Reduced representations of vector-valued coupling variables in decomposition-based design optimization

Abstract

Decomposition-based methods for system design optimization introduce consistency constraints, which contain coupling variables communicated between adjacent subproblems and link them together. When these variables are vector-valued (e.g., dynamic responses), the problem size can increase dramatically and make such methods impractical. Therefore, it is necessary to represent these vector-valued coupling variables with a reduced representation that will enable efficient optimization while maintaining an acceptable level of accuracy with respect to the original representation. This study investigates two representation techniques, radial-basis function artificial neural networks and proper orthogonal decomposition, and implements each in an analytical target cascading problem formulation for electric vehicle powertrain system optimization. Implementation of each representation technique is demonstrated and the techniques are assessed in terms of efficiency (decision vector dimensionality) and accuracy. Keywords:Decomposition-based design optimization, analytical target cascading, coupling variables, reduced representation, vector-valued target. IntroductionIn formulating design optimization problems for large-scale, complex systems, it is often practical to separate these systems into simpler, more manageable subsystem configurations. Decomposition-based optimization strategies are used frequently to solve these problems. These strategies introduce consistency constraints [1], which contain coupling variables and ensure feasible design solutions. Coupling variables appear in subproblem optimization formulations as decision variables, increasing subproblem dimension. If these coupling variables consist of a small, finite number of scalars, problem size does not increase appreciably, and efficient optimization of the system is still possible. However, if coupling comprises infinite-dimensional variables, such as functions, ensuring consistency becomes computationally challenging. Discretization is typically applied, transforming infinite-dimensional variables into finite-dimensional ones, which can be represented in vector form aswhere y is the independent variable and q is the number of discretized points. Although this transformation enables consistency constraints to be computed, it requires a large number of discretization points to ensure a sufficiently accurate representation of the function. The dimensionality (measured by q) of these vector-valued coupling variables (VVCVs) can become very large, resulting in high-dimension subproblem decision vectors. Thus, it is desirable to approximate VVCVs with a reduced dimension representation that preserves sufficient accuracy. The new coupling variables comprising these representations are known as reduced representation variables [2]. With the exception of the work by Sobieski [3], published literature has not addressed this issue. The majority of published techniques use simplified models of objective functions and constraints to approximate their more comp...

An Accuracy Assessment Method for Two-Dimensional Functional Data in Simulation-Based Design

“…Additionally, the definition for the relaxed consistency constraints is only applicable for scalar-valued linking variables; when these terms consist of functional data, the definition must be modified based on the error metric desired for measuring consistency. For example, in previous work [17][18][19] , consistency between functional data was that the functional data is inconsistent. Finally, note that for the current coordination strategy, the algorithm terminates when ||c () -c () || ∞ is within some small tolerance, where K denotes the iteration number.…”

Constraint Management of Reduced Representation Variables in Decomposition-Based Design Optimization

“…As the number of coupling variables becomes larger, ATC convergence becomes more difficult to achieve. When the coupling variables are vector-valued, ATC may converge slowly or not at all due to the large number of variables resulting from discretization [3][4][5]. In the EV problem, motor performance curves are represented as discretized function outputs to high-fidelity, "blackbox" simulations, in the form: …”

“…(1) implies that z i = f(y i ), where f is the blackbox simulation. To use ATC effectively, one may choose a reduced representation method [4,5], such as proper orthogonal decomposition (POD), to construct low-dimension VVCV approximations that improve optimization performance while maintaining accuracy [3][4][5]. However, reduced representation variables may not be physically meaningful, leading to difficulties in constraining the design space appropriately.…”

“…However, reduced representation variables may not be physically meaningful, leading to difficulties in constraining the design space appropriately. This, in turn, can inhibit ATC performance as the subsystem optimization algorithm may select values incompatible with the underlying analysis models [5].…”

“…A previous study broadly defined reduced representations as methods to decrease the dimension of VVCVs while maintaining a sufficient level of accuracy [5]. These methods include Fourier coefficients [6], weights of various basis functions [7,8], and low-dimension metamodel inputs [9].…”