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Convergence of trust region augmented Lagrangian methods using variable fidelity approximation data

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Abstract

To date the primary focus of most constrained approximate optimization strategies is that application of the method should lead to improved designs. Few researchers have focused on the development of constrained approximate optimization strategies that are assured of converging to a Karush-Kuhn-Tucker (KKT) point for the problem. Recent work by the authors based on a trust region model management strategy has shown promise in managing the convergence of constrained approximate optimization in application to a suite of single level optimization test problems. Using a trust-region model management strategy, coupled with an augmented Lagrangian approach for constrained approximate optimization, the authors have shown in application studies that the approximate optimization process converges to a KKT point for the problem. The approximate optimization strategy sequentially builds a cumulative response surface approximation of the augmented Lagrangian which is then optimized subject to a trust region constraint. In this research the authors develop a formal proof of convergence for the response surface approximation based optimization algorithm. Previous application studies were conducted on single level optimization problems for which response surface approximations were developed using conventional statistical response sampling techniques such as central composite design to query a high fidelity model over the design space. In this research the authors extend the scope of application studies to include the class of multidisciplinary design optimization (MDO) test problems. More importantly the authors show that response surface approximations constructed from variable fidelity data generated during concurrent subspace optimization (CSSOs) can be effectively managed by the trust region model management strategy. Results for two multidisciplinary test problems are presented in which convergence to a KKT point is observed. The formal proof of convergence and the successful MDO application of the algorithm using variable fidelity data generated by CSSO are original contributions to the growing body of research in MDO.

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Abbreviations

k :

Lagrangian iteration

s :

approximate minimization iteration

i, j, l :

variable indices

m :

number of inequality constraints

n :

number of design variables

p :

number of equality constraints

f(x):

objective function

g(x):

inequality constraint vector

g j (x):

j-th inequality constraint

h(x):

equality constraint vector

h j (x):

i-th equality constraint

c(x):

generalized constraint vector

c i (x):

i-th generalized constraint

c 1,c 2,c 3,c 4 :

real constants

m(x):

approximate model

q(x):

approximate model

q(x):

piecewise approximation

r p :

penalty parameter

t, t 1,t 2 :

step size length

x:

design vector, dimensionn

x l :

l-th design variable

xU :

upper bound vector, dimensionn

x Ul :

l-th design upper bound

xL :

lower bound vector, dimensionn

x Ll :

l-th design lower bound

B :

approximation of the Hessian

K :

constraints residual

S :

design space

β, β1, β2, κ:

scalars

ε1, ε2 :

convergence tolerances

γ0, γ1, γ2, η, μ:

trust region parameters

λ:

Lagrange multiplier vector, dimensionm+p

λ i :

i-th Lagrange multiplier

ρ:

trust region ratio

Ψ(x):

alternative form for inequality constraints

Φ(x, λ,r p ):

augmented Lagrangian function

\(\tilde \Phi (x,\lambda ,r_p )\) :

approximation of the augmented Lagrangian function

ϒ:

fidelity control

‖.‖:

Euclidean norm

〈,〉:

inner product

∇:

gradient operator with respect to design vector x

P(y(x)):

projection operator; projects the vector y onto the set of feasible directions at x

Δ:

trust region radius

Δx:

step size

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Authors and Affiliations

  1. Department of Aerospace and Mechanical Engineering, University of Notre Dame, 46556, Notre Dame, IN, USA

    J.F. Rodríguez & J.E. Renaud

  2. Departments of Computer Science and Mathematics, Virginia Polytechnic Institute and State University, 24061-0106, Blacksburg, VA, USA

    L.T. Watson

Authors
  1. J.F. Rodríguez
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  2. J.E. Renaud
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  3. L.T. Watson
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Rodríguez, J., Renaud, J. & Watson, L. Convergence of trust region augmented Lagrangian methods using variable fidelity approximation data. Structural Optimization 15, 141–156 (1998). https://doi.org/10.1007/BF01203525

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