OptimPackNextGen.jl

OptimPackNextGen is a Julia package for numerical optimization with particular focus on large scale problems.

Large scale problems

• Quasi-Newton methods can be used to solve nonlinear large scale optimization problems. Optionally, bounds on the variables can be taken into account. The objective function must be differentiable and the caller must provide means to compute the objective function and its gradient. If the Zygote is loaded, the gradient of the objective function may be computed by means of automatic-differentiation.

• Spectral Projected Gradient (SPG) method is provided for large-scale optimization problems with a differentiable objective function and convex constraints. The caller of spg (or spg!) shall provide a couple of functions to compute the objective function and its gradient and to project the variables on the feasible set. If the Zygote is loaded, the gradient of the objective function may be computed by means of automatic-differentiation.

• Line searches methods are used to approximately minimize the objective function along a given search direction.

• Algebra describes operations on "vectors" (that is to say the "variables" of the problem to solve).

Small to moderate size problems

For problems of small to moderate size, OptimPackNextGen provides:

• Mike Powell's COBYLA (Powell, 1994), NEWUOA (Powell, 2006), and BOBYQA (Powell, 2009) 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.

• nllsq implements non-linear (weighted) least squares fit. Powell's NEWUOA method is exploited to find the best fit parameters of given data by a user defined model function.

Univariate functions

The following methods are provided for univariate functions:

• Brent.fzero implements van Wijngaarden–Dekker–Brent method for finding a zero of a function (Brent, 1973).

• Brent.fmin implements Brent's method for finding a minimum of a function (Brent, 1973).

• Bradi.minimize (resp. Bradi.maximize) implements the BRADI ("Bracket" then "Dig"; Soulez et al., 2014) method for finding the global minimum (resp. maximum) of a function on an interval.

• Step.minimize (resp. Step.maximize) implements the STEP method (Swarzberg et al., 1994) for finding the global minimum (resp. maximum) of a function on an interval. The objective function f(x) and the variable x may have units.

Trust region

• Methods gqtpar and gqtpar! implement Moré & Sorensen algorithm for computing a trust region step (Moré & D.C. Sorensen, 1983).

Installation

The easiest way to install OptimPackNextGen is via Julia registry EmmtRegistry:

using Pkg
pkg"registry add General"  # if not yet any registries
pkg"registry add https://github.com/emmt/EmmtRegistry"
pkg"add OptimPackNextGen"

Rationale and related software

Related software are the OptimPack library which implements the C version of the algorithms and the OptimPack.jl Julia package which is a wrapper of this library for Julia. Compared to OptimPack.jl, the new OptimPackNextGen.jl implements in pure Julia the algorithms dedicated to large scale problems but still relies on the C libraries for a few algorithms (notably the Powell methods). Precompiled versions of these libraries are provided by OptimPack_jll package. The rationale is to facilitate the integration of exotic types of variables for optimization problems in Julia. Eventually, OptimPackNextGen.jl will become the next version of OptimPack.jl but, until then, it is more flexible to have two separate modules and avoid coping with compatibility and design issues.

References

• 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).

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

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

• 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).

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

• 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).

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

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

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

• 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).

• 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).

• 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).

• F. Soulez, É. Thiébaut, M. Tallon, I. Tallon-Bosc & P. Garcia, "Optimal a posteriori fringe tracking in optical interferometry" in Proc. SPIE 9146 "Optical and Infrared Interferometry IV", 91462Y (2014); doi:10.1117/12.2056590.

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

• S. Swarzberg, G. Seront & H. Bersini, "S.T.E.P.: the easiest way to optimize a function" in IEEE World Congress on Computational Intelligence, Proceedings of the First IEEE Conference on Evolutionary Computation, vol. 1, pp. 519-524 (1994).

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