OptimPack (version 3.1.0)

This is OptimPack, a C library for solving optimization problems. The library is mostly targeted at very large problems (e.g. as the ones encountered in image restoration) but also provide routines for problems of smaller size.

This document provides a general overview of OptimPack, for more specific information, see:

• Software installation.
• Developer notes.
• Software changes and history.
• Solving large scale smooth problems.

Most of the documentation is in the header files, e.g. src/optimpack.h, in Doxygen format.

Large scale problems

For large scale problems involving millions of variables (or even more), OptimPack provides:

• several non-linear conjugate gradient (NLCG) methods (see refs. [1-3]);

• limited memory variable metric (LBFGS, see Ref. [4]) possibly with bound constraints and/or preconditioning (VMLMB, see Ref. [5], or BLMVM, see Ref. [6]);

• inexact monotone and nonmonotone line searches (see Ref. [7,8]);

• linear conjugate gradients [1].

The large scale optimizers of the OptimPack library can work with the unknowns stored in almost any form (provided a minimal set of functions to manipulate them are implemented). This feature may be used to exploit hardware acceleration, multi-threading or to distribute the storage and computation across multiple machines.

See Solving large scale smooth problems for examples and more information about using Optimpack to solve large scale problems.

Problems of small to moderate size

For problems of small to moderate size, OptimPack provides:

• Moré & Sorensen method to compute a trust region step (see Ref. [13]);

• Mike Powell's COBYLA (see Ref. [10]), NEWUOA (see Ref. [11]), and BOBYQA (see Ref. [12]) algorithms for minimizing a function of many variables. These methods are derivatives free (only the function values are needed). NEWUOA is for unconstrained optimization. COBYLA accounts for general inequality constraints. BOBYQA accounts for bound constraints on the variables.

• Brent's method for the minimization of an univariate function (see Ref. [14]).

OptimPack bindings

OptimPack library is written in C but, in order to make embedding OptimPack into another language as easy as possible, most routines use reverse communication: all local variables needed by the optimizers get saved into workspaces created by the library and the optimizers never directly call the function to optimize.

The following language bindings allow OptimPack to be used in other programming languages:

• Directory yorick contains an implementation of OptimPack support in Yorick (https://github.com/emmt/OptimPack/tree/master/yorick).

• The OptimPack.jl project implements of OptimPack support for Julia.

• OptimPackNextGen.jl is a new project to provide the same features as OptimPack for Julia but with most code written in pure Julia. For now, pure Julia version of the methods devoted to large scale problems are available. To use Powell algorithms (devoted to moderate size problems) the dynamic libraries of OptimPack are still needed.

• The TiPi project provides a framework for solving inverse image reconstruction problems in Java. The optimization package of TiPi is a Java version of the OptimPack library.

References

1. M.R. Hestenes & E. Stiefel, "Methods of Conjugate Gradients for Solving Linear Systems," Journal of Research of the National Bureau of Standards 49, pp. 409-436 (1952).

2. W.W. Hager & H. Zhang, "A survey of nonlinear conjugate gradient methods," Pacific Journal of Optimization, Vol. 2, pp. 35-58 (2006).

3. W.W. Hager & H. Zhang, "A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search," SIAM J. Optim., Vol. 16, pp. 170-192 (2005).

4. D. Liu and J. Nocedal, "On the limited memory BFGS method for large scale optimization", Mathematical Programming B 45, 503-528 (1989).

5. É. Thiébaut, "Optimization issues in blind deconvolution algorithms," in Astronomical Data Analysis II, SPIE Proc. 4847, 174-183 (2002).

6. S.J. Benson & J.J. Moré, "A limited memory variable metric method in subspaces and bound constrained optimization problems", in Subspaces and Bound Constrained Optimization Problems, (2001).

7. E.G. Birgin, J.M. Martínez & M. Raydan, "Nonmonotone Spectral Projected Gradient Methods on Convex Sets," SIAM J. Optim. 10, 1196-1211 (2000).

8. Jorge J. Moré and David J. Thuente, "Line search algorithms with guaranteed sufficient decrease" in ACM Transactions on Mathematical Software (TOMS) Volume 20, Issue 3, Pages 286-307 (1994).

9. T. Steihaug, "The conjugate gradient method and trust regions in large scale optimization", SIAM Journal on Numerical Analysis, vol. 20, pp. 626-637 (1983).

10. M.J.D. Powell, "A direct search optimization method that models the objective and constraint functions by linear interpolation," in Advances in Optimization and Numerical Analysis Mathematics and Its Applications, vol. 275 (eds. Susana Gomez and Jean-Pierre Hennart), Kluwer Academic Publishers, pp. 51-67 (1994).

11. M.J.D. Powell, "The NEWUOA software for unconstrained minimization without derivatives", in Large-Scale Nonlinear Optimization, editors G. Di Pillo and M. Roma, Springer, pp. 255-297 (2006).

12. M.J.D. Powell, "The BOBYQA Algorithm for Bound Constrained Optimization Without Derivatives." Technical report, Department of Applied Mathematics and Theoretical Physics, University of Cambridge (2009).

13. J.J. Moré & D.C. Sorensen, "Computing A Trust Region Step," SIAM J. Sci. Stat. Comp. 4, 553-572 (1983).

14. R.P. Brent, "Algorithms for Minimization without Derivatives," Prentice-Hall, Inc. (1973).

Authors

• Éric Thiébaut (https://github.com/emmt)

Credits

The development of OptimPack was supported by the MiTiV project funded by the French Agence Nationale pour la Recherche (Ref. ANR-09-EMER-008).

A simpler version of OptimPack is provided by OptimPackLegacy project.

License

The OptimPack library is released under the MIT "Expat" License.