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Nonlinear Programming
Published in Albert G. Holzman, Mathematical Programming, 2020
Such an updating formula is the basis of the symmetric rank-one (SR1) algorithm, mentioned earlier. The Hk matrices are given by choosing the yk in (363) to be yk=pk−Hk−1γk(380) It is clear that, for any symmetric Hk-1, Hk will also be symmetric. Interestingly, algorithms based on this matrix updating formula possess the quadratic termination without exact line searches. A disadvantage of the SRI algorithm is that the updating formula given by (363) and (380) does not ensure positive definiteness of the updated matrix and it may become ill-conditioned.
Elements of continuum mechanics
Published in Benjamin Loret, Fernando M. F. Simões, Biomechanical Aspects of Soft Tissues, 2017
Benjamin Loret, Fernando M. F. Simões
Symmetric rank one modifications of the identity enjoy interesting properties. Let n be a unit vector, and x an arbitrary real number. The symmetric matrices, () I(x)=I+(x−1)n⊗n,
Adjoint-based SQP method with block-wise quasi-Newton Jacobian updates for nonlinear optimal control
Published in Optimization Methods and Software, 2021
Pedro Hespanhol, Rien Quirynen
Quasi-Newton optimization methods are generally popular for solving such a constrained NLP. They result in computationally efficient Newton-type methods that solve the first order necessary conditions of optimality, i.e. the Karush-Kuhn-Tucker (KKT) conditions, without evaluating the complete Hessian of the Lagrangian and/or even without evaluating the Jacobian of the constraints [34]. Instead, quasi-Newton methods are based on low-rank update formulas for the Hessian and Jacobian matrix approximations [14]. Popular examples of this approach include the Broyden-Fletcher-Goldfarb-Shanno (BFGS) [11] and the symmetric rank-one (SR1) update formula [13] for approximating the Hessian of the Lagrangian. Similarly, quasi-Newton methods can be used for approximating Jacobian matrices, e.g. of the constraint functions, such as the good and bad Broyden methods [12] as well as the more recently proposed two-sided rank-one (TR1) update formula [26].
An accurate and efficient reliability-based design optimization using the second order reliability method and improved stability transformation method
Published in Engineering Optimization, 2018
Zeng Meng, Dixiong Yang, Huanlin Zhou, Bo Yu
SORM is an accurate method by which to assess structural reliability (Zhang and Du 2010). However, the application of RBDO encounters unaffordable computation costs due to repeated Hessian matrix calculations and sensitivity analysis of probabilistic constraints during the iterative process (Lim, Lee, and Lee 2014). Thus, evaluating the Hessian matrix efficiently and accurately is the key to the SORM-based RBDO approaches. The symmetric rank-one (SR1) update is an effective tool to obtain the Hessian matrix in an iterative process. The SR1 approximates the second order derivatives using the information of first order sensitivity (Lim, Lee, and Lee 2016). This means that the second order derivatives do not require recalculation after computing the reliability index of FORM. Thus, it can be applied to enhance the efficiency of SORM-based RBDO approaches.