# Solving and Optimizing Large-Scale Nonlinear Systems

### Description

Non-monotone Barzilai-Borwein spectral methods for the solution and optimization of large-scale nonlinear systems.

### Details

A tutorial style introduction to this package is available in a vignette, which can be viewed with vignette("BB").

The main functions in this package are:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
BBsolve A wrapper function to provide a robust stategy for solving large
systems of nonlinear equations. It calls dfsane with different
algorithm control settings, until a successfully converged solution
is obtained.
BBoptim A wrapper function to provide a robust stategy for real valued function
optimization. It calls spg with different algorithm control settings,
until a successfully converged solution is obtained.
dfsane function for solving large systems of nonlinear equations using a
derivative-free spectral approach
sane function for solving large systems of nonlinear equations using
spectral approach
spg function for spectral projected gradient method for large-scale
optimization with simple constraints
``` |

### Author(s)

Ravi Varadhan

### References

J Barzilai, and JM Borwein (1988), Two-point step size gradient methods, *IMA J Numerical Analysis*, 8, 141-148.

Birgin EG, Martinez JM, and Raydan M (2000): Nonmonotone spectral projected
gradient methods on convex sets, *SIAM J Optimization*, 10, 1196-1211.

Birgin EG, Martinez JM, and Raydan M (2001): SPG: software for
convex-constrained optimization, *ACM Transactions on Mathematical Software*.

L Grippo, F Lampariello, and S Lucidi (1986), A nonmonotone line search technique for Newton's method, *SIAM J on Numerical Analysis*, 23, 707-716.

W LaCruz, and M Raydan (2003), Nonmonotone spectral methods for large-scale nonlinear systems, *Optimization Methods and Software*, 18, 583-599.

W LaCruz, JM Martinez, and M Raydan (2006), Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, *Mathematics of Computation*, 75, 1429-1448.

M Raydan (1997), Barzilai-Borwein gradient method for large-scale unconstrained minimization problem, *SIAM J of Optimization*, 7, 26-33.

R Varadhan and C Roland (2008), Simple and globally-convergent methods for accelerating the convergence of any EM
algorithm, *Scandinavian J Statistics*, doi: 10.1111/j.1467-9469.2007.00585.x.

R Varadhan and PD Gilbert (2009), BB: An R Package for Solving a Large System of Nonlinear Equations and for Optimizing a High-Dimensional Nonlinear Objective Function, *J. Statistical Software*, 32:4, http://www.jstatsoft.org/v32/i04/