A Modified BFGS Formula Using a Trust Region Model for Nonsmooth Convex Minimizations

Abstract A common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques. The use of approximation strategies is designed to reduce the number of detailed, costly computer simulations required during optimization while maintaining the pertinent features of the design problem. To date the primary focus of most approximate optimization strategies is that application of the method should lead to improved designs. This is a laudable attribute and certainly relevant for practicing designers. However to date few researchers have focused on the development of approximate optimization strategies that are assured of converging to a solution of the original problem. Recent works based on trust region model management strategies have shown promise in managing convergence in unconstrained approximate minimization. In this research we extend these well established notions from the literature on trust-region methods to manage the convergence of the more general approximate optimization problem where equality, inequality and variable bound constraints are present. The primary concern addressed in this study is how to manage the interaction between the optimization and the fidelity of the approximation models to ensure that the process converges to a solution of the original constrained design problem. Using a trust-region model management strategy, coupled with an augmented Lagrangian approach for constrained approximate optimization, one can show that the optimization process converges to a solution of the original problem. In this research an approximate optimization strategy is developed in which a cumulative response surface approximation of the augmented Lagrangian is sequentially optimized subject to a trust region constraint. Results for several test problems are presented in which convergence to a Karush-Kuhn-Tucker (KKT) point is observed.

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A Reference Error Formulation for Multi-Fidelity Design Optimization

Volume 2B: 43rd Design Automation Conference ◽

10.1115/detc2017-67843 ◽

2017 ◽

Author(s):

Ahmed H. Bayoumy ◽

Michael Kokkolaras

Keyword(s):

Design Optimization ◽

Computational Models ◽

Design Space ◽

Computational Cost ◽

Trust Region ◽

Model Management ◽

Flexible Beam ◽

Management Framework ◽

Numerical Design ◽

Region Model

We consider the problem of selecting among different computational models of various fidelity for evaluating the objective and constraint functions in numerical design optimization. Typically, higher-fidelity models are associated with higher computational cost. Therefore, it is desirable to employ them only when necessary. We introduce a reference error formulation that aims at determining whether lower-fidelity models (that are computationally cheaper) can be used in certain areas of the design space as the latter is being explored during the optimization process. The proposed approach is implemented using an existing trust region model management framework. We demonstrate the link between feasibility and fidelity and the key features of the proposed approach using the design example of a cantilever flexible beam subject to high accelerations.

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Trust Region Augmented Lagrangian Methods for Sequential Response Surface Approximation and Optimization

Journal of Mechanical Design ◽

10.1115/1.2826677 ◽

1998 ◽

Vol 120 (1) ◽

pp. 58-66 ◽

Cited By ~ 87

Author(s):

J. F. Rodri´guez ◽

J. E. Renaud ◽

L. T. Watson

Keyword(s):

Response Surface ◽

Computer Simulations ◽

Design Problem ◽

Augmented Lagrangian ◽

Original Problem ◽

Trust Region ◽

Model Management ◽

Response Surface Approximation ◽

Approximate Optimization ◽

Region Model

A common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques. The use of approximation strategies is designed to reduce the number of detailed, costly computer simulations required during optimization while maintaining the pertinent features of the design problem. To date the primary focus of most approximate optimization strategies is that application of the method should lead to improved designs. This is a laudable attribute and certainly relevant for practicing designers. However to date few researchers have focused on the development of approximate optimization strategies that are assured of converging to a solution of the original problem. Recent works based on trust region model management strategies have shown promise in managing convergence in unconstrained approximate minimization. In this research we extend these well established notions from the literature on trust-region methods to manage the convergence of the more general approximate optimization problem where equality, inequality and variable bound constraints are present. The primary concern addressed in this study is how to manage the interaction between the optimization and the fidelity of the approximation models to ensure that the process converges to a solution of the original constrained design problem. Using a trust-region model management strategy, coupled with an augmented Lagrangian approach for constrained approximate optimization, one can show that the optimization process converges to a solution of the original problem. In this research an approximate optimization strategy is developed in which a cumulative response surface approximation of the augmented Lagrangian is sequentially optimized subject to a trust region constraint. Results for several test problems are presented in which convergence to a Karush-Kuhn-Tucker (KKT) point is observed.

Download Full-text