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Gradient-Based Optimizer Solutions: LM, RLM, CG, BFGS, RG, and GRG

Excerpt

Features of a particular application (constraints, inflections, steep valleys) provide difficulty for SQ and NR, which are second-order model-based optimizers. This chapter covers several solutions that typify a myriad of similar adaptations of the basic incremental steepest descent and Newton-type approaches. The topics of this chapter are Levenberg–Marquardt (LM), conjugate gradient (CG), Broyden–Fletcher–Goldfarb–Shanno (BFGS), and generalized reduced gradient (GRG). LM blends ISD and NR to ensure it moves downhill, not to a saddle, maximum, or jump to absurd locations. CG and BFGS are for unconstrained applications with CG being a variant of Cauchy’s sequential line search method and BFGS a quasi-Newton approach. GRG is for handling constraints.

Features of a particular application (constraints, inflections, steep valleys) provide difficulty for SQ and NR, which are second-order model-based optimizers. This chapter covers several solutions that typify a myriad of similar adaptations of the basic incremental steepest descent and Newton-type approaches. The topics of this chapter are Levenberg–Marquardt (LM), conjugate gradient (CG), Broyden–Fletcher–Goldfarb–Shanno (BFGS), and generalized reduced gradient (GRG). LM blends ISD and NR to ensure it moves downhill, not to a saddle, maximum, or jump to absurd locations. CG and BFGS are for unconstrained applications with CG being a variant of Cauchy’s sequential line search method and BFGS a quasi-Newton approach. GRG is for handling constraints.

10.1Introduction
10.2Levenberg–Marquardt (LM)
10.3Scaled Variables
10.4Conjugate Gradient (CG)
10.5Broyden–Fletcher–Goldfarb–Shanno (BFGS)
10.6Generalized Reduced Gradient (GRG)
10.7Takeaway
10.8Exercises
10.1Introduction
10.2Levenberg–Marquardt (LM)
10.3Scaled Variables
10.4Conjugate Gradient (CG)
10.5Broyden–Fletcher–Goldfarb–Shanno (BFGS)
10.6Generalized Reduced Gradient (GRG)
10.7Takeaway
10.8Exercises

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