.rpi.edu/~messac

Bibliographical Information Mattson, C. A., and Messac, A., “Handling Equality Constraints in Robust Design
Optimization,” 44 th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Paper No. AIAA 2003-1780, Norfolk, VA, April 07-10, 2003.

Handling Equality Constraints in Robust Design Optimization Christopher A. Mattson ? and Achille Messac † Rensselaer Polytechnic Institute, Troy, NY Robust design optimization (RDO) is a powerful tool for managing the tradeo?s be- tween optimal performance and performance stability. A robust design is one where system
performance remains relatively unchanged (stable) when exposed to stochastic conditions.
Although many advances in the area of RDO have been realized over the past decade,
most RDO methods do not address a critical branch of design optimization – namely,
equality constrained problems. In this paper, we o?er a new perspective on handling
equality constraints in RDO. As part of the development, we identify two types of equal-
ity constraints; those that must be strictly satis?ed – regardless of stochastic conditions,
and those that cannot be strictly satis?ed because of stochastic conditions. Each con- straint type is mathematically de?ned, which allows for the process of classifying any
given equality constraint. We also provide a generic means for handling both equality
constraint types in RDO. A simple structural optimization problem is used to illustrate
our approach. 1. Introduction and Literature Survey Uncertain design information can signi?cantly im- pact the success of the engineering design process.
Such uncertainty – which is always present – comes
in numerous forms, which include imprecise mate-
rial properties, imperfect manufacturing methods, un-
known loading conditions, and over simpli?ed engi-
neering models. When these uncertainties are not considered during the optimization of a design, the
obtained solution is likely to be more “high risk” than
“optimal”. 1 For this reason, properly handling un- certainty is a critical element of design optimization.
Robust design optimization (RDO) speci?cally seeks
to account for design uncertainties and to reduce their
negative e?ect. During the past two decades, RDO has become a signi?cant area of research in the design community.
The appeal of RDO is that its solution (a robust
design) remains relatively unchanged (stable) when
exposed to uncertain conditions. The need for RDO
originates from the fact that uncertain conditions are
always present. Although many advances in the area
of RDO have been realized, most RDO methods do
not address a critical branch of design optimization
– namely, equality constrained problems. In this pa-
per we provide a new perspective on handling equality
constraints in RDO. Literature Survey In the following paragraphs, we present a brief sur- vey of the RDO literature with a focus on how various
RDO methods deal with constraint satisfaction. In the ? Doctoral Candidate, AIAA student member † Associate Professor, messac@rpi.edu, (518)276-8145, AIAA Associate Fellow Copyright c 2003 by Achille Messac. Published by the Ameri- can Institute of Aeronautics and Astronautics, Inc., with permission. early 1980’s, the ?rst RDO approaches focused pri-
marily on inequality constraint satisfaction. Balling
et. al., 2 proposed a worst-case tolerance approach to robust design. They consider worst case tolerances and shift optimization constraints such that all pos-
sible combinations of worst case tolerances result in
a feasible design. Interestingly, these authors do not
address the equality constrained problem. Parkinson et al., 3 develop a method for robust de- sign optimization called feasibility robustness. The main elements of their approach are (i) to maintain
design feasibility given input variations, and (ii) to
minimize the e?ect of variations on the performance of
the system. A sensitivity model is used to account for
variations in the constraint functions; however equal-
ity constraints are not considered. Otto and Antonsson 4 expand Taguchi’s robust de- sign 5, 6 approach by including design constraints using constrained optimization methods. They directly ad-
dress RDO with inequality constraints, but do not
provide explicit details related to equality constrained
problems. Interestingly, they say that “equality con-
straints are not discussed [here], but are easily incor-
porated.” In contrast, Ramakrishnan and Rao indi-
cate that equality constraints are generally di?cult to
handle in RDO. 7 In their work, they start with a con- ventional (non-robust) optimization problem that in-
cludes both inequality and equality constraints. They
address the equality constrained problem by adding
new constraints that are obtained by relaxing the
equality constraints. According to the authors, this
is done to make the non-linear optimization problem
tractable and easier to solve. The past decade has seen continued successful e?orts to improve RDO approaches – but with little emphasis
on handling equality constrained problems. Similar to 1 of 10 American Institute of Aeronautics and Astronautics Paper 2003–1780 the early developments in robust design optimization,
numerous recent publications do not provide speci?c
details for handling equality constraints in RDO. 1, 8–16 Most RDO approaches start by converting a con- ventional (deterministic) design optimization (CDO)
problem into a RDO problem where objectives have
been added and/or constraints have been modi?ed to
account for uncertainties. In this paper, we too con-
sider the important task of CDO to RDO conversion.
We note that one of the greatest challenges of this
conversion is to ensure that we have not inadvertently
changed the problem we wish to solve during the con-
version process. It is our tenet that, although many
approaches do not attempt to do so, it is crucial that
during this conversion some of the equality constraints
be retained in their exact and strict form. Anything
short of this retention could be a signi?cant departure
from the original CDO problem being solved. We believe that as advances in RDO methods con- tinue to come forth, it is critical that problems with
equality constraints be adequately addressed. Any other approach should be considered incomplete and
potentially incorrect. A limited number of publications provide details for handling equality constraints in RDO problems.
In these publications, three distinct and con?icting
approaches for handling equality constraints can be
found. They are: (a) to relax the equality constraint,
(b) to satisfy the equality constraint in a probabilis-
tic sense, and (c) to remove the equality constraint
through substitution. Each is brie?y discussed. Su and Renaud, 17 Ramakrishanah and Rao, 7 Fares et al, 18 and Messac and Ismail-Yahaya, 19 all take an equality constraint relaxation approach. Su and Re-
naud 17 indicate that it is nearly impossible to satisfy equality constraints in RDO, and that in order to
solve such problems, equality constraints must ?rst be
relaxed. Ramakrishanah and Rao 7 minimize the vari- ation of the system performance using techniques of
stochastic non-linear programming. They use a Tay-
lor series expansion of the objective function about the
mean values of the design variables. This results in the
expected value of the objective function. Similarly,
the inequality and equality constraints are expanded
about the mean values of the design variables. Fares et. al, 18 convert a deterministic linear pro- gramming formulation with equality constraints into
another linear programming formulation with slack
variables that augment the equality constraint. Mes-
sac and Ismail-Yahaya 19 suggest that equality con- straints theoretically leave no room for the ?exibility
inherent in non-deterministic problems. Furthermore,
they suggest that some compromise must take place.
Each equality constraint is then converted into two in-
equality constraints. Others have handled equality constraints in RDO by satisfying the constraint in a probabilistic sense. Sundaresan, et al, 20 and Putko, et al. 21 use this ap- proach. Satisfaction of the equality constraint is only
enforced at the “target” design – or the expected value
of the design parameters. These authors do indicate,
however, that these constraints may be violated when
exposed to uncertainties. Sundaresan, et al. 20 de?ne ?ve classes of inequality constraint violation. Only one class is given for equality constraints – the vio-
lated class, since any variation in the design causes
the equality constraint to be violated. In all of the cases described above, it has been em- phasized that equality constraints cannot be satis?ed
under stochastic conditions. Das 22 supports a slightly di?erent idea. He suggests that there is a special type
of equality constraint that must be satis?ed; such as a
physics based equality constraint. To ensure satisfac-
tion of this special equality constraint, Das substitutes
the equality constraint back into the objective func-
tion, which converts the equality constrained RDO
problem into an unconstrained RDO problem. Observations from Survey We make the following observations from our sur- vey of the RDO literature. Observation 1: Most of
the literature does not address the handling of equal-
ity constraints in RDO problems. Observation 2: The
limited number of publications that do address equal-
ity constrained problems separately provide three dis-
tinct approaches for handling the equality constraint.
They are (a) to relax the equality constraint, (b) to sat-
isfy the equality constraint in a probabilistic sense, and
(c) to remove the equality constraint through substitu-
tion. Observation 3: These three basic approaches for
handling equality constraints in RDO are con?icting
in principle. Such con?icts in proposed methods for handling equality constraints lead to the research question ad-
dressed in this paper: What are the conditions that
govern the strictness to which equality constraints
should be enforced in RDO? In this paper, we seek
to answer this question by providing a new perspec-
tive on handling equality constraints in RDO. As part
of the development, we identify two types of equal-
ity constraints; those that must be strictly satis?ed
– regardless of stochastic conditions, and those that
cannot be strictly satis?ed because of stochastic condi-
tions. Each constraint type is mathematically de?ned,
which allows for the process of classifying any given
equality constraint. We also provide a generic means
for handling both equality constraint types in RDO. The remainder of this paper is presented as follows. In Section 2, we provide an analytical development
for solving the RDO problem – with explicit details
regarding the handling of equality constraints. In Sec-
tion 3, we make important observations about the
analytical development presented in Section 2, and
about RDO in general. Section 4 presents a simple 2 of 10 American Institute of Aeronautics and Astronautics Paper 2003–1780 structural optimization example that illustrates our
approach, and concluding remarks are given in Sec-
tion 5. 2. Analytical Development: Equality Constraint Handling in RDO In this section, we analytically develop an approach for handling equality constraints in robust design opti-
mization (RDO). As discussed in the introduction, we
seek to account for uncertainty during the optimiza-
tion process; otherwise, the ?nal design might be at
risk of failure when exposed to uncertainties. The analytical development presented in this section starts at the root objective of RDO, which is to opti-
mize a design while accounting for uncertainties. That
is, we wish to solve the following conventional (non-
robust) optimization problem when the parameters x and p are stochastic. Problem 1: Conventional Design Optimization (CDO) min x f(x, p) (1) subject to g k (x, p) ? 0 (k = 1, 2, ..., n g ) (2) h k (x, p) = 0 (k = 1, 2, ..., n h ) (3) x i min ? x i ? x i max (i = 1, 2, ..., n x ) (4) where x is a vector of design parameters that are ac-
tively changed during the optimization process, and p
is a vector of ?xed (constant valued) parameters. The
number of inequality constraints, equality constraints,
and design parameters are denoted as n g , n h , and n x , respectively. Numerous approaches have been developed that seek to solve this optimization problem; a few of them
are discussed in our literature survey. As part of this
development, we provide a new prospective on solv-
ing Problem 1 under stochastic conditions. We begin
the development by examining the objective function
and the inequality constraints and make appropri-
ate changes to them that account for uncertainty. A
framework for handling the equality constraints is then
developed. Our development proceeds based on the
following assumptions. Assumptions: 1. All x and p are independent parameters. 2. The stochastic natures of x and p are charac- terized by prescribed variations with rectangular
distributions of the form shown in Fig. 1. The
parameter x is along the horizontal axis and the
probability of x is on the vertical axis. The mean
value of x is denoted by ¯ x, and the maximum vari- ation thereof by the tilde. x x-x x+x x x x P(x) Fig. 1 Rectangular Probability Distribution 3. The maximum variations of x and p, denoted as ˜ x and ˜ p, are small valued (so as to allow for linear approximations). 4. The functions f (x, p), g(x, p), and h(x, p) are dif- ferentiable. 5. When accounting for the stochastic nature of x and p, we wish to optimize the original function
of Problem 1 and its variation. We note that the assumed rectangular probability distribution is only invoked for simplicity of presenta-
tion. The study of more realistic distributions would
be an important next step. Based on Assumptions 2 and 5, the objective func- tion (Eq. 1) can be transformed to the following equation when the stochastic natures of x and p are
considered in the optimization problem. min x J = f(x, p) + ? ˜ f(x, p, ˜x, ˜ p) (5) The tilde over the variable represents its variations,
and ? is a scalar weight. Based on Assumptions 1–
4, we can use a ?rst-order Taylor series expansion
about ¯ x and ¯ p to determine the variations transmit- ted from the stochastic parameters to the functions
f(x, p), g(x, p), and h(x, p). The ?rst-order Taylor se-
ries expansion leads to the following expression for the
variation of f (x, p). For notation simplicity, we de?ne ˜ f = ˜ f(x, p, ˜x, ˜ p). ˜ f = n x i=1 ?f(x, p) ?x i 2 ˜ x i + n p i=1 ?f(x, p) ?p i 2 ˜ p i (6) To ensure design feasibility under stochastic condi- tions, we can shift (make more stringent) the inequal-
ity constraints in Problem 1 so that given worst-case
variations, ˜ x and ˜ p, the constraint is still satis?ed. With this shift, Eq. 2 becomes g k (x, p) + ˜ g k (x, p, ˜ x, ˜ p) ? 0 (k = 1, 2, ..., n g ) (7) where ˜ g k (x, p, ˜ x, ˜ p) is found using a ?rst-order Taylor series expansion as discussed for the case of f (x, p)
above. When in the form of Eq. 6, we note that ˜ g k (x, p, ˜ x, ˜ p) is conservative. Likewise, to ensure de- sign feasibility we can shift the side constraints (Eq. 3 of 10 American Institute of Aeronautics and Astronautics Paper 2003–1780 4) away from the upper and lower limits, so that given
worst case variations, ˜ x and ˜ p, the constraint is still satis?ed. As such, Eq. 4 becomes x i min + ˜ x i ? x i ? x i max ? ˜x i (i = 1, 2, ..., n x ) (8) Recall that our main objective is to solve Problem 1 when x and p are stochastic. Equations 5–8 and the
related discussions show how Eqs. 1, 2, and 4 can be
handled when the stochastic natures of x and p are
included in the optimization problem. In the next sec-
tion, we develop a framework for handling the equality
constraint (Eq. 3) under stochastic conditions. Handling Equality Constraints Before we can determine how the equality con- straints of Eq. 3 should be handled when x and p are
stochastic, we must ?rst consider the notion that dif-
ferent types of equality constraints exist and that each
type should be handled accordingly when x and p are
stochastic. Figure 2 shows a taxonomy of optimization con- straints, as we de?ne it in this paper. It also shows
(on the right) the proposed approach for handling each
type of optimization constraint. In the previous sub-
section we have shown that the inequality constraints
are shifted away from deterministic constraint bound-
aries (made more stringent) in order to maintain fea-
sibility under stochastic conditions. The remainder of this development focuses on the equality constraint
branch of the taxonomy chart. We de?ne two types of equality constraints in the following. Each requires di?erent treatment when un-
certainties are introduced. Type 1 Equality Constraint: A Type 1 equality constraint is a constitutive equality relationship
that is strictly satis?ed – regardless of all stochas-
tic conditions. Type 1 equality constraints are
denoted as h T 1 . Type 2 Equality Constraint: A Type 2 equality constraint is an equality relationship that cannot
be strictly satis?ed due to stochastic conditions.
Type 2 equality constraints are denoted as h T 2 . To exemplify and further explore these two equality constraint types, let us consider the design of a three-
piece modular bridge shown spanning an open channel
in Fig. 3. The length of each modular section is de-
noted as x i for i = 1, 2, 3. The total length of the bridge is indicated as L and the width of the channel
is given as L c . Let us assume that the following equal- ity constraints are part of a deterministic optimization
bridge problem. L ? x 1 ? x 2 ? x 3 = 0 (9) ?L c ? x 1 ? x 2 ? x 3 = 0 (10) Optimization Constraints Inequality Equality Type 1 Type 2 Shift* Constraint Satisfy at Mean Relax+ Constraint Satisfy Constraint *Shift = To make more stringent
+Relax = To make less stringent Fig. 2 Optimization Constraint Taxonomy L x</i>2 x</i>1 x</i>3 Lc Fig. 3 Three Piece Modular Bridge where ? is a scalar parameter with a value greater
than 1. In this hypothetical example, Eq. 9 serves to
keep the modular sections together at the joints during
the optimization process, and Eq. 10 ensures that the
bridge is su?ciently long. Importantly, we note that
Eqs. 9 and 10 are fundamentally di?erent, and require
fundamentally di?erent approaches for satisfying them
under stochastic conditions. Each of these constraints
is further discussed in the following. Assuming that x i is stochastic and satis?es Assump- tions 1–3, and that L is not an independent parameter
(rather it is a function of x i ), we make an impor- tant observation about Eq. 9. Regardless of how the lengths x i vary, the total length of the bridge will always be L = x 1 + x 2 + x 3 . That is, Eq. 9 is al- ways satis?ed – regardless of all stochastic conditions.
Therefore by de?nition, Eq. 9 is a Type 1 equality
constraint. As a note, Type 1 equality constraints are
also often included in the optimization problem state-
ment to ensure that the optimal solution satis?es the
physical laws of nature. Examples of Type 1 equality
constraints include F = ma, and force and moment
equilibrium equations. If by de?nition, Type 1 equality constraints are strictly satis?ed regardless of the stochastic nature of
x and p, then one of two things must be true. Either
h T 1 is not a function of x or p, and therefore its vari- ation due to ˜ x and ˜ p is zero. Or, h T 1 is a function of x or p such that, whatever variations of x and p
exist, the net change in h T 1 is zero. In both cases the 4 of 10 American Institute of Aeronautics and Astronautics Paper 2003–1780 following property is true. ?h T 1 (x, p, ?x, ?p) = 0 (11) where ?h T 1 (x, p, ?x, ?p) is the actual variation of the constraint h T 1 when x and p are stochastic, and where ?x is any variation of x and ?p is any variation of p –
be it the maximum variations (˜ x, ˜ p) or not. For nota- tion simplicity, we de?ne ?h T 1 = ?h T 1 (x, p, ?x, ?p). The actual constraint variation can be written as ?h T 1 = h T 1 (x + ?x, p + ?p) ? h T 1 (x, p) (12) A key observation in the modular bridge example (Eq. 9) is that although x i are independent param- eters, L is NOT independent of x i . Mutual depen- dence between terms is another characteristic of Type
1 equality constraints. Now let us consider the equality constraint given in Eq. 10. Equation 10 is fundamentally di?erent than
Eq. 9 because there is no mutual dependence between
terms. The term ?L c is an independent parameter, which is also used in other relationships governing the
optimization problem. As such, the variations in x 1 are not guaranteed or even likely to be cancelled by
equal and opposite variations in x 2 or x 3 . As a result Eq. 10 cannot be strictly satis?ed when uncertainties
are included as part of the optimization. Therefore Eq.
10 is a Type 2 equality constraint. Mathematically,
Type 2 equality constraints are de?ned as having the
following property: ?h T 2 (x, p, ?x, ?p) = 0 (13) where ?h T 2 is the actual variation of the constraint h T 2 , and is of the same form given in Eq. 12. Given the existence of at least Type 1 and Type 2 equality constraints, we return to Problem 1 and our
main RDO objective, which was to solve the optimiza-
tion problem while including the e?ects of uncertainty.
If Eq. 3 of Problem 1 is a Type 1 equality constraint,
then it must be strictly satis?ed, therefore it stays
in its present form when uncertainties are considered.
That is, h T 1 k (x, p) = 0 (k = 1, 2, ..., n h T 1 ) (14) If Eq. 3 is a Type 2 equality constraint, then there are two possibilities for handling it under uncertainty;
(i) the constraint can be relaxed (made less stringent),
or (ii) the constraint can be satis?ed at the mean pa-
rameter values only. Type 2 equality constraints that
are handled using the former are denoted as ¯ h T 2r and referred to as Type 2r, and those using the latter are
denoted as ¯ h T 2m and referred to as Type 2m. The decision to model Type 2 equality constraints by re-
laxing or satisfying them at the mean should be made
after assessing why the given constraint is included
in the original problem. For example, if a market study showed that the ideal surface area of desk is
2 m 2 then perhaps it can be concluded that satisfying this constraint only that the mean parameter values is
su?cient – since small variations in the actual surface
area are not likely to negatively impact the success of
the product. When Eq. 3 is relaxed, it can be transformed into two inequality constraints of the following form h T 2r k (x, p) + |?h T 2r k (x, p, ?x, ?p) | ? 0 (15) h T 2r k (x, p) ? |?h T 2r k (x, p, ?x, ?p) | ? 0 (16) where k = 1, ..., n h T 2r . When Eq. 3 is to be satis?ed only at the mean value of the parameters, then it is
written as follows h T 2m k (x, p) = 0 (17) where x and p are the mean parameter values, and k =
1, ..., n h T 2m . Satisfying Type 2 equality constraints at the mean parameter values only does not imply that
?h T 2 (x, p, ?x, ?p) = 0. Rather it means that the designer declares ?h T 2 to be of little consequence. We note that the notion of satisfying the constraint only at the mean parameter values stems from the
work of Sundaresan, et al, 20 and Putko, et al., 21 which was the approach they used to handle equality con-
straints under uncertainty. With Eqs. 5–17 and the related discussion, we have shown how Problem 1 can be solved when the stochas-
tic nature of x and p are included in the optimization.
Below in Problem 2, the main equations from the
preceding discussion are rewritten as constituent com-
ponents of the RDO problem statement. Problem 2: Robust Design Optimization (RDO) min x J = f(x, p) + ? ˜ f(x, p, ˜x, ˜ p) (18) subject to g k (x, p) + ˜ g k (x, p, ˜ x, ˜ p) ? 0 (k = 1, 2, ..., n g ) (19) h T 1 k (x, p) = 0 (k = 1, 2, ..., n h T 1 ) (20) h T 2m k (x, p) = 0 (k = 1, 2, ..., n h T 2m ) (21) h T 2r k (x, p) + |?h T 2r k (x, p, ?x, ?p) | ? 0 (22) h T 2r k (x, p) ? |?h T 2r k (x, p, ?x, ?p) | ? 0 (23) x i min + ˜ x i ? x i ? x i max ? ˜x i (i = 1, 2, ..., n x ) (24) where k = 1, ..., n h T 2r for Eqs. 22 and 23, and where x is a vector of design parameters that are actively
changed during the optimization process, and p is a
vector of ?xed (but stochastic) parameters. The vari-
able n ( ) represents the number of ( ). 5 of 10 American Institute of Aeronautics and Astronautics Paper 2003–1780 3. Observations and Discussion In this section, we make important observations about the analytical development presented in Sec-
tion 2, and comment on pertinent aspects of robust
design optimization (RDO). This section is divided
into three small subsections, which discuss the fol-
lowing topics. (1) A single approach for handling both Type 1 and Type 2r equality constraints. (2)
Prescribed versus non-prescribed variations, and (3)
formulating the aggregate objective function in RDO. A Single Approach for Handling Equalities We make an interesting observation regarding the analytical development presented in Section 2. Be-
cause Type 1 equality constraints are such that
?h T 1 (x, p, ?x, ?p) = 0, a single computational ap- proach can be used to handle both Type 1 and Type 2r
equality constraints under stochastic conditions. That
is, for Type 1 and Type 2r equality constraints, h k (x, p) + |?h k (x, p, ?x, ?p) | ? 0 (25) h k (x, p) ? |?h k (x, p, ?x, ?p) | ? 0 (26) for (k = 1, 2, ..., n h T 1 + n h T 2r ). Given that the varia- tion of h T 1 is zero (?h T 1 (x, p, ?x, ?p) = 0), Eqs. 25 and 26 collapse to the following for Type 1 equality
constraints. h T 1 k (x, p) = 0 (k = 1, 2, ..., n h T 1 ) (27) The collapsing of Eqs. 25 and 26 to Eq. 27 for Type 1 equality constraints makes this single approach
equivalent to the approach developed in Section 2.
Using the approach discussed in the current section
eliminates the need to identify the constraint type,
which may be particularly useful for some complex
problems. Even so, we advocate identifying the equal-
ity constraint type and treating each type as discussed
in the analytical development of Section 2. Doing so allows the designer to better understand the opti-
mization problem being solved, and reduces unneeded
function evaluations. Prescribed versus Non-Prescribed Variations In the analytical development of Section 2, we as- sumed that the stochastic natures of x and p where
characterized by prescribed variations. Although pre-
scribed variations represent an important and practi-
cal branch of RDO problems, we note that under many
other practical circumstances, the variations are not
prescribed; further, the designer is often required to
specify them. In such cases, optimal variations for the
parameters are sought during the optimization pro-
cess. The important related point is that allowable
variation levels signi?cantly impact cost. We now restate Problem 2 for non-prescribed vari- ations in the parameters. For notation simplicity, we
let f = f (x, p) and ˜ f = ˜ f(x, p, ˜x, ˜ p). Problem 3: RDO with Non-Prescribed Variations min x,˜ x,˜ p J = f + ? ˜ f + ?(x, p, ˜x, ˜ p) + ?˜h T 2r (28) subject to Eqs. 19–24. When comparing Problem 3
to Problem 2, it can be seen that the third and fourth
terms of the aggregate objective function have been
added and that the optimization is now over x, ˜ x, and ˜ p. The third term of Eq. 28 is a function that accounts for the cost of decreasing parameter varia-
tion (or tightening manufacturing tolerances). The fourth term is critical in that it keeps Type 2r equality
constraints from being excessively relaxed. The vari-
able ? is a scalar weight. Messac and Ismail-Yahaya 19 o?er more on the topic of non-prescribed variations
in RDO, including the combination of prescribed and
non-prescribed variations. The RDO Aggregate Objective Function We now comment on the important task of for- mulating the RDO aggregate objective function for
multiobjective optimization. For simplicity we discuss
the prescribed variations case, although the same basic
principle applies to the case with non-prescribed vari-
ations. A deterministic multiobjective optimization problem of n m metrics may result in a RDO problem with 2 ×n m metrics – one metric for each of the original metrics, and one for each of the variations thereof. Whatever approach is used to formulate the aggre- gate objective function for the 2 × n m metrics, it is important that no particular set of metrics becomes
prematurely combined into small sub-objectives such
as “minimize variation”, which is one approach typi-
cally found in the literature. Importantly, the designer
should formulate the aggregate objective function so
that he or she can have adequate control over changing
the degree to which each individual metric (includ-
ing individual variations) should be optimized. The
premise for this is that one may wish to signi?cantly
minimize the variation of say metric 1, while at the
same time have little desire to minimize the variation
of metric 2. The approach used in Eq. 5 in fact suf-
fers from this de?ciency. In addition, the preference
embodied in the original aggregate objective function
does not necessarily translate to preference for the
variations of the objectives. For example, one may
wish to signi?cantly minimize say objective 7, with
little desire to minimize its variation. This issue is
addressed by Messac and Ismail-Yahaya, 19 where a physical programming based approach to formulate
the RDO aggregate objective function is developed. 4. A Simple Structural Example In this section, a simple two-bar truss is used to il- lustrate the developments presented in this paper. We
present the example by ?rst providing two equivalent
deterministic optimization problem statements for the
truss – one with equality constraints, and one without. 6 of 10 American Institute of Aeronautics and Astronautics Paper 2003–1780 2<i>L b W1 W2 P L a 1 , E a 2 , E Fig. 4 Two Bar Truss Schematic The two formulations indeed show (i) that Type 1 equality constraints do not need to be relaxed in the
robust design optimization (RDO) formulation, and
(ii) that the basic approach for handling the conver-
sion from the conventional (non-robust) to the RDO
formulation works well. Two Equivalent Deterministic Formulations for
the Truss Optimization Problem In this section, we provide two equivalent optimiza- tion formulations for optimizing the truss shown in
Figure 4. Note that the equivalent formulations are
for the deterministic case, where uncertainties are not
considered. As part of the optimization problem we
seek to minimize the squared nodal de?ection at point
P and minimize the total structural volume, subject to
the normal stress and beam cross-sectional areas being
within acceptable limits. The structure is exposed to
constant horizontal and vertical loads as indicated in
the ?gure. The following detailed truss information is
used throughout the entire example, unless otherwise
noted. We will subsequently refer to the following in-
formation as the “detailed truss information”. Design Parameters
a 1 cross-sectional area of bar 1 (m 2 ) a 2 cross-sectional area of bar 2 (m 2 ) b horizontal distance from the left most
part of the truss to node P (m) Constant Parameters
L height of structure; 18.288 m W 1 horizontal load; 4.45 × 10 5 N W 2 vertical load; 4.45 × 10 6 N E modulus of elasticity; 1.99 × 10 11 Pa S max maximum allowable stress; 3.79 × 10 9 Pa a i min lower bound for a i ; 5.16 × 10 ?4 m 2 (i=1,2) a i max upper bound for a i ; 1.94 × 10 ?3 m 2 (i=1,2) b min lower bound for b; 9.144 m b max upper bound for b; 27.432 m w 1 scalar weight; 0.3 w 2 scalar weight; 0.7 Design Metrics f 1 squared nodal displacement at node P (m 2 ) f 2 structural volume (m 3 ) We now state the ?rst truss optimization formula- tion, which is for the case where there are no equality
constraints. Although we generally do not recommend
the use of weighted sum aggregate objective functions,
we have chosen to use one for this particular case –
where it is already known that the Pareto frontier is
convex. Truss Formulation 1: No Equality Constraints min a,b J = w 1 f 1 + w 2 f 2 (29) subject to S i ? S max (i = 1, 2) (30) a i min ? a i ? a i max (i = 1, 2) (31) b min ? b ? b max (32) where the normal stress in each bar is denoted as S i . Note that Truss Formulation 1 has no equality con-
straints. We now consider the second truss optimization for- mulation. For this formulation, the information given
as “detailed truss information” is used with the excep-
tion of the following additional information. Additional Parameters for Truss Formulation 2
u 1 horizontal de?ection at node P (m) u 2 vertical de?ection at node P (m) ? acute angle between the horizontal and bar 1 ? acute angle between the horizontal and bar 2 Importantly, we note that the additional parameters
are not independent. That is, each is a function of the
original design parameters. We will show shortly the
e?ect of such dependencies. Truss Formulation 2: With Equality Constraints min a,b,u,?,? J = w 1 f 1 + w 2 f 2 (33) subject to S i ? S max (i = 1, 2) (34) a i min ? a i ? a i max (i = 1, 2) (35) b min ? b ? b max (36) W 1 = F 1 cos (?) ? F 2 cos (?) (37) W 2 = F 1 sin (?) + F 2 sin (?) (38) ? = arctan (L/b) (39) ? = arctan (L/(2L ? b)) (40) where the normal stress in each bar is denoted as S i , and F i is the normal forces acting in bar i for i = 1, 2. 7 of 10 American Institute of Aeronautics and Astronautics Paper 2003–1780 Table 1 Results for Deterministic Case Parameter Formulation 1 Formulation 2 J 0.0800 0.0800 f 1 0.0889 0.0889 f 2 0.0762 0.0762 w 1 0.3000 0.7000 w 2 0.3000 0.7000 Equations 37 and 38 are force equilibrium constraints
that keep the structure in static equilibrium. Equa-
tions 39 and 40 keep the structure connected a node P . Both Truss Formulation 1 and Truss Formulation 2 yield the same optimal solution. The results for this
deterministic case are shown in Table 1 and in Fig. 5.
In the ?gure the optimal solution for both cases is
shown as a triangle, and the Pareto frontier is shown
as a solid curve. We note that the di?erence between
Truss Formulation 1 and Truss Formulation 2 is that
Eqs. 37–40 have been substituted into the objective
function of Truss Formulation 1 (Eq. 29). Optimizing the Truss under Stochastic Conditions Recall that our main objective is to solve the opti- mization problem under stochastic conditions. There-
fore, we use the developments presented in this paper
to restate the truss formulations while accounting for
uncertainties. As part of the RDO problem we wish to
minimize the squared nodal de?ection at node P , the
structural volume, the variation of the de?ection, and
the variation of the volume, subject to the constraints
of the original problem. To avoid a cumbersome example, we will present de- tails on solving Truss Formulation 2 (which is the truss
formulation with equality constraints) under stochas-
tic conditions, while we simply use the solution to the
RDO problem stemming from Truss Formulation 1 as a
reference. The detailed truss information given above
is used again here with the following added informa-
tion. Additional Parameters for RDO Truss Formulation 2
˜ a 1 variation of a 1 ; 1.00 × 10 ?4 (m 2 ) ˜ a 2 variation of a 2 ; 1.00 × 10 ?4 (m 2 ) ˜ b variation of b; 1.00 (m) w 3 scalar weight; 4 w 4 scalar weight; 181 Additional Metrics for RDO Truss Formulation 2 ˜ f 1 variation of squared nodal
displacement at node P (m 2 ) ˜ f 2 variation of structural volume (m 3 ) Revisiting the taxonomy chart given in Fig. 2, we can see that the inequality constraints in Eqs. 34–36
can be shifted so that given worst case variation, the
constraints will still be satis?ed. With such a shifting,
the inequality constraints are written as shown below
in Eqs. 42–44. 0 0.1 0.2 0.3 0.4 0.5 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 Squared Nodal Displacement, m 2 Structural Volume, m 3 Deterministic Optimum
RDO Formulation 1
RDO Formulation 2
Deterministic Pareto Frontier Fig. 5 Optimal Solutions for Truss Example Before being able to properly handle the equality constraints (Eqs. 37–40) under stochastic conditions,
we need to identify which type of equality constraint
they are. Using the mathematical de?nitions of Type
1 and Type 2 equality constraints, as given in Section
2, we can see that ?h = 0 for each equality constraint
in this problem. Therefore all four of these constraints
are Type 1 equality constraints. Based on the develop-
ments provided in Section 2, these constraints are to
be strictly satis?ed in the RDO problem. Equations
45–48 shown that these Type 1 equality constraints
are kept in their exact form when uncertainties are
considered. We note that this particular example does
not have Type 2 equality constraints. Had there been
Type 2 equality constraints, we would have chosen to
either relax those equality constraints, or satisfy them
only at the mean parameter values as discussed in Sec-
tion 2. The optimization problem statement for Truss For- mulation 2 is now given as follows when uncertainties
are considered during the optimization process. RDO Truss Formulation 2: With Equality Constraints min a,b,u,?,? J = w 1 f 1 + w 2 f 2 + w 3 ˜ f 1 + w 4 ˜ f 2 (41) subject to S i + ˜ S i ? S max (i = 1, 2) (42) a i min + ˜ a i ? a i ? a i max ? ˜a i (i = 1, 2) (43) b min + ˜ b ? b ? b max ? ˜b (44) W 1 = F 1 cos(?) ? F 2 cos(?) (45) W 2 = F 1 sin(?) + F 2 sin(?) (46) ? = arctan(L/b) (47) ? = arctan(L/(2L ? b)) (48) where the normal stress in each bar is given as S i and F i is the normal force applied to the ith bar. 8 of 10 American Institute of Aeronautics and Astronautics Paper 2003–1780 Table 2 Results for Non-Deterministic Case Parameter RDO Formul. 1 RDO Formul. 2 J 0.7929 0.7929 f 1 0.0973 0.0973 f 2 0.0950 0.0950 ˜ f 1 0.0077 0.0077 ˜ f 2 0.0037 0.0037 w 1 0.3000 0.3000 w 2 0.7000 0.7000 w 3 4.0000 4.0000 w 4 181.00 181.00 The results for this case are shown in Table 2 and compared to the results obtained from the RDO for-
mulation stemming from Truss Formulation 1. The
results for de?ection and volume are also shown in
Fig. 5. Note that the solutions for RDO Truss Formu-
lation 1 (circle) and RDO Truss Formulation 2 (star)
are identical, which is further indication that certain
equality constraints should be strictly satis?ed regard-
less of all stochastic conditions. We note that the obtained solutions are expected in that the overall performance for the de?ection and
volume has decreased from the deterministic to the
non-deterministic problem. This is because minimiz-
ing the variation of de?ection and volume comes at a
cost of reduced performance. We also note that the
dependencies in the equality constraints for Truss For-
mulation 2 (only L, b, W 1 , and W 2 are independent) are a good indication that the constraints are Type 1
equality constraints. 5. Concluding Remarks In this paper, we have presented an approach to solve the equality constrained design optimization
problem under stochastic conditions. A taxonomy of optimization constraint types is developed, and an
approach for handling each constraint under stochas-
tic conditions is presented. As part of the develop-
ment, we identify two types of equality constraints;
those that must be strictly satis?ed – regardless of all
stochastic conditions, and those that cannot be strictly
satis?ed because of stochastic conditions. Mathemat-
ical de?nitions for each equality constraint type are
provided, thus allowing for any given equality con-
straint to be classi?ed. A simple structural optimiza-
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