A Note on the Convergence of
Analytical Target Cascading With
Infinite Norms
Jeongwoo Han
Argonne National Laboratory,
9700 S. Cass Avenue, Building 362,
Argonne, IL 60439
e-mail: jhan@anl.gov
Panos Y. Papalambros
Department of Mechanical Engineering,
University of Michigan,
2250 GG Brown Building,
Ann Arbor, MI 48104
e-mail: pyp@umich.edu
Analytical target cascading (ATC) is a multidisciplinary design
optimization method for multilevel hierarchical systems. To im-
prove computational efficiency, especially for problems under un-
certainty or with strong monotonicity, a sequential linear pro-
gramming (SLP) algorithm was previously employed as an
alternate coordination strategy to solve ATC and probabilistic
ATC problems. The SLP implementation utilizes L
norms to
maintain the linearity of SLP subsequences. This note offers a
proof that there exists a set of weights such that the ATC algo-
rithm converges when L
norms are used. Examples are also pro-
vided to illustrate the effectiveness of using L
norms as a penalty
function to maintain the formulation linear and differentiable. The
examples show that the proposed method provides more robust
results for linearized ATC problems due to the robustness of the
linear programming solver. DOI: 10.1115/1.4001001
1 Introduction
Design optimization of complex systems often requires multi-
disciplinary analyses involving significant interactions. Such
problems may be solved with an all-in-one AIO method in
which the system is treated as a fully integrated single problem.
When the analysis models used at each optimization iteration are
computationally expensive or difficult to solve, the AIO method
may not be practical or reliable. In such cases, decomposition
strategies can be used: the problem is partitioned into several sub-
problems that are solved with an appropriate coordination strat-
egy. Decomposition strategies are classified as nonhierarchical or
hierarchical Figs. 1a and 1b. Nonhierarchical partitions often
use two levels: subproblems, typically representing different sub-
systems or disciplinary analyses, are optimized concurrently,
while a system-level problem coordinates the interactions among
the subsystems 1–8. On the other hand, multilevel hierarchical
systems are typically partitioned into physical subsystems or ob-
jects. Hierarchical systems can be optimized using coordination
strategies such as analytical target cascading ATC and its exten-
sion, probabilistic analytical target cascading PATC9,10.
In Figs. 1a and 1b, each block in the hierarchical structure is
referred to as an element and is an optimization subproblem. Let
quantities with indices ij be related to element j at level i. An
element is coupled with its parent and children via targets t
ij
and
responses r
ij
, and ATC allows relaxed consistency constraints
t
ij
= r
ij
using penalty functions t
ij
- r
ij
. Then, a subproblem of
element j at level i can be expressed as follows:
min
x
ij
f
ij
x
ij
+ t
ij
- r
ij
subject to g
ij
x
ij
0, h
ij
x
ij
= 0
where x
ij
= x
ij
, r
ij
, t
i+1k
, ∀ k C
ij
∀ j E
i
, i = p,..., s 1
where f
ij
, g
ij
, and h
ij
are the separated objective, inequality, and
equality constraints of element ij, respectively; C
ij
is the set of
children of element ij.
Using a penalty function that decreases to zero monotonically
as the inconsistency becomes smaller, ATC enforces the consis-
tency of values shared between elements. The proper choice of
penalty functions and associated weights is critical for solution
convergence. For quadratic penalty QP functions, large weights
are required to obtain accurate and consistent solutions 11. Simi-
lar to other decomposition strategies, ATC typically is more ex-
pensive than the AIO method, if the latter could be used to obtain
a solution, due to the coordination overhead. Michalek and Pa-
palambros 12 proposed an efficient weight update method
WUM that finds minimal weights to achieve a given level of
consistency, especially important for problems with unattainable
system targets. Still, the inner loop coordination, where the de-
composedATC problems are solved iteratively, is computationally
expensive. To address this, Tosserams et al. 13 introduced a
separable augmented Lagrangian AL penalty function and an
alternating direction solution method ALAD, resulting in signifi-
cant computational cost savings. Also, an ATC dual coordination
algorithm was proposed using the Lagrangian duality theory,
where Lagrangian dual problems are solved by subgradient opti-
mization to update Lagrange multipliers 14,15. In order to take
advantage of parallel computing, diagonal quadratic approxima-
tion DQA and truncated diagonal quadratic approximation
TDQA were applied by linearizing the cross term of the aug-
mented Lagrangian function 16.
To improve the computational efficiency further, especially for
problems under uncertainty or with strong monotonicity, a se-
quential linear programming SLP algorithm by Fletcher et al.
17 was previously adapted as an alternate coordination strategy
to solve ATC and PATC problems 18–20. This coordination
strategy was able to reduce the computational cost by taking ad-
vantage of the strong monotonicity of problems and the simplicity
and ease of uncertainty propagation for a linear system. The SLP-
based ATC algorithm utilizes L
norms to maintain the linearity of
SLP subsequences. With L
norms, a subproblem of element j at
level i can be expressed as follows:
min
x
ij
,
ij
,
i+1k
f
ij
x
ij
+
ij
+
kC
ij
i+1k
subject to g
ij
x
ij
0, h
ij
x
ij
= 0
-
ij
w t - r
ij
ij
Contributed by the Design Automation Committee of ASME for publication in the
JOURNAL OF MECHANICAL DESIGN. Manuscript received January 14, 2009; final manu-
script received December 18, 2009; published online March 1, 2010. Special Editor:
Timothy W. Simpson.
Fig. 1 „a… Nonhierarchical and „b… hierarchical
decompositions
Journal of Mechanical Design MARCH 2010, Vol. 132 / 034502-1 Copyright © 2010 by ASME
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